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__NOTOC__
 
 
 
= Mathematics Colloquium =
 
= Mathematics Colloquium =
  
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
  
== Fall 2013 ==
+
== Spring 2018 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
!align="left" | date
+
!align="left" | date  
 
!align="left" | speaker
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | title
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 6
+
|January 29 (Monday)
|[http://people.math.gatech.edu/~mbaker/ Matt Baker] (Georgia Institute of Technology)
+
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
|Riemann-Roch for Graphs and Applications
+
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture  ]]
|Ellenberg
+
| Jordan Ellenberg
|-
+
|Sept 13
+
|[http://math.wisc.edu/~andrews/ Uri Andrews] (University of Wisconsin)
+
|
+
 
|
 
|
 
|-
 
|-
|Sept 20
+
|February 2 (Room: 911)
|[http://www.math.neu.edu/people/profile/valerio-toledano-laredo Valerio Toledano Laredo] (Northeastern)
+
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
 +
|[[#February 2 Thomas Fai (Harvard)|  The Lubricated Immersed Boundary Method ]]
 +
| Spagnolie, Smith
 
|
 
|
|Gurevich
 
 
|-
 
|-
|'''Wed, Sept 25, 2:30PM'''
+
|February 5 (Monday, Room: 911)
|[http://mypage.iu.edu/~alindens/ Ayelet Lindenstrauss]
+
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)
 +
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
 +
| Ellenberg, Gurevitch
 
|
 
|
|Meyer
 
 
|-
 
|-
|'''Wed, Sept 25''' (LAA lecture)
+
|February 6 (Tuesday 2 pm, Room 911)
|[http://www.cs.berkeley.edu/~demmel/ Jim Demmel] (Berkeley)
+
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
 +
|[[#February 6 Alex Lubotzky (Hebrew University)|  Groups' approximation, stability and high dimensional expanders ]]
 +
| Ellenberg, Gurevitch
 
|
 
|
|Gurevich
 
 
|-
 
|-
|'''Thurs, Sept 26''' (LAA lecture)
+
|February 9
|[http://www.cs.berkeley.edu/~demmel/ Jim Demmel] (Berkeley)
+
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
 +
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
 +
| Roch
 
|
 
|
|Gurevich
 
 
|-
 
|-
|Sept 27 (LAA lecture)
+
|March 2
|[http://www.cs.berkeley.edu/~demmel/ Jim Demmel] (Berkeley)
+
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
 +
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]
 +
| Caldararu
 
|
 
|
|Gurevich
 
 
|-
 
|-
|Oct 4
+
| March 16  (Room: 911)
|[http://www.math.tamu.edu/~sottile/ Frank Sottile] (Texas A&M)
+
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
 +
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]
 +
| WIMAW
 
|
 
|
|Caldararu
 
 
|-
 
|-
|Oct 11
+
|April 5 (Thursday)
|[http://math.uchicago.edu/~wilkinso/ Amie Wilkinson] (Chicago)
+
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
 +
|[[# TBA|  TBA  ]]
 +
| Craciun
 
|
 
|
|WIMAW (Cladek)
 
 
|-
 
|-
|'''Tues, Oct 15''' (Distinguished Lecture)
+
| April 6
|[http://math.mit.edu/people/profile.php?pid=1222 Alexei Borodin] (MIT)
+
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)
 +
|[[# Edray Goins|  Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups  ]]
 +
| Melanie
 
|
 
|
|Valko
 
 
|-
 
|-
|'''Wed, Oct 16''' (Distinguished Lecture)
+
| April 13
|[http://math.mit.edu/people/profile.php?pid=1222 Alexei Borodin] (MIT)
+
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
 +
|[[# TBA|  TBA  ]]
 +
| WIMAW
 
|
 
|
|Valko
 
 
|-
 
|-
|<strike>Oct 18</strike>
+
|April 16 (Monday)
|No colloquium due to the distinguished lecture
+
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)
|
+
|[[# TBA|  TBA  ]]
 +
| Erman, Sam
 
|
 
|
 
|-
 
|-
|Oct 25
+
| April 25 (Wednesday)
|[http://www.math.umn.edu/~garrett/ Paul Garrett] (Minnesota)
+
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture
|
+
|[[# TBA| TBA  ]]
|Gurevich
+
| Tran
|
+
 
|
 
|
 
|-
 
|-
|Nov 1
+
| May 4
|[http://www.cs.utexas.edu/~alewko/ Allison Lewko] (Microsoft Research New England)
+
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)
 +
|[[# TBA|  TBA  ]]
 +
| Ellenberg
 
|
 
|
|Stovall
 
 
|-
 
|-
|Nov 8
+
|date
|[http://www.math.cornell.edu/~riley/ Tim Riley] (Cornell)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|Dymarz
 
 
|-
 
|-
|Nov 15 and later
+
|date
|Reserved
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|Street
 
