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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'''.


==[[Tentative Colloquia|Tentative schedule for next semester]] ==
== Spring 2018 ==


== Spring 2016  ==
 
{| 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)
|-
|-
| '''January 22'''
|January 29 (Monday)
|<!--[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)-->
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
| <!-- [[Colloquia#September 11:  Speaker (University) | title]] -->
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture  ]]
| <!--Host-->
| Jordan Ellenberg
|
|-
|February 2 (Room: 911)
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
|[[#February 2 Thomas Fai (Harvard)|  The Lubricated Immersed Boundary Method ]]
| Spagnolie, Smith
|
|-
|February 5 (Monday, Room: 911)
| [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
|
|-
|-
| '''January 28 (Th 4pm VV901)'''
|February 6 (Tuesday 2 pm, Room 911)
| [https://web.math.princeton.edu/~ssivek/ Steven Sivek] (Princeton)
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
|     [[Colloquia#September 11:  Speaker (University) | The augmentation category of a Legendrian knot]]
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]
| Ellenberg
| Ellenberg, Gurevitch
|
|-
|-
| '''January 29'''
|February 9
|[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
| [[Colloquia#September 11:  Ana Caraiani (Princeton) | Locally symmetric spaces, torsion classes, and the geometry of period domains]]
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]
| Ellenberg
| Roch
|
|-
|March 2
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]
| Caldararu
|
|-
|-
| '''February 5'''
| March 16  (Room: 911)
|[http://math.uchicago.edu/~souganidis/ Takis Souganidis] (University of Chicago)
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
| [[Colloquia#September 11:  Takis Souganidis (University of Chicago) | Scalar Conservation Laws with Rough Dependence]]
|[[#March 16 Anne Gelb (Dartmouth)| Reducing the effects of bad data measurements using variance based weighted joint sparsity ]]
| Lin
| WIMAW
|
|-
|-
| '''February 12'''
|April 5 (Thursday, Room: 911)
|[http://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)  
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
| [[Colloquia#February 12:  Gautam Iyer (CMU)| Homogenization and Anomalous Diffusion]]
|[[#April 5 John Baez (UC Riverside)| Monoidal categories of networks  ]]
| Jean-Luc
| Craciun
|
|-
|-
| '''February 19'''
| April 6
| [https://people.math.osu.edu/lafont.1/ Jean-François Lafont] (Ohio State)  
| [https://www.math.purdue.edu/~egoins Edray Goins] (Purdue)
| [[Colloquia#February 19: Jean-François Lafont (Ohio State) | Rigidity and flexibility of almost-isometries]]
|[[# Edray Goins| Toroidal Bely&#301; Pairs, Toroidal Graphs, and their Monodromy Groups  ]]
| Dymarz
| Melanie
|
|-
|-
| '''February 26'''
| April 13 (911 Van Vleck)
|Hiroyoshi Mitake (Hiroshima university)    
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
| [[Colloquia#February 26Hiroyoshi Mitake (Hiroshima university| On asymptotic speed of the crystal growth]]
|[[#April 13, Jill Pipher, Brown University|  Mathematical ideas in cryptography  ]]
| WIMAW
|
|-
| April 16 (Monday)
| [http://www-users.math.umn.edu/~cberkesc/ Christine Berkesch Zamaere ] (University of Minnesota)
|[[#April 16, Christine Berkesch Zamaere (University of Minnesota)|  Free complexes on smooth toric varieties ]]
| Erman, Sam
|
|-
| April 25 (Wednesday, Room: 911)
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Tsuda University) Wasow lecture
|[[#April 25, Hitoshi Ishii (Tsuda University)|  Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory ]]
| Tran
| Tran
|
|-
|-
| '''March 4'''
| May 1 (Tuesday, 4:30pm, Room: B102 VV)
| [http://www.columbia.edu/~gb2030/ Guillaume Bal] (Columbia University)
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University Chicago and Imperial College London) Distinguished lecture
| [[Colloquia#September 11:  Guillaume Bal (Columbia University) | Inverse and Control Transport Problems]]
|[[#May 1, Andre Neves (University Chicago and Imperial College London)| Wow, so many minimal surfaces! (Part I)]]
| Li, Jin
| Lu Wang
|
|-
|-
| '''March 11'''
| May 2 (Wednesday, 3pm, Room: B325 VV)
| [http://math.umn.edu/~luskin Mitchell Luskin] (University of Minnesota)
| [https://math.uchicago.edu/~aneves/ Andre Neves] (University of Chicago and Imperial College London) Distinguished lecture
| [[Colloquia#March 11: Mitchell Luskin (UMN) | Mathematical Modeling of Incommensurate 2D Materials]]
|[[#May 2, Andre Neves (University Chicago and Imperial College London)| Wow, so many minimal surfaces! (Part II) ]]
| Li
| Lu Wang
|
|-
|-
| '''March 18'''
| May 4
| [http://www.math.lsa.umich.edu/~spatzier/ Ralf Spatzier] (University of Michigan)  
| [http://math.mit.edu/~cohn/ Henry Cohn] (Microsoft Research and MIT)
CANCELED
|[[# TBA|  The sphere packing problem in dimensions 8 and 24 ]]
| Dymarz
| Ellenberg
|
|-
|-
| '''March 25'''
|date
| Spring Break<!-- [webpage Speaker Name] (University) -->   
| person (institution)
| <!-- [[Colloquia#September 11: Speaker (University) | title]] -->
|[[# TBA|  TBA ]]
| <!-- host -->
| hosting faculty
|
|-
|-
| '''April 1'''
|date
|  
| person (institution)
|   
|[[# TBA|  TBA ]]
| hosting faculty
|
|
|-
|-
| '''April 8'''
|date
| [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton)  
| person (institution)
| [[Colloquia#April 8: Alexandru Ionescu (Princeton) | On long-term existence of solutions of water wave models]]  
|[[# TBA| TBA  ]]
| Wainger/Seeger
| hosting faculty
|
|-
|date
| person (institution)
|[[# TBA|  TBA  ]]
| hosting faculty
|
|-
|-
| '''April 15'''
|date
| [https://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/wigmani.aspx Igor Wigman] (King's College - London)  
| person (institution)
| [[Colloquia#September 11: Speaker (University) |Nodal Domains of Eigenfunctions]]
|[[# TBA|  TBA ]]
| Gurevich/Marshall
| hosting faculty
|
|-
|-
| '''April 22'''
|date
| [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade] (NYU)
| person (institution)
| [[Colloquia#April 22: Paul Bourgade (NYU) | TBA]]
|[[# TBA| TBA ]]
| Seppalainen/Valko
| hosting faculty
|
|-
|-
| '''April 29'''
|date
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn)  
| person (institution)
| [[Colloquia#April 29: Randall Kamien (U Penn) | TBA]]  
|[[# TBA| TBA ]]
| Spagnolie
| hosting faculty
|
|-
|-
| '''May 6'''
|date
| [https://www.math.upenn.edu/~shaneson/ Julius Shaneson] (University of Pennsylvania)    
| person (institution)
| [[Colloquia#September 11: Julius Shaneson (University of Pennsylvania) | TBA]]  
|[[# TBA| TBA ]]
| Maxim/Kjuchukova
| hosting faculty
|
|}
|}


