https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Bwang&feedformat=atomMath - User contributions [en]2018-09-23T14:33:25ZUser contributionsMediaWiki 1.26.0https://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=15259Colloquia2018-03-15T18:37:00Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16 (Room: 911)<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)<br />
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 16 (Monday)<br />
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)<br />
|[[# TBA| TBA ]]<br />
| Erman, Sam<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
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|date<br />
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|[[# TBA| TBA ]]<br />
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|[[# TBA| TBA ]]<br />
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|[[# TBA| TBA ]]<br />
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|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 2 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
===March 16 Anne Gelb (Dartmouth)===<br />
<br />
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity<br />
<br />
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.<br />
<br />
<br />
===April 6 Edray Goins (Purdue)===<br />
<br />
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups<br />
<br />
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math> A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math> Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math> Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math><br />
<br />
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math> Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math> We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus. <br />
<br />
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=15243Colloquia2018-03-13T17:42:25Z<p>Bwang: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)<br />
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 16 (Monday)<br />
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)<br />
|[[# TBA| TBA ]]<br />
| Erman, Sam<br />
|<br />
|-<br />
| April 20<br />
| [http://www.math.stonybrook.edu/~xiu/ Xiuxiong Chen] (Stony Brook University, CANCELLED)<br />
|[[# Xiuxiong Chen| TBA ]]<br />
| Bing Wang<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
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|[[# TBA| TBA ]]<br />
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|[[# TBA| TBA ]]<br />
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<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 2 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
===March 16 Anne Gelb (Dartmouth)===<br />
<br />
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity<br />
<br />
Abstract: We introduce the variance based joint sparsity (VBJS) method for sparse signal recovery and image reconstruction from multiple measurement vectors. Joint sparsity techniques employing $\ell_{2,1}$ minimization are typically used, but the algorithm is computationally intensive and requires fine tuning of parameters. The VBJS method uses a weighted $\ell_1$ joint sparsity algorithm, where the weights depend on the pixel-wise variance. The VBJS method is accurate, robust, cost efficient and also reduces the effects of false data.<br />
<br />
<br />
===April 6 Edray Goins (Purdue)===<br />
<br />
Title: Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups<br />
<br />
Abstract: A Bely&#301; map <math> \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) </math> is a rational function with at most three critical values; we may assume these values are <math> \{ 0, \, 1, \, \infty \}. </math> A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). </math> Replacing <math> \mathbb P^1 </math> with an elliptic curve <math>E </math>, there is a similar definition of a Bely&#301; map <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). </math> Since <math> E(\mathbb C) \simeq \mathbb T^2(\mathbb R) </math> is a torus, we call <math> (E, \beta) </math> a toroidal Bely&#301; pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: <math> \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R). </math><br />
<br />
This project seeks to create a database of such Bely&#301; pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer <math> N </math>, there are only finitely many toroidal Bely&#301; pairs <math> (E, \beta) </math> with <math> \deg \, \beta = N. </math> Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences <math> \mathcal D </math> on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups <math> G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; </math> they are the ``Galois closure'' of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Bely&#301; maps <math> \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) </math> associated to some elliptic curve <math> E: \ y^2 = x^3 + A \, x + B. </math> We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus. <br />
<br />
This work is part of PRiME (Purdue Research in Mathematics Experience) with Chineze Christopher, Robert Dicks, Gina Ferolito, Joseph Sauder, and Danika Van Niel with assistance by Edray Goins and Abhishek Parab.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=15227Geometry and Topology Seminar2018-03-09T22:10:41Z<p>Bwang: /* Yu Li */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 9<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|[[#Yu Li|"The Rigidity of Ricci shrinkers of dimension four"]]<br />
