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Mathematics Colloquium

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

Fall 2016

date speaker title host(s)
September 9
September 16 Po-Shen Loh (CMU) Directed paths: from Ramsey to Pseudorandomness Ellenberg
September 23 Gheorghe Craciun (UW-Madison) Toric Differential Inclusions and a Proof of the Global Attractor Conjecture Street
September 30 Akos Magyar (University of Georgia) Geometric Ramsey theory Cook
October 7
October 14 Ling Long (LSU) Hypergeometric functions over finite fields Yang
October 21 No colloquium this week
Tuesday, October 25, 9th floor Stefan Steinerberger (Yale) Three Miracles in Analysis Seeger
October 28, 9th floor Linda Reichl (UT Austin) Microscopic hydrodynamic modes in a binary mixture Minh-Binh Tran
Monday, October 31, B239 Kathryn Mann (Berkeley) Groups acting on the circle Smith
November 4
Monday, November 7 at 4:30, 9th floor (AMS Maclaurin lecture) Gaven Martin (New Zealand Institute for Advanced Study) Siegel's problem on small volume lattices Marshall
November 11 Reserved for possible job talks
Wednesday, November 16, 9th floor Kathryn Lindsey (U Chicago) Shapes of Julia Sets Michell
November 18, B239 Andrew Snowden (University of Michigan) Recent progress in representation stability Ellenberg
Monday, November 21, 9th floor Mariya Soskova (University of Wisconsin-Madison) Definability in degree structures Smith
November 25 Thanksgiving break
December 2, 9th floor Hao Shen (Columbia) Singular Stochastic Partial Differential Equations - How do they arise and what do they mean? Roch
Monday, December 5, B239 Botong Wang (UW Madison) Enumeration of points, lines, planes, etc. Maxim
December 9, B239 Aaron Brown (U Chicago) Lattice actions and recent progress in the Zimmer program Kent
Monday, December 19 Andrew Zimmer (U Chicago) Metric spaces of non-positive curvature and applications in several complex variables Gong

Spring 2017

date speaker title host(s)
January 20 Reserved for possible job talks
January 27 Reserved for possible job talks
February 3
February 6 (Wasow lecture) Benoit Perthame (University of Paris VI) TBA Jin
February 10 (WIMAW lecture) Alina Chertock (NC State Univ.) WIMAW
February 17 Gustavo Ponce (UCSB) Minh-Binh Tran
February 24
March 3 Ken Bromberg (University of Utah) Dymarz
Tuesday, March 7, 4PM (Distinguished Lecture) Roger Temam (Indiana University) Smith
Wednesday, March 8, 2:25PM Roger Temam (Indiana University) Smith
March 10 No Colloquium
March 17 Lillian Pierce (Duke University) TBA M. Matchett Wood
March 24 Spring Break
Wednesday, March 29 (Wasow) Sylvia Serfaty (NYU) TBA Tran
March 31 No Colloquium
April 7 Hal Schenck Erman
April 14 Wilfrid Gangbo Feldman & Tran
April 21 Mark Andrea de Cataldo (Stony Brook) TBA Maxim
April 28 Thomas Yizhao Hou TBA Li


September 16: Po-Shen Loh (CMU)

Title: Directed paths: from Ramsey to Pseudorandomness

Abstract: Starting from an innocent Ramsey-theoretic question regarding directed paths in graphs, we discover a series of rich and surprising connections that lead into the theory around a fundamental result in Combinatorics: Szemeredi's Regularity Lemma, which roughly states that every graph (no matter how large) can be well-approximated by a bounded-complexity pseudorandom object. Using these relationships, we prove that every coloring of the edges of the transitive N-vertex tournament using three colors contains a directed path of length at least sqrt(N) e^{log^* N} which entirely avoids some color. The unusual function log^* is the inverse function of the tower function (iterated exponentiation).

September 23: Gheorghe Craciun (UW-Madison)

Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

Abstract: The Global Attractor Conjecture says that a large class of polynomial dynamical systems, called toric dynamical systems, have a globally attracting point within each linear invariant space. In particular, these polynomial dynamical systems never exhibit multistability, oscillations or chaotic dynamics.

The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem.

We discuss the history of this problem, including the connection between this conjecture and the Boltzmann equation. Then, we introduce toric differential inclusions, and describe how they can be used to prove this conjecture in full generality.

September 30: Akos Magyar (University of Georgia)

Title: Geometric Ramsey theory

Abstract: Initiated by Erdos, Graham, Montgomery and others in the 1970's, geometric Ramsey theory studies geometric configurations, determined up to translations, rotations and possibly dilations, which cannot be destroyed by finite partitions of Euclidean spaces. Later it was shown by ergodic and Fourier analytic methods that such results are also possible in the context of sets of positive upper density in Euclidean spaces or the integer lattice. We present a new approach, motivated by developments in arithmetic combinatorics, which provide new results as well new proofs of some classical results in this area.

October 14: Ling Long (LSU)

Title: Hypergeometric functions over finite fields

Abstract: Hypergeometric functions are special functions with lot of symmetries. In this talk, we will introduce hypergeometric functions over finite fields, originally due to Greene, Katz and McCarthy, in a way that is parallel to the classical hypergeometric functions, and discuss their properties and applications to character sums and the arithmetic of hypergeometric abelian varieties. This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu.