|}
 
 
== Spring 2014 ==
 
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
 
|-
 
|-
|Jan 24
+
|date
|
+
| person (institution)
|
+
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
 
|-
 
|-
|Jan 31
+
|date
|[http://csi.usc.edu/~ubli/ubli.html Urbashi Mitra] (USC)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|Gurevich
 
 
|-
 
|-
|Feb 7
+
|date
|David Treumann (Boston College)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|Street
 
 
|-
 
|-
|Feb 14
+
|date
|
+
| person (institution)
|
+
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
 
|-
 
|-
|Feb 21
+
|date
|
+
| person (institution)
|
+
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
 
|-
 
|-
|Feb 28
+
|date
|
+
| person (institution)
|
+
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|-
 
|March 7
 
|
 
|
 
|
 
|-
 
|March 14
 
|
 
|
 
|
 
|-
 
|<strike>March 21</strike>
 
|'''Spring Break'''
 
|No Colloquium
 
|
 
|-
 
|March 28
 
|[http://people.math.gatech.edu/~lacey/ Michael Lacey] (GA Tech)
 
|The Two Weight Inequality for the Hilbert Transform
 
|Street
 
|-
 
|April 4
 
|[https://sites.google.com/site/katejuschenko/ Kate Jushchenko] (Northwestern)
 
|
 
|Dymarz
 
|-
 
|April 11
 
|[http://www.cs.uchicago.edu/people/risi Risi Kondor] (Chicago)
 
|
 
|Gurevich
 
|-
 
|April 18 (Wasow Lecture)
 
|[http://mathnt.mat.jhu.edu/sogge/ Christopher Sogge] (Johns Hopkins)
 
|
 
|A. Seeger
 
|-
 
|April 25
 
|[http://www.charlesdoran.net Charles Doran](University of Alberta)
 
|
 
|Song
 
|-
 
|May 2
 
|[http://www.stat.uchicago.edu/~lekheng/ Lek-Heng Lim] (Chicago)
 
|
 
|Boston
 
|-
 
|May 9
 
|[http://www.ma.utexas.edu/users/rward/ Rachel Ward] (UT Austin)
 
|
 
|WIMAW
 
 
|}
 
|}
  
== Abstracts ==
+
== Spring Abstracts ==
 +
 
 +
 
 +
===January 29 Li Chao (Columbia)===
 +
 
 +
Title: Elliptic curves and Goldfeld's conjecture
 +
 
 +
Abstract:
 +
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.
 +
 
 +
=== February 2 Thomas Fai (Harvard) ===
 +
 
 +
Title: The Lubricated Immersed Boundary Method
 +
 
 +
Abstract:
 +
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.
 +
 
 +
===February 5 Alex Lubotzky (Hebrew University)===
 +
 
 +
Title:  High dimensional expanders: From Ramanujan graphs to Ramanujan complexes
 +
 
 +
Abstract:
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
 
 +
===February 6 Alex Lubotzky (Hebrew University)===
 +
 
 +
Title:  Groups' approximation, stability and high dimensional expanders
 +
 
 +
Abstract:
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
All notions will be explained.      Joint work with M, De Chiffre, L. Glebsky and A. Thom.
 +
 
 +
===February 9 Wes Pegden (CMU)===
 +
 
 +
Title: The fractal nature of the Abelian Sandpile
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
===March 2 Aaron Bertram (Utah)===
 +
 
 +
Title: Stability in Algebraic Geometry
 +
 
 +
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.
 +
 
 +
===March 16 Anne Gelb (Dartmouth)===
 +
 
 +
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity
 +
 
 +
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.
 +
 
 +
 
 +
===April 6 Edray Goins (Purdue)===
 +
 
 +
Title: Toroidal Bely&#301;  Pairs, Toroidal Graphs, and their Monodromy Groups
 +
 
 +
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>
 +
 
 +
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.
 +
 
 +
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.
 +
 
 +
== Past Colloquia ==
 +
 
 +
[[Colloquia/Blank|Blank Colloquia]]
 +
 
 +
[[Colloquia/Fall2017|Fall 2017]]
 +
 
 +
[[Colloquia/Spring2017|Spring 2017]]
 +
 
 +
[[Archived Fall 2016 Colloquia|Fall 2016]]
 +
 
 +
[[Colloquia/Spring2016|Spring 2016]]
 +
 
 +
[[Colloquia/Fall2015|Fall 2015]]
  
===Sep 6: Matt Baker (GA Tech) ===
+
[[Colloquia/Spring2014|Spring 2015]]
''Riemann-Roch for Graphs and Applications''
+
  
We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications include a new proof of the Brill-Noether theorem in algebraic geometry (work of by Cools-Draisma-Payne-Robeva), a "volume-theoretic proof" of Kirchhoff's Matrix-Tree Theorem (work of An, Kuperberg, Shokrieh, and the speaker), and a new Chabauty-Coleman style bound for the number of rational points on an algebraic curve over the rationals (work of Katz and Zureick-Brown).
+
[[Colloquia/Fall2014|Fall 2014]]
  