== 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.


=== January 28: Steven Sivek (Princeton) ===
Title: The augmentation category of a Legendrian knot


Abstract: A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle.  One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.
===February 6 Alex Lubotzky (Hebrew University)===


The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space.  This construction, called the augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially.  I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Maslow.
Title: Groups' approximation, stability and high dimensional expanders


=== January 29: Ana Caraiani (Princeton) ===
Abstract:  
Title:  Locally symmetric spaces, torsion classes, and the geometry of period domains


Abstract:  The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.
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.


I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.
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 5: Takis Souganidis (University of Chicago) ===  
===February 9 Wes Pegden (CMU)===
Title:  Scalar Conservation Laws with Rough Dependence


I will present a recently developed theory for scalar conservation laws with nonlinear multiplicative rough signal dependence. I will describe the difficulties, introduce the notion of pathwise entropy/kinetic solution and its well-posedness. I will also talk about the long time behavior of the solutions as well as some regularization by noise type results.
Title: The fractal nature of the Abelian Sandpile


=== February 12: Gautam Iyer (CMU) ===
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.


Homogenization and Anomalous Diffusion
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.


Homogenization is a well known technique used to approximate the macroscopic behaviour of a material with microscopic impurities.
===March 2 Aaron Bertram (Utah)===
While this originally arose in the study of composite materials, it has applications to various other fields, and I will focus on a few results
motivated by fluid dynamics. One well known result in this direction is by GI Taylor estimating the dispersion rate of a solute in a pipe. The
length scales involved in typical pipelines, however, are too short for this result to apply. I will conclude with a few recent "intermediate time" results describing the effective behaviour in scaling regimes outside those of standard homogenization results.


=== February 19: Jean-François Lafont (Ohio State) ===
Title: Stability in Algebraic Geometry


Rigidity and flexibility of almost-isometries
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.