Tuesday, October 25, 9th floor: Stefan Steinerberger (Yale)

Title: Three Miracles in Analysis

Abstract: I plan to tell three stories: all deal with new points of view on very classical objects and have in common that there is a miracle somewhere. Miracles are nice but difficult to reproduce, so in all three cases the full extent of the underlying theory is not clear and many interesting open problems await. (1) An improvement of the Poincare inequality on the Torus that encodes a lot of classical Number Theory. (2) If the Hardy-Littlewood maximal function is easy to compute, then the function is sin(x). (Here, the miracle is both in the statement and in the proof). (3) Bounding classical integral operators (Hilbert/Laplace/Fourier-transforms) in L^2 -- but this time from below (this problem originally arose in medical imaging). Here, the miracle is also known as 'Slepian's miracle' (this part is joint work with Rima Alaifari, Lillian Pierce and Roy Lederman).

October 28: Linda Reichl (UT Austin)

Title: Microscopic hydrodynamic modes in a binary mixture

Abstract: Expressions for propagation speeds and decay rates of hydrodynamic modes in a binary mixture can be obtained directly from spectral properties of the Boltzmann equations describing the mixture. The derivation of hydrodynamic behavior from the spectral properties of the kinetic equation provides an alternative to Chapman-Enskog theory, and removes the need for lengthy calculations of transport coefficients in the mixture. It also provides a sensitive test of the completeness of kinetic equations describing the mixture. We apply the method to a hard-sphere binary mixture and show that it gives excellent agreement with light scattering experiments on noble gas mixtures.

Monday, October 31: Kathryn Mann (Berkeley)

Title: Groups acting on the circle

Abstract: Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic and natural question from geometric topology, but also a very difficult one -- even in the case where M is the circle, and G is a familiar, finitely generated group.

In this talk, I’ll introduce you to the theory of groups acting on the circle, building on the perspectives of Ghys, Calegari, Goldman and others. We'll see some tools, old and new, some open problems, and some connections between this theory and themes in topology (like foliated bundles) and dynamics.

November 7: Gaven Martin (New Zealand Institute for Advanced Study)

Title: Siegel's problem on small volume lattices

Abstract: We outline in very general terms the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram. This gives us the smallest regular tessellation of hyperbolic 3-space. This solves (in three dimensions) a problem posed by Siegel in 1945. Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area.

There are strong connections with arithmetic hyperbolic geometry in the proof, and the result has applications in the maximal symmetry groups of hyperbolic 3-manifolds in much the same way that Hurwitz's 84g-84 theorem and Siegel's result do.

Wednesday, November 16 (9th floor): Kathryn Lindsey (U Chicago)

Title: Shapes of Julia Sets

Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. William Thurston asked "What are the possible shapes of polynomial Julia sets?" For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is "yes!" I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.

November 18: Andrew Snowden (University of Michigan)

Title: Recent progress in representation stability

Abstract: Representation stability is a relatively new field that studies somewhat exotic algebraic structures and exploits their properties to prove results (often asymptotic in nature) about objects of interest. I will describe some of the algebraic structures that appear (and state some important results about them), give a sampling of some notable applications (in group theory, topology, and algebraic geometry), and mention some open problems in the area.

Monday, November 21: Mariya Soskova (University of Wisconsin-Madison)

Title: Definability in degree structures

Abstract: Some incomputable sets are more incomputable than others. We use Turing reducibility and enumeration reducibility to measure the relative complexity of incomputable sets. By identifying sets of the same complexity, we can associate to each reducibility a degree structure: the partial order of the Turing degrees and the partial order of the enumeration degrees. The two structures are related in nontrivial ways. The first has an isomorphic copy in the second and this isomorphic copy is an automorphism base. In 1969, Rogers asked a series of questions about the two degree structures with a common theme: definability. In this talk I will introduce the main concepts and describe the work that was motivated by these questions.

Friday, December 2: Hao Shen (Columbia)

Title: Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?

Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.

Monday, December 5: Botong Wang (UW-Madison)

Title: Enumeration of points, lines, planes, etc.

Abstract: It is a theorem of de Bruijn and Erdos that n points in the plane determine at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization of this theorem, which confirms a “top-heavy” conjecture of Dowling and Wilson in 1975. I will give a sketch of the key ideas of the proof, which are the hard Lefschetz theorem and the decomposition theorem in algebraic geometry. I will also talk about a log-concave conjecture on the number of independent sets. These are joint works with June Huh.

Friday, December 9: Aaron Brown (U Chicago)

Lattice actions and recent progress in the Zimmer program

Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds. For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. In particular—on manifolds whose dimension is below the dimension of all algebraic examples—Zimmer’s conjecture asserts that every action is finite.

I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my results within the Zimmer program: (1) a solution to Zimmer’s conjecture for actions of cocompact lattices in SL(n,R) (joint with D. Fisher and S. Hurtado); (2) a classification (up to topological semiconjugacy) of all actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).

Past Colloquia

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012