Department of

Mathematics


Seminar Calendar
for Mathematics Colloquium - Special Lecture 2015-16 events the year of Friday, December 18, 2015.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, December 8, 2015

4:00 pm in 245 Altgeld Hall,Tuesday, December 8, 2015

Quantization, reduction mod p, and Autoequivalences of the Weyl algebra

Christopher Dodd (Perimeter Institute for Theoretical Physics, Waterloo, Ontario)

Abstract: The Weyl algebra of polynomial differential operators is a basic object which appears in algebraic geometry, representation theory, and mathematical physics. In this talk, I will discuss my recent proof of some conjectures of A. Belov-Kanel and M. Kontsevich concerning the structure of the automorphism group of the Weyl algebra. The question turns out to be related to defining an appropriate notion of "support cycle" for a differential equation, which, in turn, involves techniques from positive characteristic. In particular, we shall explain a "quantization correspondence" which is based on reducing differential equations to finite characteristic. View talk at https://youtu.be/cKNa6rQW1RQ

Wednesday, December 9, 2015

4:00 pm in 245 Altgeld Hall,Wednesday, December 9, 2015

Factors in graphs, weighted graphs and directed graphs

Theodore Molla (University of Illinois at Urbana-Champaign)

Abstract: A factor is a subgraph that contains all of the vertices of its host graph, so a perfect matching is a factor consisting entirely of vertex disjoint edges and a Hamiltonian cycle is a factor that is a cycle. Many celebrated theorems in graph theory give sufficient conditions for the existence of a specific factor. For example, Dirac's Theorem states that if $G$ is a graph on $n$ vertices, $n \ge 3$ and the minimum degree of $G$ is at least $n/2$, then $G$ contains a Hamiltonian cycle. In this talk, we will discuss several theorems on the existence of factors in graphs, weighted graphs and directed graphs that are similar to Dirac's Theorem.

Thursday, December 10, 2015

4:00 pm in 245 Altgeld Hall,Thursday, December 10, 2015

Multilinear phenomena in analysis and related areas

Eyvindur Ari Palsson (Williams College)

Abstract: Many concepts in mathematics, from boundary value problems in partial differential equations and mathematical physics to finite point configurations in geometric combinatorics, are fundamentally tied to various operator bounds. This frequently leads to the estimation of linear and multilinear integral operators with techniques from harmonic analysis, often combined with geometric and combinatorial principles. In this talk I shall describe two examples from my research program united by the common theme of multilinearity. I will begin by discussing finite point configuration questions that can be thought of as continuous analogs of the Erdos distinct distance problem. Multilinear analogs of the linear generalized Radon transforms arise naturally here and are of independent interest. I will then discuss singular integral operators motivated by Calderon's commutators and present estimates for an operator in between the bilinear Hilbert transform and the trilinear one. View talk at https://youtu.be/YPfbIyZ5bKo

Friday, December 11, 2015

4:00 pm in 245 Altgeld Hall,Friday, December 11, 2015

The dynamics of Type II solutions to energy critical wave equations

Hao Jia (University of Chicago)

Abstract: The study of dynamics of energy critical wave equations has seen remarkable progresses in recent years, resulting in deeper understanding of the singularity formation, soliton dynamics, and global large data theory. I will firstly review some of the landmark results, with emphasis on the channel of energy inequalities discovered by Duyckaerts, Kenig and Merle. Applications in the study of global dynamics of defocusing energy critical wave equation with a trapping potential in the radial case will be presented in some detail. We remark that the channel of energy argument provides crucial control on the global dynamics of the solution, and seems to be the only tool currently available to measure dispersion in this context, when we do not assume any smallness condition. The channel of energy argument is however sensitive to dimensions, and in higher dimensions, it is less powerful. We will mention a new approach to eliminate the dispersive energy when the channel of energy argument fails. Lastly, a new Morawetz estimate in the context of focusing energy critical wave equations will be discussed. This estimate allows us to study the singularity formation in more details in the non-radial case, without size restriction. As a result, we can characterize the solution along a sequence of times approaching the singular time, up to every nontrivial scale, as modulated solitons. View talk at https://youtu.be/rIesHSI8c24

Monday, December 14, 2015

4:30 pm in 245 Altgeld Hall,Monday, December 14, 2015

The core entropy of quadratic polynomials

Giulio Tiozzo (Yale)

Abstract: The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter. Recently, W. Thurston has introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomials as the entropy of a certain forward-invariant set of the Julia set (the Hubbard tree). As we shall see, the core entropy is a purely topological / combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we shall see how to relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.

