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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. | 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. | ||
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=== Friday, December 9: Aaron Brown (U Chicago) === | === Friday, December 9: Aaron Brown (U Chicago) === |
Revision as of 14:24, 23 November 2016
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) | TBA | Maxim | |||
December 9 | Aaron Brown (U Chicago) | Lattice actions and recent progress in the Zimmer program | Kent |
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 |
Abstracts
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.
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).