|Bing Wang<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|Wei Ho<br />
|TBA<br />
|Daniel Erman<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|Pei-Ken Hung (Columbia Univ.)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
===Yu Li===<br />
<br />
"The rigidity of Ricci shrinkers of dimension four"<br />
<br />
In dimension 4, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov- Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curva- ture and potential functions on Ricci shrinkers satisfying some natural geometric properties.<br />
This is a joint work with Bing Wang.<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=15226Geometry and Topology Seminar2018-03-09T22:10:24Z<p>Bwang: /* Spring Abstracts */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 9<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|[[#Yu Li|"The Rigidity of Ricci shrinkers of dimension four"]]<br />
|Bing Wang<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|Wei Ho<br />
|TBA<br />
|Daniel Erman<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|Pei-Ken Hung (Columbia Univ.)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
===Yu Li===<br />
<br />
"The rigidity of Ricci shrinkers of dimension 4"<br />
<br />
In dimension 4, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov- Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curva- ture and potential functions on Ricci shrinkers satisfying some natural geometric properties.<br />
This is a joint work with Bing Wang.<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=15225Geometry and Topology Seminar2018-03-09T22:06:47Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 9<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|[[#Yu Li|"The Rigidity of Ricci shrinkers of dimension four"]]<br />
|Bing Wang<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|Wei Ho<br />
|TBA<br />
|Daniel Erman<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|Pei-Ken Hung (Columbia Univ.)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=15074Colloquia2018-02-09T16:32:09Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[# TBA| TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
| April 20<br />
| Xiuxiong Chen(Stony Brook University)<br />
|[[# Xiuxiong Chen| TBA ]]<br />
| Bing Wang<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=15073Colloquia2018-02-09T16:29:35Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[# TBA| TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
| April 20<br />
| Reserved<br />
|[[# Xiuxiong Chen| TBA ]]<br />
| Bing Wang<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=15072Colloquia2018-02-09T16:27:30Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[# TBA| TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
| April 20<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Bing Wang<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=15071Colloquia2018-02-09T16:26:29Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[# TBA| TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
| April 20<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Bing Wang<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=15004Geometry and Topology Seminar2018-02-03T15:34:46Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 9<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|Wei Ho<br />
|TBA<br />
|Daniel Erman<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=15003Geometry and Topology Seminar2018-02-03T15:34:26Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|Local<br />
|-<br />
|February 9<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence" (continued)]]<br />
|TBA<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|Wei Ho<br />
|TBA<br />
|Daniel Erman<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14929Geometry and Topology Seminar2018-01-30T03:46:33Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|TBA<br />
|Local<br />
|-<br />
|February 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14928Geometry and Topology Seminar2018-01-30T03:45:58Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|[[#Jingrui Cheng|"Estimates for constant scalar curvature Kahler metrics with applications to existence"]]<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|TBA<br />
|Local<br />
|-<br />
|February 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|Yu Li<br />
|TBA<br />
|TBA<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== Jingrui Cheng ===<br />
<br />
"Estimates for constant scalar curvature Kahler metrics with applications to existence"<br />
<br />
We develop new a priori estimates for scalar curvature type of equations on a compact Kahler manifold. As an application, we show that the properness of K-energy implies the existence of constant scalar curvature Kahler metrics. I will also talk about other applications if time permits. This is joint work with Xiuxiong Chen.<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14812Geometry and Topology Seminar2018-01-23T04:50:53Z<p>Bwang: /* Spring 2018 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 26<br />
|Jingrui Cheng<br />
|TBA<br />
|Local<br />
|-<br />
