Department Colloquium
Forthcoming colloquia (in chronological order)
May 2013
Date: Thursday, May 23, 2013
Time: 4:30-5:30 PM
Venue: Eckhart 206
Speaker: Luis Caffarelli
Abstract: We will discuss several moments involving segregation: adjacent ones, where particles of different kind annihilate on contact, and models where the segregation occurs at a distance We will describe some simple mathematical models and their analytical and geometric properties.
Date: Friday, May 24, 2013
Time: 3-4PM
Venue: Eckhart 206
Speaker: Thomas Scanlon
Title: Diophantine geometry of algebraic dynamics
Abstract: Generalizing from the case of sequences defined by linear recurrence relations, one may consider sequences obtained by iteratively applying polynomials. Several authors have raised a conjecture about iterates of polynomial functions inspired by Mordell's conjecture (Faltings' Theorem) about rational points on curves ... (full abstract at http://math.uchicago.edu/research/abstracts/Cohenlecture2013.pdf).
Past colloquia (in reverse chronological order)
May 2013
This colloquium is jointly sponsored by the Association for Women in Mathematics:
Date: Friday, May 10, 2013
Time: 4:30 - 5:30 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Lai-Sang Young (Courant Institute/NYU)
Talk title: Measuring dynamical complexity
Abstract: I will discuss three ways to capture dynamical complexity: (A) hyperbolicity, a geometric characterization of instability, (B) entropy, an information-theoretical approach to capturing the randomness of dynamical events, and (C) correlation decay or memory loss as a function of time. I will review these ideas, discuss how they are related, and give a very brief (and somewhat personal) survey of the progress made in the last decades. For illustration, I will use a concrete example, namely that of shear-induced chaos in periodically kicked oscillators. The idea of this example goes back to van der Pol nearly 100 years ago, but the mathematics was done only recently.
April 2013
This colloquium is jointly sponsored by the Association for Women in Mathematics:
Date: Friday, April 19, 2013
Time: 4 - 5 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Laura DeMarco (UIC, works in dynamics and complex analysis)
Talk title: Complex and Arithmetic Dynamics.
Abstract: Questions about complex dynamical systems have traditionally been approached with techniques from analysis (complex or geometric). In the last 5 years or so, methods from arithmetic and algebraic geometry have played a central role -- and the result is an active new research area, the "arithmetic of dynamical systems" (to borrow the title of Silverman's textbook on the subject). The questions themselves have evolved, inspired by results from arithmetic geometry. In this talk, I will present joint work with Matt Baker, where we study "special points" within the moduli space of complex polynomial dynamical systems.
Date: Wednesday, April 3, 2013
Time: 3 - 4 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Marston Conder. He is the AMS-NZMS Maclaurin lecturer.
Talk title: Some unexpected theoretical consequences of computations involving groups
Abstract: In this talk I will give some instances of experimental computations involving groups (in the context of their actions on graphs and maps) that have led to unexpected theoretical discoveries. These include new presentations for 3-dimensional special linear groups, a closed-form definition for the binary reflected Gray codes, a new theorem on groups expressible as a product of an abelian group and a cyclic group, and revealing observations about the genus spectra of particular classes of regular maps on surfaces. Such examples highlight the value of computational algebra as a tool for experimental work, which can have surprising outcomes.
March 2013
Date: Tuesday, March 19, 2013
Time: 3 - 4 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Jeff Harvey (University of Chicago Physics Department)
Talk title: Mock Modular Forms, K3 surfaces, and Umbral Moonshine
Abstract: Monstrous moonshine is a phenomenon linking modular forms with the representation theory of the Monster, the largest sporadic group. Mock modular forms are generalizations of modular forms introduced by Ramanujan, but a coherent definition of them appeared only in the last ten years due to work of Zwegers connecting them to indefinite theta functions, Appell-Lerch sums and meromorphic Jacobi forms. Recently evidence for a new kind of moonshine, dubbed Umbral Moonshine, has emerged involving connections between certain of these mock modular forms and a sequence of finite groups that arise as subgroups of the Conway group. The first of these examples, discovered by Eguchi, Ooguri and Tachikawa, involves a connection between the largest Mathieu group M24 and a mock modular form which is related to the elliptic genus of K3 surfaces. The origin and explanation of Umbral Moonshine are at the moment a mystery. I will survey these new developments and suggest that the eventual explanation is likely to involve aspects of string theory and the AdS/CFT correspondence. This talk is based on arXiv preprint 1204.2779 with Miranda Cheng and John Duncan.
This colloquium is jointly sponsored by the Association for Women in Mathematics:
Date: Wednesday, March 6, 2013
Time: 3 - 4 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Karin Melnick (University of Maryland)
Talk title: A rigidity theorem in conformal geometry and beyond
Abstract: The classical exponential map in Riemannian geometry has the following very important implications: if an isometry f fixes a point and has trivial derivative there, then f is trivial; moreover, the differential gives a simple normal form for all isometries fixing a given point. Conformal transformations of a Riemannian manifold are required only to preserve angles, not distances. These have no exponential map. Nontrivial conformal transformations can have differential equal the identity at a fixed point, but this occurrence has very strong implications for the underlying manifold.
