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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
+
|January 29 (Monday)
| [https://sites.google.com/a/wisc.edu/theresa-c-anderson/home/ Tess Anderson] (Madison)
+
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)
|[[#September 8: Tess Anderson (Madison) |  A Spherical Maximal Function along the Primes ]]
+
|[[#January 29 Li Chao (Columbia)|  Elliptic curves and Goldfeld's conjecture ]]
| Yang
+
| Jordan Ellenberg
 
|
 
|
 
|-
 
|-
|September 15
+
|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
|
+
 
|
 
|
 
|-
 
|-
|September 22, '''9th floor'''
+
|February 5 (Monday, Room: 911)
| Jaeyoung Byeon (KAIST)
+
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University)  
|[[#September 22: Jaeyoung Byeon (KAIST) | Patterns formation for elliptic systems with large interaction forces  ]]
+
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]
| Rabinowitz & Kim
+
| Ellenberg, Gurevitch
 
|
 
|
 
|-
 
|-
|September 29
+
|February 6 (Tuesday 2 pm, Room 911)
|
+
| [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
 
|
 
|
 
|-
 
|-
|October 6,  '''9th floor'''
+
|February 9
| [http://www3.nd.edu/~jhauenst/ Jonathan Hauenstein] (Notre Dame)
+
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)
|[[#October 6: Jonathan Hauenstein (Notre Dame) |  Real solutions of polynomial equations ]]
+
|[[#February 9 Wes Pegden (CMU)|  The fractal nature of the Abelian Sandpile ]]
| Boston
+
| Roch
|
+
|-
+
|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
+
|March 2
| [http://cims.nyu.edu/~pgermain/ Pierre Germain] (Courant, NYU)
+
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)
|[[#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, B239
+
|[http://pages.iu.edu/~shaoguo/  Shaoming Guo] (Indiana)
+
|[[#November 1: Shaoming Guo (Indiana)|  Parsell-Vinogradov systems in higher dimensions  ]]
+
|Seeger
+
|
+
|-
+
|November 3
+
 
|[[# TBA|  TBA  ]]
 
|[[# TBA|  TBA  ]]
 +
| Caldararu
 
|
 
|
|
 
|-
 
|November 10
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|November 17
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|November 24
 
|'''Thanksgiving break'''
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|December 1
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|-
 
|December 5
 
| Ryan Hynd (U Penn)
 
|[[#December 5: Ryan Hynd (U Penn)|  TBA  ]]
 
|
 
|
 
|-
 
|December 8
 
| Reserved for possible job talks
 
|[[# TBA|  TBA  ]]
 
|
 
|
 
|-
 
|December 11
 
| Connor Mooney (ETH Zurich)
 
|[[#December 11: Connor Mooney (ETH Zurich)|  Finite time blowup for parabolic systems in the plane]]
 
|
 
|
 
|-
 
 
|}
 
 
== 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.
 
 
===#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.
 
 
== Spring 2018 ==
 
 
{| cellpadding="8"
 
!align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
 
|-
 
|-
 
| March 16
 
| March 16
Line 222: 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 287: 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 296: Line 192:
  
 
[[Colloquia/Blank|Blank Colloquia]]
 
[[Colloquia/Blank|Blank Colloquia]]
 +
 +
[[Colloquia/Fall2017|Fall 2017]]
  
 
[[Colloquia/Spring2017|Spring 2017]]
 
[[Colloquia/Spring2017|Spring 2017]]

Latest revision as of 10:32, 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