|February 2<br />
|Jingrui Cheng<br />
|TBA<br />
|Local<br />
|-<br />
|February 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|February 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 2<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 9<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 16<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|March 23<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b> Spring Break </b><br />
|<br />
|<br />
|-<br />
|April 6<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|April 27<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|May 4<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
=== TBA ===<br />
<br />
TBA<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|[[#Marco Mendez-Guaraco| "Some geometric aspects of the Allen-Cahn equation"]]<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Deformation Formulas for Parameterizable Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"Some geometric aspects of the Allen-Cahn equation"<br />
<br />
In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation from the theory of phase transitions has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding this analogy including a new min-max proof of the celebrated Almgren-Pitts theorem.<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Deformation Formulas for Parameterizable Hypersurfaces"<br />
<br />
We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the multiple-point complex, a perverse sheaf naturally associated to any parameterized hypersurface.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14380Geometry and Topology Seminar2017-10-17T01:37:02Z<p>Bwang: /* Fall Abstracts */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Perverse Results on Parameterized Hypersurfaces"<br />
<br />
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14379Geometry and Topology Seminar2017-10-17T01:35:08Z<p>Bwang: /* Ovidiu Munteanu */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.<br />
<br />
=== Brian Hepler ===<br />
"Perverse Results on Parameterized Hypersurfaces"<br />
<br />
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14378Geometry and Topology Seminar2017-10-17T01:34:01Z<p>Bwang: /* Ovidiu Munteanu */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons.<br />
<br />
=== Brian Hepler ===<br />
"Perverse Results on Parameterized Hypersurfaces"<br />
<br />
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14377Geometry and Topology Seminar2017-10-17T01:33:49Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|[[#Ovidiu Munteanu| "The geometry of four dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Perverse Results on Parameterized Hypersurfaces"<br />
<br />
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14376Geometry and Topology Seminar2017-10-17T01:31:32Z<p>Bwang: /* Fall Abstracts */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|The geometry of four dimensional shrinking Ricci solitons<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"The geometry of four dimensional shrinking Ricci solitons"<br />
<br />
I will present several results, joint with Jiaping Wang, about the asymptotic structure of four dimensional gradient shrinking Ricci solitons. <br />
<br />
=== Brian Hepler ===<br />
"Perverse Results on Parameterized Hypersurfaces"<br />
<br />
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14375Geometry and Topology Seminar2017-10-17T01:30:29Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|[[#Sebastian Baader| "A filtration of the Gordian complex via symmetric groups"]]<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|[[#Shengwen Wang| "Hausdorff stability of round spheres under small-entropy perturbation"]]<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|The geometry of four dimensional shrinking Ricci solitons<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|[[#Brian Hepler| "Perverse Results on Parameterized Hypersurfaces"]]<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Sebastian Baader ===<br />
"A filtration of the Gordian complex via symmetric groups"<br />
<br />
The Gordian complex is a countable graph whose vertices correspond to knot types and whose edges correspond to pairs of knots that are related by a crossing change in a suitable diagram. For every natural number n, we consider the subgraph of the Gordian complex defined by restricting to the knot types whose fundamental group surjects onto S_n. We will prove that the various inclusion maps from these subgraphs into the Gordian complex are isometric embeddings. From this, we obtain a simple metric filtration of the Gordian complex.<br />
<br />
=== Shengwen Wang ===<br />
"Hausdorff stability of round spheres under small-entropy perturbation"<br />
<br />
Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round n-spheres minimize entropy among all closed hypersurfaces for n less than or equal to 6, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Brian Hepler ===<br />
"Perverse Results on Parameterized Hypersurfaces"<br />
<br />