An almost isometry (AI) is a quasi-isometry (QI) with multiplicative
===March 16 Anne Gelb (Dartmouth)===
constant =1. Given two metrics on a closed manifold, Milnor-Swarc implies
that the lifted metrics on the universal cover are QI to each other. When are
they AI to each other? In the rigidity direction, we give various examples
where the only time such lifts are AI is when they are isometric (joint with
Kar and Schmidt). In the flexible direction, we show that for higher genus
surfaces, any two metrics have lifts which, after possibly scaling one of the
lifted metrics, are AI to each other (joint with Schmidt and van Limbeek). In
the latter examples, one can further show that the AI is usually not equivariant
with respect to the group actions.


=== February 26: Hiroyoshi Mitake (Hiroshima university) ===
Title: Reducing the effects of bad data measurements using variance based weighted joint sparsity
In the talk, I will propose a model equation to study the crystal growth as a prototype, which is described by a level-set mean curvature flow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears because of the double nonlinear effects coming from the surface evolution and the source term, which is sensitive to the shapes of source terms. This is a joint work with Y. Giga (U. Tokyo), and H. V. Tran (U. Wisconsin-Madison).


=== March 11: Mitchell Luskin (UMN) ===
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.
Title: Mathematical Modeling of Incommensurate 2D Materials


Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation.


Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.


===March 18: Ralf Spatzier (UMichigan)===


CANCELED: Rigidity in Geometry and Dynamics
===April 5 John Baez (UC Riverside)===


I will survey some rigidity phenomena in dynamics and also geometry, with emphasis on the notion of higher rank.
Title: Monoidal categories of networks
This first emerged in Margulis’ celebrated work on superrrigidity but has also been important in more recent work on symmetry in dynamical systems.
 
How special is it for maps commute with each other? Smale asked this problem fifty years ago, and answers are finally emergingMuch depends on the differentiability
Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.
of the maps: it gets harder the more differentiable the map is. Sometimes we can even classify such maps.  I’ll discuss this and
 
related phenomena.
 
====   ====
 
 
 
===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.
 
===April 13, Jill Pipher, Brown University===
 
Title:  Mathematical ideas in cryptography
 
Abstract:  This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,
including homomorphic encryption.
 
===April 16, Christine Berkesch Zamaere (University of Minnesota)===
Title: Free complexes on smooth toric varieties
 
Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.
 
 
===April 25, Hitoshi Ishii (Tsuda University)===
Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory
 
Abstract:  In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.
 
===May 1 and 2, Andre Neves (University of Chicago and Imperial College London)===
Title: Wow, so many minimal surfaces!
 
Abstract: Minimal surfaces are ubiquitous in geometry and  applied science but their existence theory is rather mysteriousFor instance, Yau in 1982 conjectured that any 3-manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three.
 
After a brief historical account, I will talk about my ongoing work with Marques  and  the progress we made on this question jointly with Irie and Song: we showed that for generic metrics, minimal hypersurfaces are dense and equidistributed. In particular, this settles Yau’s conjecture for generic metrics.
 
The first talk will be more general and the second talk will contain  proofs of the denseness and equidistribution results. This part is joint work with Irie, Marques, and Song.
 
===May 4, Henry Cohn (Microsoft Research and MIT)===
 
Title: The sphere packing problem in dimensions 8 and 24
 
Abstract:
What is the densest packing of congruent spheres in Euclidean space? This problem arises naturally in geometry, number theory, and information theory, but it is notoriously difficult to solve, and until recently no sharp bounds were known above three dimensions. In 2016 Maryna Viazovska found a remarkable solution of the sphere packing problem in eight dimensions, which is much simpler than the proof in three dimensions but tells us nothing about dimensions four through seven. In this talk I'll describe how her breakthrough works and where it comes from, as well as follow-up work extending it to twenty-four dimensions (joint work with Kumar, Miller, Radchenko, and Viazovska).
 
== Future Colloquia ==
[[Colloquia/Blank|Fall 2018]]


== Past Colloquia ==
== Past Colloquia ==
[[Colloquia/Blank|Blank]]
[[Colloquia/Fall2017|Fall 2017]]
[[Colloquia/Spring2017|Spring 2017]]
[[Archived Fall 2016 Colloquia|Fall 2016]]
[[Colloquia/Spring2016|Spring 2016]]


[[Colloquia/Fall2015|Fall 2015]]
[[Colloquia/Fall2015|Fall 2015]]

Revision as of 19:38, 30 April 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, Room: 911) John Baez (UC Riverside) Monoidal categories of networks Craciun
April 6 Edray Goins (Purdue) Toroidal Belyĭ Pairs, Toroidal Graphs, and their Monodromy Groups Melanie
April 13 (911 Van Vleck) Jill Pipher (Brown) Mathematical ideas in cryptography WIMAW
April 16 (Monday) Christine Berkesch Zamaere (University of Minnesota) Free complexes on smooth toric varieties Erman, Sam
April 25 (Wednesday, Room: 911) Hitoshi Ishii (Tsuda University) Wasow lecture Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory Tran
May 1 (Tuesday, 4:30pm, Room: B102 VV) Andre Neves (University Chicago and Imperial College London) Distinguished lecture Wow, so many minimal surfaces! (Part I) Lu Wang
May 2 (Wednesday, 3pm, Room: B325 VV) Andre Neves (University of Chicago and Imperial College London) Distinguished lecture Wow, so many minimal surfaces! (Part II) Lu Wang
May 4 Henry Cohn (Microsoft Research and MIT) The sphere packing problem in dimensions 8 and 24 Ellenberg
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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 5 John Baez (UC Riverside)