I will present this rigidity phenomonenon in conformal geometry and a wide range of generalizations. The key to these results is the notion of Cartan geometry, which infinitesimally models a manifold on a homogeneous space. This point of view leads to a normal forms theorem for conformal Lorentzian flows. It also leads to a suite of results on a seemingly widespread rigidity phenomenon for flows on parabolic geometries, a rich family of geometric structures whose homogeneous models include flag varieties and boundaries of symmetric spaces.
February 2013
Date: Tuesday, February 19, 2013
Time: 4:30 - 5:30 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Leonid Polterovich (Tel Aviv University; visiting the University of Chicago)
Title: Symplectic topologist's tale of quantum mechanics
Abstract: We focus on constraints on the Poisson brackets found within Symplectic Topology. Their interpretation and proof are related to Quantum Mechanics. In the talk we discuss an exchange of ideas between these fields.
November 2012
Date: Wednesday, November 28, 2012
Time: 5:00-6:00 PM
Venue: Eckhart 202 (Room capacity: 72)
Speaker: Luis Silvestre (University of Chicago)
Title: On the continuity of solutions to drift-diffusion equations
Abstract: A drift-diffusion equation is like the heat equation with an extra first order term. In some cases, the Laplacian is replaced by a fractional Laplacian. There are several nonlinear models in a variety of contexts that fit into this scheme. In order to understand the solvability of the non linear models, it is essential to obtain a priori estimates on the smoothness of the solution for linear drift-diffusion equations when the first order term is given by a very irregular vector field times the gradient. We will analyze different smoothness estimates in different situations. It is particularly important to understand the consequences of assuming that the drift is divergence free, given their applications to models related to incompressible fluids.
October 2012
Date: Wednesday, October 3, 2012
Time: 4:30-5:30 PM
Venue: Eckhart 133 (Room capacity: 106)
Speaker: Maryanthe Malliaris (University of Chicago)
Title: Comparing the complexity of unstable theories
Abstract: Model theory brings a unique and powerful point of view to bear on many fundamental structural questions in mathematics. This talk will have two interrelated aims. The first is to motivate and present some recent progress of Malliaris on Keisler's order: a suggested program of comparing the complexity of theories from asymptotic (ultrapower) points of view. These results, the first in decades in this program, build towards a true classification of the unstable theories. The stable theories have been well understood thanks to Morley's 1962 Chicago PhD thesis and to Shelah's significant generalizations of Morley's work.
The second is to indicate how our work arising from this model theoretic program has led to theorems in Szemeredi regularity and in set theory and general topology. Cantor proved in 1874 that the continuum is uncountable, and Hilbert's first problem asked whether it is the smallest uncountable cardinal. A program developed over the course of the last century to measure the continuum in terms of various related cardinals. By Godel 1939 and Cohen 1963, Hilbert's first problem was proved independent of ZFC. However, several basic questions about these cardinal invariants of the continuum remained open despite much investigation. The oldest and probably the most famous of these is whether "p = t", proved in a special case by Rothberger 1948, building on Hausdorff 1934. Very recently, Malliaris and Shelah resolved this question, in a surprising way, using model-theoretic tools developed for the study of Keisler's order.
AWM Colloquium, April 2012
The colloquium is jointly sponsored by the Department of Mathematics and the Association for Women in Mathematics.
Date: Friday, April 6, 2012
Time: 3 - 4 PM
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Melanie Wood (University of Michigan)
Title: Counting polynomials and motivic stabilization
Abstract: We will begin with the problem of counting polynomials modulo a prime p with a given pattern of root multiplicity. Here we will discover phenomena that point to vastly more general patterns in configuration spaces of points. To see these patterns, one has to work in the ring of motives — so we will describe this place where a space is equivalent to the sum of its pieces. We will then be able to describe how these patterns in the ring of motives are related to theorems in topology on the homological stability of configuration spaces. This talk is based on joint work with Ravi Vakil.
November 2011
Date: Friday, November 11, 2011
Time: 4 PM onward
Venue: Eckhart 206 (Room capacity: 72)
Speaker: Viatcheslav Kharlamov (IRMA, Strasbourg)
Title: First steps in real enumerative geometry.
Abstract: Surprisingly, in a quite a number of real enumerative problems the number of real solutions happens to be comparable (for example, in logarithmic scale) with the number of complex ones. For the moment, such a phenomenon is studied quite in depth in the case of interpolation of real points on a real rational surface by real rational curves. The key tool here are the Welschinger invariants (which can be seen as a real analogue of genus zero Gromov-Witten invariants). In this talk, based on joint works with I. Itenberg and E. Shustin, I will remind how Welschinger invariants look like in the case of surfaces, point certain modifications, and present some recursive formulas that allow to control Welschinger invariants in the case of Del Pezzo surfaces and by these means to establish some basic properties of the invariants that imply, in particular, the abundance of real solutions.