Difference between revisions of "Colloquia"

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= 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)
|Flat connections and quantum groups
+
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]
|Gurevich
+
| Spagnolie, Smith
|-
+
|'''Wed, Sept 25, 2:30PM'''
+
|[http://mypage.iu.edu/~alindens/ Ayelet Lindenstrauss]
+
 
|
 
|
|Meyer
 
 
|-
 
|-
|'''Wed, Sept 25''' (LAA lecture)
+
|February 5 (Monday, Room: 911)
|[http://www.cs.berkeley.edu/~demmel/ Jim Demmel] (Berkeley)
+
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
|Communication Avoiding Algorithms for Linear Algebra and Beyond
+
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
|Gurevich
+
| Ellenberg, Gurevitch
|-
+
|'''Thurs, Sept 26''' (LAA lecture, Joint with Applied Algebra Seminar)
+
|[http://www.cs.berkeley.edu/~demmel/ Jim Demmel] (Berkeley)
+
|Implementing Communication Avoiding Algorithms
+
|Gurevich
+
|-
+
|Sept 27 (LAA lecture)
+
|[http://www.cs.berkeley.edu/~demmel/ Jim Demmel] (Berkeley)
+
|Communication Lower Bounds and Optimal Algorithms for Programs that Reference Arrays
+
|Gurevich
+
|-
+
|Oct 4
+
|[http://www.math.tamu.edu/~sottile/ Frank Sottile] (Texas A&M)
+
 
|
 
|
|Caldararu
 
 
|-
 
|-
|Oct 11
+
|February 6 (Tuesday 2 pm, Room 911)
|[http://math.uchicago.edu/~wilkinso/ Amie Wilkinson] (Chicago)
+
| [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
 
|
 
|
|WIMAW (Cladek)
 
|-
 
|'''Tues, Oct 15, 4PM''' (Distinguished Lecture)
 
|[http://math.mit.edu/people/profile.php?pid=1222 Alexei Borodin] (MIT)
 
|Integrable probability I
 
|Valko
 
|-
 
|'''Wed, Oct 16, 2:30PM''' (Distinguished Lecture)
 
|[http://math.mit.edu/people/profile.php?pid=1222 Alexei Borodin] (MIT)
 
|Integrable probability II
 
|Valko
 
 
|-
 
|-
|<strike>Oct 18</strike>
+
|February 9
|No colloquium due to the distinguished lecture
+
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
|
+
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
 +
| Roch
 
|
 
|
 
|-
 
|-
|Oct 25
+
|March 2
|[http://www.math.umn.edu/~garrett/ Paul Garrett] (Minnesota)
+
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
|
+
|[[# TBA| TBA  ]]
|Gurevich
+
| Caldararu
|
+
 
|
 
|
 
|-
 
|-
|Nov 1
+
| March 16
|[http://www.cs.utexas.edu/~alewko/ Allison Lewko] (Microsoft Research New England)
+
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)
 +
|[[# TBA|  TBA  ]]
 +
| WIMAW
 
|
 
|
|Stovall
 
 
|-
 
|-
|Nov 8
+
|April 4 (Wednesday)
|[http://www.math.cornell.edu/~riley/ Tim Riley] (Cornell)
+
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)
 +
|[[# TBA|  TBA  ]]
 +
| Craciun
 
|
 
|
|Dymarz
 
 
|-
 
|-
|Nov 15 and later
+
| April 6
|Reserved
+
| Reserved
 +
|[[# TBA|  TBA  ]]
 +
| Melanie
 
|
 
|
|Street
 
|}
 
 
== Spring 2014 ==
 
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
 
|-
 
|-
|Jan 24
+
| April 13
|
+
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)
|
+
|[[# TBA|  TBA  ]]
 +
| WIMAW
 
|
 
|
 
|-
 
|-
|Jan 31
+
| April 20
|[http://csi.usc.edu/~ubli/ubli.html Urbashi Mitra] (USC)
+
| Xiuxiong Chen(Stony Brook University)
 +
|[[# Xiuxiong Chen|  TBA  ]]
 +
| Bing Wang
 
|
 
|
|Gurevich
 
 
|-
 
|-
|Feb 7
+
| April 25 (Wednesday)
|David Treumann (Boston College)
+
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture
 +
|[[# TBA|  TBA  ]]
 +
| Tran
 
|
 
|
|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
+
|date
|
+
| person (institution)
|
+
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
 
|-
 
|-
|March 14
+
|date
 +
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|
 
|
 
|-
 
|<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
+
|date
|[https://sites.google.com/site/katejuschenko/ Kate Jushchenko] (Northwestern)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|Dymarz
 
 
|-
 
|-
|April 11
+
|date
|[http://www.cs.uchicago.edu/people/risi Risi Kondor] (Chicago)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|Gurevich
 
 
|-
 
|-
|April 18 (Wasow Lecture)
+
|date
|[http://mathnt.mat.jhu.edu/sogge/ Christopher Sogge] (Johns Hopkins)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|A. Seeger
 
 
|-
 
|-
|April 25
+
|date
|[http://www.charlesdoran.net Charles Doran](University of Alberta)
+
| person (institution)
 +
|[[# TBA|  TBA  ]]
 +
| hosting faculty
 
|
 
|
|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)===
  
===Sep 6: Matt Baker (GA Tech) ===
+
Title: The fractal nature of the Abelian Sandpile
''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).
+
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.  
  