We discuss some results for the cohomology of Milnor fibers inside parameterized hypersurfaces which follow quickly from results in the category of perverse sheaves. In particular, we define a new perverse sheaf called the multiple-point complex of the parameterization, which naturally arises when investigating how the multiple-point set influences the topology of the Milnor fiber. Time Permitting, we will discuss applications to one-parameter deformations of such hypersurfaces. This is joint work with David Massey.<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14216Geometry and Topology Seminar2017-09-22T20:46:15Z<p>Bwang: /* Fall Abstracts */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|TBA<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"<br />
<br />
A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an \epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant. <br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14215Geometry and Topology Seminar2017-09-22T20:45:14Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han (University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|Sigurd Angenent (UW-Madison)<br />
|[[#Sigurd Angenent| "Topology of closed geodesics on surfaces and curve shortening"]]<br />
|Local<br />
|-<br />
|September 29<br />
|Ke Zhu (Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang (Stony Brook)<br />
|[[#Shaosai Huang| "\epsilon-Regularity for 4-dimensional shrinking Ricci solitons"]]<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|Sebastian Baader (Bern)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|Brian Hepler (Northeastern University)<br />
|TBA<br />
|Max<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Sigurd Angenent ===<br />
"Topology of closed geodesics on surfaces and curve shortening"<br />
<br />
A closed geodesic on a surface with a Riemannian metric defines a knot in the unit tangent bundle of that surface. Which knots can occur? Given a particular knot type, what is the lowest number of closed geodesics a surface must have if you are allowed to pick the metric on the surface? Curve shortening allows you to define an invariant for each knot type (called the Conley index) which gives some answers to these questions.<br />
<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14106Geometry and Topology Seminar2017-09-08T20:23:23Z<p>Bwang: /* Fall Abstracts */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han(University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"<br />
<br />
Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK<br />
metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with<br />
Jeff Viaclovsky.<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14105Geometry and Topology Seminar2017-09-08T20:22:33Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han(University of Wisconsin-Madison)<br />
|[[#Jiyuan Han| "On closeness of ALE SFK metrics on minimal ALE Kahler surfaces"]]<br />
|Local <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Jiyuan Han ===<br />
"TBA"<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14065Geometry and Topology Seminar2017-09-05T23:34:15Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|Jiyuan Han<br />
|TBA<br />
|Local <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14062Geometry and Topology Seminar2017-09-05T18:47:47Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|[[#Ke Zhu| "Isometric Embedding via Heat Kernel"]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14060Geometry and Topology Seminar2017-09-05T18:46:11Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|[[#Ke Zhu| ""]]<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14059Geometry and Topology Seminar2017-09-05T18:32:07Z<p>Bwang: /* Ke Zhu */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|Isometric Embedding via Heat Kernel<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding. In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14058Geometry and Topology Seminar2017-09-05T18:31:45Z<p>Bwang: /* Fall Abstracts */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|Isometric Embedding via Heat Kernel<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Ke Zhu===<br />
"Isometric Embedding via Heat Kernel"<br />
<br />
The Nash embedding theorem states that every Riemannian manifold can be isometrically embedded into some Euclidean space with dimension bound. Isometric means preserving the length of every path. Nash's proof involves sophisticated perturbations of the initial embedding, so not much is known about the geometry of the resulted embedding.<br />
In this talk, using the eigenfunctions of the Laplacian operator, we construct canonical isometric embeddings of compact Riemannian manifolds into Euclidean spaces, and study the geometry of embedded images. They turn out to have large mean curvature (intuitively, very bumpy), but the extent of oscillation is about the same at every point. This is a joint work with Xiaowei Wang.<br />
<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14057Geometry and Topology Seminar2017-09-05T18:30:17Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu(Minnesota State University)<br />
|Isometric Embedding via Heat Kernel<br />
|Bing Wang<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14056Geometry and Topology Seminar2017-09-05T18:27:07Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|Ke Zhu<br />
|TBA<br />
|TBA<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|(reserved)<br />
|TBA<br />
|Kjuchukova<br />
|-<br />
|October 20<br />
|Shengwen Wang (Johns Hopkins)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu (University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