Tuesday, December 15, 2015

4:30 pm in 245 Altgeld Hall,Tuesday, December 15, 2015

Random walk parameters and the geometry of groups

Tianyi Zheng (Stanford University)

Abstract: The first characterization of groups by an asymptotic description of random walks on their Cayley graphs dates back to Kestenís criterion of amenability. I will first review some connections between the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of simple random walks on groups of exponential volume growth. Based on joint work with Jeremie Brieussel. View talk at https://youtu.be/ScOxXUOm02g

Wednesday, December 16, 2015

4:00 pm in 245 Altgeld Hall,Wednesday, December 16, 2015

Introducing realism into mathematical models of malaria

Lauren M. Childs (Harvard)

Abstract: Half of the world population remains at risk of malaria infection, and each year of the 200 million people infected nearly a half million die. Mathematical models have been used for over 100 years to study the spread of malaria at the population level. Analysis of these initial models suggested interventions against its necessary mosquito host, which proved highly effective in the original malaria eradication campaigns in the US and Europe. The persistence of malaria in many regions of the world, however, indicates the necessity for more complex models of malaria dynamics incorporating realism through heterogeneities at a variety of biological scales. In this talk, I present a suite of models examining the human immune response, variability in the malaria parasite, and mosquito population control. I begin with a stochastic energy landscape model of the development of immunity to the diverse set of antigens expressed by the malaria parasite. Next, I present a complicated difference equation model of interactions between the malaria parasite and the human immune response. I conclude with an analysis of an ordinary differential equation model of mosquito population dynamics involving co-infection of the malaria parasite and Wolbachia bacteria, known to inhibit the development of infectious diseases. The range of mathematical techniques utilized reflects the variation in biological questions addressed in this talk. View talk at https://youtu.be/Lj739EqTA4Y

Thursday, December 17, 2015

4:30 pm in 245 Altgeld Hall,Thursday, December 17, 2015

Rigidity in von Neumann algebras

Ionut Chifan (University of Iowa)

Abstract: Back in the 30's F. Murray and J. von Neumann discovered a natural way to associate a von Neumann algebra, $L^\infty(X)\rtimes \Gamma $, to any measure preserving action of a countable group on a probability space $\Gamma \curvearrowright X$. The general paradigm of classifying $L^\infty(X)\rtimes \Gamma $ in terms of $\Gamma \curvearrowright X$ has emerged overtime as an interesting yet very challenging theme of study as these algebras tend to have very limited "memory" of the initial data. This is best illustrated by A. Connes' celebrated result which asserts that all free ergodic actions of amenable groups on probability spaces give rise to isomorphic algebras. The rigidity phenomena in this subject, i.e. instances when $L^\infty(X)\rtimes \Gamma $ remembers well $\Gamma \curvearrowright X$, emerged only over the last decade via S. Popa's deformation/rigidity theory and represent a very active area of research in operator algebras. In the first part of my talk I will survey these developments, commenting on the methods involved and their importance and also pointing out my contributions to the subject. The second part of the talk focuses on my recent results regarding the classification of ultrapower II$_1$ factors. Precisely, I will show that the group factors introduced by D. McDuff in '69 have non-isomorphic ultrapowers with respect to any ultrafilter on any set. This entails the existence of a continuum of pairwise non-elementarily equivalent II_1 factors, settling a well-known open problem in continuous model theory. view talk at https://youtu.be/iOLdoSwyQAw

Friday, December 18, 2015

4:00 pm in 245 Altgeld Hall,Friday, December 18, 2015

Locally symmetric spaces and torsion classes

Ana Caraiani (Princeton University)

Abstract: The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields. I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry. View talk at https://youtu.be/x0QtCA-A73s