Title: Monoidal categories of networks

Abstract: Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, chemical reaction networks, signal-flow graphs, Bayesian networks, food webs, Feynman diagrams and the like. Far from mere informal tools, many of these diagrammatic languages fit into a rigorous framework: category theory. I will explain a bit of how this works and discuss some applications.



April 6 Edray Goins (Purdue)

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

Abstract: A Belyĭ map [math]\displaystyle{ \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]\displaystyle{ \{ 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]\displaystyle{ \beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). }[/math] Replacing [math]\displaystyle{ \mathbb P^1 }[/math] with an elliptic curve [math]\displaystyle{ E }[/math], there is a similar definition of a Belyĭ map [math]\displaystyle{ \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C). }[/math] Since [math]\displaystyle{ E(\mathbb C) \simeq \mathbb T^2(\mathbb R) }[/math] is a torus, we call [math]\displaystyle{ (E, \beta) }[/math] a toroidal Belyĭ pair. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: [math]\displaystyle{ \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 Belyĭ pairs, their corresponding Dessins d'Enfant, and their monodromy groups. For each positive integer [math]\displaystyle{ N }[/math], there are only finitely many toroidal Belyĭ pairs [math]\displaystyle{ (E, \beta) }[/math] with [math]\displaystyle{ \deg \, \beta = N. }[/math] Using the Hurwitz Genus formula, we can begin this database by considering all possible degree sequences [math]\displaystyle{ \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]\displaystyle{ 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 Belyĭ maps [math]\displaystyle{ \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) }[/math] associated to some elliptic curve [math]\displaystyle{ 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.

April 13, Jill Pipher, Brown University

Title: Mathematical ideas in cryptography

Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research, including homomorphic encryption.

April 16, Christine Berkesch Zamaere (University of Minnesota)

Title: Free complexes on smooth toric varieties

Abstract: Free resolutions have been a key part of using homological algebra to compute and characterize geometric invariants over projective space. Over more general smooth toric varieties, this is not the case. We will discuss the another family of complexes, called virtual resolutions, which appear to play the role of free resolutions in this setting. This is joint work with Daniel Erman and Gregory G. Smith.


April 25, Hitoshi Ishii (Tsuda University)

Title: Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory

Abstract: In the lecture, I discuss two asymptotic problems related to Hamilton-Jacobi equations. One concerns the long-time behavior of solutions of time evolutionary Hamilton-Jacobi equations and the other is the so-called vanishing discount problem for stationary Hamilton-Jacobi equations. The last two decades have seen a fundamental importance of weak KAM theory in the asymptotic analysis of Hamilton-Jacobi equations. I explain briefly the Aubry sets and Mather measures from weak KAM theory and their use in the analysis of the two asymptotic problems above.

May 1 and 2, Andre Neves (University of Chicago and Imperial College London)

Title: Wow, so many minimal surfaces!

Abstract: Minimal surfaces are ubiquitous in geometry and applied science but their existence theory is rather mysterious. For instance, Yau in 1982 conjectured that any 3-manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three.

After a brief historical account, I will talk about my ongoing work with Marques and the progress we made on this question jointly with Irie and Song: we showed that for generic metrics, minimal hypersurfaces are dense and equidistributed. In particular, this settles Yau’s conjecture for generic metrics.

The first talk will be more general and the second talk will contain proofs of the denseness and equidistribution results. This part is joint work with Irie, Marques, and Song.

May 4, Henry Cohn (Microsoft Research and MIT)

Title: The sphere packing problem in dimensions 8 and 24

Abstract: What is the densest packing of congruent spheres in Euclidean space? This problem arises naturally in geometry, number theory, and information theory, but it is notoriously difficult to solve, and until recently no sharp bounds were known above three dimensions. In 2016 Maryna Viazovska found a remarkable solution of the sphere packing problem in eight dimensions, which is much simpler than the proof in three dimensions but tells us nothing about dimensions four through seven. In this talk I'll describe how her breakthrough works and where it comes from, as well as follow-up work extending it to twenty-four dimensions (joint work with Kumar, Miller, Radchenko, and Viazovska).

Future Colloquia

Fall 2018

Past Colloquia

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