=== Shaosai Huang ===<br />
"TBA"<br />
<br />
=== Ovidiu Munteanu ===<br />
"TBA"<br />
<br />
=== Shengwen Wang ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14034Geometry and Topology Seminar2017-09-03T14:55:31Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu(University of Connecticut)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14033Geometry and Topology Seminar2017-09-03T14:54:23Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|Ovidiu Munteanu(University of Connecticut)<br />
|TBA<br />
|TBA<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14032Geometry and Topology Seminar2017-09-03T14:53:48Z<p>Bwang: /* Fall 2017 */</p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|Ovidiu Munteanu(University of Connecticut)<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Geometry_and_Topology_Seminar&diff=14031Geometry and Topology Seminar2017-09-02T14:50:20Z<p>Bwang: </p>
<hr />
<div>The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.<br />
<br> <br />
For more information, contact [http://www.math.wisc.edu/~kjuchukova Alexandra Kjuchukova] or [https://sites.google.com/a/wisc.edu/lu-wang/ Lu Wang] .<br />
<br />
[[Image:Hawk.jpg|thumb|300px]]<br />
<br />
<br />
<br />
== Fall 2017 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 15<br />
|TBA<br />
|TBA<br />
|TBA <br />
|-<br />
|September 22<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|September 29<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 6<br />
|Shaosai Huang(Stony Brook)<br />
|TBA<br />
|Bing Wang<br />
|-<br />
|October 13<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 20<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|October 27<br />
|Marco Mendez-Guaraco (Chicago)<br />
|TBA<br />
|Lu Wang<br />
|-<br />
|November 3<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 10<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|November 17<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|<b>Thanksgiving Recess</b><br />
| <br />
| <br />
| <br />
|-<br />
|December 1<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|December 8<br />
|TBA<br />
|TBA<br />
|TBA<br />
|-<br />
|}<br />
<br />
== Fall Abstracts ==<br />
<br />
=== Marco Mendez-Guaraco ===<br />
"TBA"<br />
<br />
== Archive of past Geometry seminars ==<br />
2016-2017 [[Geometry_and_Topology_Seminar_2016-2017]]<br />
<br><br><br />
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]<br />
<br><br><br />
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]<br />
<br><br><br />
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]<br />
<br><br><br />
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]<br />
<br><br><br />
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]<br />
<br><br><br />
2010: [[Fall-2010-Geometry-Topology]]</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13629PDE Geometric Analysis seminar2017-04-05T19:30:58Z<p>Bwang: /* PDE GA Seminar Schedule Spring 2017 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2017 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu<br />
|[[#Jianfeng Lu | TBA]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | TBA]]<br />
| Lin<br />
<br />
|-<br />
|May 1st(Special time: 4:00-5:00pm)<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.<br />
<br />
===Jeffrey Streets===<br />
Generalized Kahler Ricci flow and a generalized Calabi conjecture<br />
<br />
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13628PDE Geometric Analysis seminar2017-04-05T19:29:32Z<p>Bwang: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2017 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu<br />
|[[#Jianfeng Lu | TBA]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | TBA]]<br />
| Lin<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.<br />
<br />
===Jeffrey Streets===<br />
Generalized Kahler Ricci flow and a generalized Calabi conjecture<br />
<br />
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13627PDE Geometric Analysis seminar2017-04-05T19:29:01Z<p>Bwang: /* Jeffery Streets */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2017 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu<br />
|[[#Jianfeng Lu | TBA]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | TBA]]<br />
| Lin<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.<br />
<br />
===Jeffrey Streets===<br />
<br />
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13626PDE Geometric Analysis seminar2017-04-05T19:27:54Z<p>Bwang: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2017 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu<br />
|[[#Jianfeng Lu | TBA]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | TBA]]<br />
| Lin<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.<br />
<br />
===Jeffery Streets===<br />
<br />
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13625PDE Geometric Analysis seminar2017-04-05T19:26:31Z<p>Bwang: /* PDE GA Seminar Schedule Spring 2017 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2017 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6 - Wasow lecture<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Jin<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Eric Baer (UW)<br />
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]<br />
| Smith <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Analysis/Applied math/PDE seminar<br />