===Sep 25: Jim Demmel (Berkeley) ===
+
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.
''Communication Avoiding Algorithms for Linear Algebra and Beyond''
+
  
Algorithm have two costs: arithmetic and communication, i.e. moving data between levels of a memory hierarchy or processors over a network. Communication costs (measured in time or energy per operation) already greatly exceed arithmetic costs, and the gap is growing over time following technological trends. Thus our goal is to design algorithms that minimize communication. We present algorithms that attain provable lower bounds on communication, and show large speedups compared to their conventional counterparts. These algorithms are for direct and iterative linear algebra, for dense and sparse matrices, as well as direct n-body simulations. Several of these algorithms exhibit perfect strong scaling, in both time and energy: run time (resp. energy) for a fixed problem size drops proportionally to the number of processors p (resp. is independent of p). Finally, we describe extensions to algorithms involving arbitrary loop nests and array accesses, assuming only that array subscripts are affine functions of the loop indices.
 
  
===Sep 26: Jim Demmel (Berkeley) ===
+
== Past Colloquia ==
''Implementing Communication Avoiding Algorithms''
+
  
Designing algorithms that avoiding communication, attaining
+
[[Colloquia/Blank|Blank Colloquia]]
lower bounds if possible, is critical for algorithms to minimize runtime and
+
energy on current and future architectures. These new algorithms can have
+
new numerical stability properties, new ways to encode answers, and new data
+
structures, not just depend on loop transformations (we need those too!).
+
We will illustrate with a variety of examples including direct linear algebra
+
(eg new ways to perform pivoting, new deterministic and randomized
+
eigenvalue algorithms), iterative linear algebra (eg new ways to reorganize
+
Krylov subspace methods) and direct n-body algorithms, on architectures
+
ranging from multicore to distributed memory to heterogeneous.
+
The theory describing communication avoiding algorithms can give us a large
+
design space of possible implementations, so we use autotuning to find
+
the fastest one automatically. Finally, on parallel architectures one can
+
frequently not expect to get bitwise identical results from multiple runs,
+
because of dynamic scheduling and floating point nonassociativity;
+
this can be a problem for reasons from debugging to correctness.
+
We discuss some techniques to get reproducible results at modest cost.
+
  
===Sep 27: Jim Demmel (Berkeley) ===
+
[[Colloquia/Fall2017|Fall 2017]]
''Communication Lower Bounds and Optimal Algorithms for Programs that Reference Arrays''
+
  
Our goal is to minimize communication, i.e. moving data, since it increasingly
+
[[Colloquia/Spring2017|Spring 2017]]
dominates the cost of arithmetic in algorithms. Motivated by this, attainable
+
communication lower bounds have been established by many authors for a
+
variety of algorithms including matrix computations.
+
  
The lower bound approach used initially by Irony, Tiskin and Toledo
+
[[Archived Fall 2016 Colloquia|Fall 2016]]
for O(n^3)  matrix multiplication, and later by Ballard et al
+
for many other linear algebra algorithms, depends on a geometric result by
+
Loomis and Whitney: this result bounds the volume of a 3D set
+
(representing multiply-adds done in the inner loop of the algorithm)
+
using the product of the areas of certain 2D projections of this set
+
(representing the matrix entries available locally, i.e., without communication).
+
  
Using a recent generalization of Loomis' and Whitney's result, we generalize
+
[[Colloquia/Spring2016|Spring 2016]]
this lower bound approach to a much larger class of algorithms,
+
that may have arbitrary numbers of loops and arrays with arbitrary dimensions,
+
as long as the index expressions are affine combinations of loop variables.
+
In other words, the algorithm can do arbitrary operations on any number of
+
variables like A(i1,i2,i2-2*i1,3-4*i3+7*i_4,…).
+
Moreover, the result applies to recursive programs, irregular iteration spaces,
+
sparse matrices,  and other data structures as long as the computation can be
+
logically mapped to loops and indexed data structure accesses.
+
  
We also discuss when optimal algorithms exist that attain the lower bounds;
+
[[Colloquia/Fall2015|Fall 2015]]
this leads to new asymptotically faster algorithms for several problems.
+
  
 +
[[Colloquia/Spring2014|Spring 2015]]
  
 +
[[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 11:32, 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