===March 28: Michael Lacey (GA Tech) ===
+
[[Colloquia/Spring2014|Spring 2014]]
''The Two Weight Inequality for the Hilbert Transform ''
+
  
The individual two weight inequality for the Hilbert transform
+
[[Colloquia/Fall2013|Fall 2013]]
asks for a real variable characterization of those pairs of weights
+
(u,v) for which the Hilbert transform H maps L^2(u) to L^2(v).
+
This question arises naturally in different settings, most famously
+
in work of Sarason. Answering in the positive a deep
+
conjecture of Nazarov-Treil-Volberg, the mapping property
+
of the Hilbert transform is characterized by a triple of conditions,
+
the first being a two-weight Poisson A2 on the pair of weights,
+
with a pair of so-called testing inequalities, uniform over all
+
intervals.  This is the first result of this type for a singular
+
integral operator.  (Joint work with Sawyer, C.-Y. Shen and Uriate-Tuero)
+
  
== Past talks ==
+
[[Colloquia 2012-2013|Spring 2013]]
  
Last year's schedule: [[Colloquia 2012-2013]]
+
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]

Revision as of 12:20, 19 March 2018

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Spring 2018

date speaker title host(s)
January 29 (Monday) Li Chao (Columbia) Elliptic curves and Goldfeld's conjecture Jordan Ellenberg
February 2 (Room: 911) Thomas Fai (Harvard) The Lubricated Immersed Boundary Method Spagnolie, Smith
February 5 (Monday, Room: 911) Alex Lubotzky (Hebrew University) High dimensional expanders: From Ramanujan graphs to Ramanujan complexes Ellenberg, Gurevitch
February 6 (Tuesday 2 pm, Room 911) Alex Lubotzky (Hebrew University) Groups' approximation, stability and high dimensional expanders Ellenberg, Gurevitch
February 9 Wes Pegden (CMU) The fractal nature of the Abelian Sandpile Roch
March 2 Aaron Bertram (University of Utah) Stability in Algebraic Geometry Caldararu
March 16 (Room: 911) Anne Gelb (Dartmouth) Reducing the effects of bad data measurements using variance based weighted joint sparsity WIMAW
April 5 (Thursday) John Baez (UC Riverside) TBA Craciun
April 6 Edray Goins (Purdue) Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups Melanie
April 13 Jill Pipher (Brown) TBA WIMAW
April 16 (Monday) Christine Berkesch Zamaere (University of Minnesota) TBA Erman, Sam
April 25 (Wednesday) Hitoshi Ishii (Waseda University) Wasow lecture TBA Tran
May 4 Henry Cohn (Microsoft Research and MIT) TBA Ellenberg
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty
date person (institution) TBA hosting faculty

Spring Abstracts

January 29 Li Chao (Columbia)

Title: Elliptic curves and Goldfeld's conjecture

Abstract: 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.

February 2 Thomas Fai (Harvard)

Title: The Lubricated Immersed Boundary Method

Abstract: 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.

February 5 Alex Lubotzky (Hebrew University)

Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes

Abstract:

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.

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.

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.


February 6 Alex Lubotzky (Hebrew University)

Title: Groups' approximation, stability and high dimensional expanders

Abstract:

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.

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.

All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.

February 9 Wes Pegden (CMU)

Title: The fractal nature of the Abelian Sandpile

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.

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.

March 2 Aaron Bertram (Utah)

Title: Stability in Algebraic Geometry

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.

March 16 Anne Gelb (Dartmouth)

Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity

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.


April 6 Edray Goins (Purdue)

Title: Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups

Abstract: A Belyĭ map  \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) is a rational function with at most three critical values; we may assume these values are  \{ 0, \, 1, \, \infty \}. 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:  \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). Replacing  \mathbb P^1 with an elliptic curve E , there is a similar definition of a Belyĭ map  \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). Since  E(\mathbb C) \simeq \mathbb T^2(\mathbb R) is a torus, we call  (E, \beta) a toroidal Belyĭ pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm:  \beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R).

This project seeks to create a database of such Belyĭ pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer  N , there are only finitely many toroidal Belyĭ pairs  (E, \beta) with  \deg \, \beta = N. Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences  \mathcal D on the ramification indices as multisets on three partitions of N. For each degree sequence, we compute all possible monodromy groups  G = \text{im} \, \bigl[ \pi_1 \bigl( \mathbb P^1(\mathbb C) - \{ 0, \, 1, \, \infty \} \bigr) \to S_N \bigr]; they are the ``Galois closure of the group of automorphisms of the graph. Finally, for each possible monodromy group, we compute explicit formulas for Belyĭ maps  \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) associated to some elliptic curve  E: \ y^2 = x^3 + A \, x + B. We will discuss some of the challenges of determining the structure of these groups, and present visualizations of group actions on the torus.

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.

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