| Roger Temam (Indiana University) <br />
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]<br />
| Smith<br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]<br />
| Tran<br />
<br />
|-<br />
|March 29 - Wasow lecture<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]<br />
| <br />
<br />
<br />
|-<br />
|March 30 <br>Special day (Thursday) and location:<br> B139 Van Vleck<br />
| Gui-Qiang Chen (Oxford)<br />
|[[#Gui-Qiang Chen | Supersonic Flow onto Solid Wedges,<br />
Multidimensional Shock Waves and Free Boundary Problems ]]<br />
| Feldman<br />
<br />
<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|April 17<br />
| Siao-Hao Guo (Rutgers)<br />
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]<br />
| Lu Wang<br />
<br />
<br />
|-<br />
|April 24<br />
| Jianfeng Lu<br />
|[[#Jianfeng Lu | TBA]]<br />
| Li<br />
<br />
|-<br />
|April 25- joint Analysis/PDE seminar<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | TBA]]<br />
| Lin<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.<br />
<br />
===Eric Baer===<br />
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.<br />
<br />
===Ben Seeger===<br />
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.<br />
<br />
===Sona Akopian===<br />
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.<br />
<br />
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation. <br />
<br />
===Sylvia Serfaty===<br />
Mean-Field Limits for Ginzburg-Landau vortices<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
===Gui-Qiang Chen===<br />
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems<br />
<br />
When an upstream steady uniform supersonic flow, governed by the Euler equations,<br />
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock<br />
configurations if the wedge angle is less than the detachment angle -- the steady weak shock<br />
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger<br />
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which<br />
satisfy the entropy conditions.<br />
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically<br />
admissible solutions -- has been vigorously debated over the past eight decades.<br />
In this talk, we discuss some of the most recent developments on the stability analysis<br />
of the steady shock solutions in both the steady and dynamic regimes.<br />
The corresponding stability problems can be formulated as free boundary problems<br />
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose<br />
solutions are fundamental for multidimensional hyperbolic conservation laws.<br />
Some further developments, open problems, and mathematical challenges in this direction<br />
are also addressed.<br />
<br />
===Zhenfu Wang===<br />
<br />
Title: Mean field limit for stochastic particle systems with singular forces<br />
<br />
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution<br />
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos<br />
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Siao-hao Guo===<br />
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity<br />
<br />
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13249PDE Geometric Analysis seminar2017-02-04T19:52:14Z<p>Bwang: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Wasow lecture<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Hans-Joachim Hein (Fordham)<br />
|[[#Hans-Joachim Hein | ]]<br />
| Viaclovsky<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Applied math/PDE/Analysis seminar<br />
| Roger Temam (Indiana University) <br />
|[[#| ]]<br />
| Mathematics Department Distinguished Lecture <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Applied math/PDE/Analysis seminar<br />
| Roger Temam (Indiana University) <br />
|[[#| ]]<br />
| Mathematics Department Distinguished Lecture <br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | ]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Tran<br />
<br />
|-<br />
|March 29<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Wasow lecture<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | ]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | ]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=13248PDE Geometric Analysis seminar2017-02-04T19:50:44Z<p>Bwang: /* PDE GA Seminar Schedule Spring 2017 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Wasow lecture<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| Hans-Joachim Hein (Fordham)<br />
|[[#Hans-Joachim Hein | ]]<br />
| Viaclovsky<br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Applied math/PDE/Analysis seminar<br />
| Roger Temam (Indiana University) <br />
|[[#| ]]<br />
| Mathematics Department Distinguished Lecture <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Applied math/PDE/Analysis seminar<br />
| Roger Temam (Indiana University) <br />
|[[#| ]]<br />
| Mathematics Department Distinguished Lecture <br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | ]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Tran<br />
<br />
|-<br />
|March 29<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Wasow lecture<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | ]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | ]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=Fall_2016&diff=12833Fall 20162016-12-17T16:40:31Z<p>Bwang: /* PDE GA Seminar Schedule Spring 2017 */</p>
