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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 ==
 
==Fall 2017==


{| 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)
|-
|September 8
| [https://sites.google.com/a/wisc.edu/theresa-c-anderson/home/ Tess Anderson] (Madison)
|[[#September 8: Tess Anderson (Madison) |  A Spherical Maximal Function along the Primes  ]]
| Yang
|
|-
|September 15
|
|[[#|  ]]
|
|
|
|-
|September 22, '''9th floor'''
| Jaeyoung Byeon (KAIST)
|[[#September 22: Jaeyoung Byeon (KAIST) |  Patterns formation for elliptic systems with large interaction forces  ]]
| Rabinowitz & Kim
|
|-
|September 29
|
|[[# TBA|  TBA  ]]
|
|
|-
|October 6,  '''9th floor'''
| [http://www3.nd.edu/~jhauenst/ Jonathan Hauenstein] (Notre Dame)
|[[#October 6: Jonathan Hauenstein (Notre Dame) |  Real solutions of polynomial equations ]]
| Boston
|
|-
|October 13, '''9th floor'''
| [http://www.tomokokitagawa.com/ Tomoko L. Kitagawa] (Berkeley)
|[[#October 13: Tomoko Kitagawa (Berkeley) |  A Global History of Mathematics from 1650 to 2017 ]]
| Max
|
|-
|October 20
|  [http://cims.nyu.edu/~pgermain/ Pierre Germain] (Courant, NYU)
|[[#October 13: Pierre Germain (Courant, NYU) |  Stability of the Couette flow in the Euler and Navier-Stokes equations ]]
|  Minh-Binh Tran
|
|-
|October 27
|Stefanie Petermichl (Toulouse)
|[[#October 27: Stefanie Petermichl (Toulouse)  |  Higher order Journé commutators  ]]
| Stovall, Seeger
|
|-
|We, November 1,
|[http://pages.iu.edu/~shaoguo/  Shaoming Guo] (Indiana)
|[[#November 1: Shaoming Guo (Indiana)|  Parsell-Vinogradov systems in higher dimensions  ]]
|Seeger
|
|
|
|
|
|-
|November 17
| [http://math.mit.edu/~ylio/  Yevgeny Liokumovich] (MIT)
|[[#November 17:Yevgeny Liokumovich (MIT)|  Recent progress in Min-Max Theory  ]]
|Sean Paul
|-
|-
|November 21, '''9th floor'''
|January 29 (Monday)
| [https://web.stanford.edu/~mkemeny/homepage.html  Michael Kemeny] (Stanford)
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
|[[#November 21:Michael Kemeny (Stanford)|  The equations defining curves and moduli spaces ]]
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture ]]
|Jordan Ellenberg
| Jordan Ellenberg
|
|
|-
|-
|November 24
|February 2 (Room: 911)
|'''Thanksgiving break'''
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)
|
|[[#February 2 Thomas Fai (Harvard)|  The Lubricated Immersed Boundary Method ]]
| Spagnolie, Smith
|
|
|-
|-
|November 27,  
|February 5 (Monday, Room: 911)
| [http://www.math.harvard.edu/~tcollins/homepage.html  Tristan Collins] (Harvard)
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
|[[#November 27:Tristan Collins (Harvard)| The J-equation and stability  ]]
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
|Sean Paul
| Ellenberg, Gurevitch
|
|
|-
|-
|December 1
|February 6 (Tuesday 2 pm, Room 911)
| Reserved for possible job talks
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)
|[[# TBATBA  ]]
|[[#February 6 Alex Lubotzky (Hebrew University)Groups' approximation, stability and high dimensional expanders ]]
| Ellenberg, Gurevitch
|
|
|-
|-
|December 5 (Tuesday)
|February 9
| [http://web.sas.upenn.edu/rhynd/ Ryan Hynd] (U Penn)
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
|[[#December 5: Ryan Hynd (U Penn)|  TBA  ]]
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
|Sigurd Angenent
| Roch
|
|
|-
|-
|December 8
|March 2
| Reserved for possible job talks
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| Caldararu
|
|
|
|-
|December 11 (Monday)
| [https://people.math.ethz.ch/~mooneyc/  Connor Mooney] (ETH Zurich)
|[[#December 11: Connor Mooney (ETH Zurich)|  Finite time blowup for parabolic systems in the plane]]
|Sigurd Angenent
|
|-
|December 18 (Monday)
| [https://web.stanford.edu/~jchw/ Jenny Wilson] (Stanford)
|[[#December 18: Jenny Wilson (Stanford)|  Stability in the homology of configuration spaces]]
|Jordan Ellenberg
|
|-
|December 19 (Tuesday)
| [https://web.stanford.edu/~amwright/  Alex Wright] (Stanford)
|[[#December 19: Alex Wright (Stanford)|  Dynamics, geometry, and the moduli space of Riemann surfaces]]
|Jordan Ellenberg
|}
== Fall Abstracts ==
=== September 8: Tess Anderson (Madison) ===
Title: A Spherical Maximal Function along the Primes
Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior.  The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example.  In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to.  We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory.  This is joint work with Cook, Hughes, and Kumchev.
=== September 22: Jaeyoung Byeon (KAIST) ===
Title: Patterns formation for elliptic systems with large interaction forces
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
===October 6: Jonathan Hauenstein (Notre Dame) ===
Title: Real solutions of polynomial equations
Abstract: Systems of nonlinear polynomial equations arise frequently in applications with the set of real solutions typically corresponding to physically meaningful solutions.  Efficient algorithms for computing real solutions are designed by exploiting structure arising from the application.  This talk will highlight some of these algorithms for various applications such as solving steady-state problems of hyperbolic conservation laws, solving semidefinite programs, and computing all steady-state solutions of the Kuramoto model.
===October 13: Tomoko Kitagawa (Berkeley) ===
Title: A Global History of Mathematics from 1650 to 2017
Abstract: This is a talk on the global history of mathematics. We will first focus on France by revisiting some of the conversations between Blaise Pascal (1623–1662) and Pierre de Fermat (1607–1665). These two “mathematicians” discussed ways of calculating the possibility of winning a gamble and exchanged their opinions on geometry. However, what about the rest of the world? We will embark on a long oceanic voyage to get to East Asia and uncover the unexpected consequences of blending foreign mathematical knowledge into domestic intelligence, which was occurring concurrently in Beijing and Kyoto. How did mathematicians and scientists contribute to the expansion of knowledge? What lessons do we learn from their experiences?
===October 20: Pierre Germain (Courant, NYU) ===
Title: Stability of the Couette flow in the Euler and Navier-Stokes equations
Abstract: I will discuss the question of the (asymptotic) stability of the Couette flow in Euler and Navier-Stokes. The Couette flow is the simplest nontrivial stationary flow, and the first one for which this question can be fully answered. The answer involves the mathematical understanding of important physical phenomena such as inviscid damping and enhanced dissipation. I will present recent results in dimension 2 (Bedrossian-Masmoudi) and dimension 3 (Bedrossian-Germain-Masmoudi).
===October 27: Stefanie Petermichl (Toulouse)===
Title: Higher order Journé commutators
Abstract: We consider questions that stem from operator theory via Hankel and
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
more basic terms, let us consider a function on the unit circle in its
Fourier representation. Let P_+ denote the projection onto
non-negative and P_- onto negative frequencies. Let b denote
multiplication by the symbol function b. It is a classical theorem by
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
only if b is in an appropriate space of functions of bounded mean
oscillation. The necessity makes use of a classical factorisation
theorem of complex function theory on the disk. This type of question
can be reformulated in terms of commutators [b,H]=bH-Hb with the
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
as in the real variable setting, in the multi-parameter setting or
other, these classifications can be very difficult.
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
spaces of bounded mean oscillation via L^p boundedness of commutators.
We present here an endpoint to this theory, bringing all such
characterisation results under one roof.
The tools used go deep into modern advances in dyadic harmonic
analysis, while preserving the Ansatz from classical operator theory.
===November 1: Shaoming Guo (Indiana) ===
Title: Parsell-Vinogradov systems in higher dimensions
Abstract:
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
===November 17:Yevgeny Liokumovich (MIT)===
Title: Recent progress in Min-Max Theory
Abstract:
Almgren-Pitts Min-Max Theory is a method of constructing minimal hypersurfaces in Riemannian manifolds. In the last few years a number of long-standing open problems in Geometry, Geometric Analysis and 3-manifold Topology have been solved using this method. I will explain the main ideas and challenges in Min-Max Theory with an emphasis on its quantitative aspect: what quantitative information about the geometry and topology of minimal hypersurfaces can be extracted from the theory?
===November 21:Michael Kemeny (Stanford)===
Title: The equations defining curves and moduli spaces
Abstract:
A projective variety is a subset of projective space defined by polynomial equations. One of the oldest problems in algebraic geometry is to give a qualitative description of the equations defining a variety, together with
the relations amongst them. When the variety is an algebraic curve (or Riemann surface), several conjectures
made since the 80s give a fairly good picture of what we should expect. I will describe a new variational approach to these conjectures,
which reduces the problem to studying cycles on Hurwitz space or on the moduli space of curves.
===November 27:Tristan Collins (Harvard)===
Titile: The J-equation and stability
Abstract: Donaldson and Chen introduced the J-functional in '99, and explained its importance in the existence problem for constant scalar curvature metrics on compact Kahler manifolds. An important open problem is to find algebro-geometric conditions under which the J-functional has a critical point.  The critical points of the J-functional are described by a fully-nonlinear PDE called the J-equation.  I will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the J-equation.
===December 5: Ryan Hynd (U Penn)===
Title: TBA.
===December 11: Connor Mooney (ETH Zurich)===
Title: Finite time blowup for parabolic systems in the plane
Abstract:
Hilbert's 19th problem asks about the smoothness of solutions to nonlinear elliptic PDE that arise in the calculus of variations. This problem leads naturally to the question of continuity for solutions to linear elliptic and parabolic systems with measurable coefficients. We will first discuss some classical results on this topic, including Morrey's result that solutions to linear elliptic systems in two dimensions are continuous. We will then discuss surprising recent examples of finite time blowup from smooth data for linear parabolic systems in two dimensions, and important open problems.
===December 18: Jenny Wilson (Stanford)===
Title: Stability in the homology of configuration spaces
Abstract:
This talk will illustrate some patterns in the homology of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.
===December 19: Alex Wright (Stanford)===
Title: Dynamics, geometry, and the moduli space of Riemann surfaces
Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.
== Spring 2018 ==
{| cellpadding="8"
!align="left" | date 
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
| March 16
| March 16
Line 267: Line 69:
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| WIMAW
| WIMAW
|
|-
| April 20
| Xiuxiong Chen(Stony Brook University)
|[[# Xiuxiong Chen|  TBA  ]]
| Bing Wang
|
|
|-
|-
| April 25 (Wednesday)
| April 25 (Wednesday)
| Hitoshi Ishii (Waseda University) Wasow lecture
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture
|[[# TBA|  TBA  ]]
|[[# TBA|  TBA  ]]
| Tran
| Tran
Line 332: Line 140:
== Spring Abstracts ==
== Spring Abstracts ==


=== <DATE>: <PERSON> (INSTITUTION) ===
Title: <TITLE>


Abstract: <ABSTRACT>
===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.




Line 341: Line 192:


[[Colloquia/Blank|Blank Colloquia]]
[[Colloquia/Blank|Blank Colloquia]]
[[Colloquia/Fall2017|Fall 2017]]


[[Colloquia/Spring2017|Spring 2017]]
[[Colloquia/Spring2017|Spring 2017]]

Revision as of 16:32, 9 February 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) TBA Caldararu
March 16 Anne Gelb (Dartmouth) TBA WIMAW
April 4 (Wednesday) John Baez (UC Riverside) TBA Craciun
April 6 Reserved TBA Melanie
April 13 Jill Pipher (Brown) TBA WIMAW
April 20 Xiuxiong Chen(Stony Brook University) TBA Bing Wang
April 25 (Wednesday) Hitoshi Ishii (Waseda University) Wasow lecture TBA Tran
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
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.


Past Colloquia

Blank Colloquia

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012