<hr />
<div>= PDE GA Seminar Schedule Spring 2017 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|#date<br />
| #speaker<br />
|[[# | #title ]]<br />
| #host<br />
|-}<br />
<br />
|-<br />
|January 23<br />
| Sigurd Angenent (UW)<br />
|[[# Sigurd Angenent | ]]<br />
| Local<br />
|-}<br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[# Serguei Denissov | ]]<br />
| Local<br />
|-}<br />
<br />
<br />
|-<br />
|February 6<br />
| Benoit Perthame (University of Paris VI) -- Wasow lecture.<br />
|[[#| ]]<br />
| <br />
|-}<br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[# Bing Wang | ]]<br />
| Local<br />
|-}<br />
<br />
|-<br />
|February 20<br />
| Hans-Joachim Hein (Fordham)<br />
|[[# Hans-Joachim Hein | ]]<br />
| Viaclovsky<br />
|-}<br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | ]]<br />
| Tran<br />
|-}<br />
<br />
|-<br />
|March 6<br />
| No seminar because of Distinguished lectures by Temam on March 7 and March 8.<br />
|[[#| ]]<br />
| <br />
|-}<br />
<br />
<br />
|-<br />
|March 27<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Tran<br />
<br />
|-<br />
|May 1st<br />
| Jeffery Streets (UC-Irvine)<br />
|[[#Jeffery Streets | ]]<br />
| Bing Wang</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12806PDE Geometric Analysis seminar2016-12-05T15:21:23Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5 ('''Special time and room''': 3PM-4PM, B313VV)<br />
| Brian Weber (University of Pennsylvania)<br />
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12805PDE Geometric Analysis seminar2016-12-05T15:20:53Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5 ('''Special time and room''': 3PM-4PM, B313)<br />
| Brian Weber (University of Pennsylvania)<br />
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12802PDE Geometric Analysis seminar2016-12-01T19:00:10Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5<br />
| Brian Weber (University of Pennsylvania)<br />
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang(Notice the special time of this talk: 3:00-3:50pm)<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12801PDE Geometric Analysis seminar2016-12-01T18:57:01Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5<br />
| Brian Weber (University of Pennsylvania)<br />
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12800PDE Geometric Analysis seminar2016-12-01T18:55:58Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5<br />
| Brian Weber (University of Pennsylvania)<br />
|[[# Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12799PDE Geometric Analysis seminar2016-12-01T18:54:16Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5<br />
| Brian Weber (University of Pennsylvania)<br />
|[[# |Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwanghttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12798PDE Geometric Analysis seminar2016-12-01T18:53:08Z<p>Bwang: /* PDE GA Seminar Schedule Fall 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[#Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[#Alexis Vasseur | Compressible Navier-Stokes equations with degenerate viscosities ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[#Minh-Binh Tran | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[#David Kaspar | Kinetics of shock clustering ]]<br />
|Tran<br />
|-<br />
|December 5<br />
| Brian Weber (University of Pennsylvania)<br />
|[[# |Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.<br />
<br />
===Alexis Vasseur===<br />
<br />
Compressible Navier-Stokes equations with degenerate viscosities <br />
<br />
We will discuss recent results on the construction of weak solutions for<br />
3D compressible Navier-Stokes equations with degenerate viscosities.<br />
The method is based on the Bresch and Desjardins entropy. The main<br />
contribution is to derive MV type inequalities for the weak solutions,<br />
even if it is not verified by the first level of approximation. This<br />
provides existence of global solutions in time, for the compressible<br />
Navier-Stokes equations, in three dimensional space, with large initial<br />
data, possibly vanishing on the vacuum.<br />
<br />
===Minh-Binh Tran===<br />
<br />
Quantum kinetic problems<br />
<br />
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.<br />
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).<br />
<br />
===David Kaspar===<br />
<br />
Kinetics of shock clustering<br />
<br />
Suppose we solve a (deterministic) scalar conservation law<br />
with random initial data. Can we describe the probability law of the<br />
solution as a stochastic process in x for fixed later time t? The<br />
answer is yes, for certain Markov initial data, and the probability<br />
law factorizes as a product of kernels. These kernels are obtained by<br />
solving a mean-field kinetic equation which most closely resembles the<br />
Smoluchowski coagulation equation. We discuss prior and ongoing work<br />
concerning this and related problems.<br />
<br />
===Brian Weber===<br />
<br />
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds<br />
<br />
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations <br />
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results <br />
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.</div>Bwang