# Department of Mathematics

## Forthcoming colloquia and special talks (in chronological order)

### January 2015

#### Department Colloquium

Date: Wednesday, January 7, 2015

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: Henry Cohn (Microsoft Research, New England)

Title: Sums of squares, correlation functions, and exceptional geometric structures

Abstract: Some exceptional structures such as the icosahedron or E_8 root system have remarkable optimality properties in settings such as packing, energy minimization, or coding. How can we understand and prove their optimality? In this talk, I'll interweave this story with two other developments in recent mathematics (without assuming familiarity with either): how semidefinite optimization and sums of squares have expanded the scope of optimization, and how representation theory has shed light on higher correlation functions for particle systems.

## Past colloquia and special talks (in reverse chronological order)

### November 2014

#### Department Colloquium

Date: Wednesday, November 19, 2014

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: Miguel Walsh (University of Oxford)

Title: The algebraicity of sieved sets and rational points on curves

Abstract: We will discuss some connections between the polynomial method, sieve theory, inverse problems in arithmetic combinatorics and the estimation of rational points on curves. Our motivating questions will be to classify those sets that are irregularly distributed in residue classes and to understand how many rational points of bounded height can a curve of fixed degree have.

#### Special Colloquium

Date: Thursday, November 6, 2014 (UNUSUAL DAY)

Time: 1:30pm--2:30pm (UNUSUAL TIME)

Venue: Ryerson 352 (Barn)

Speaker: Charles Smart (Cornell)

Title: The scaling limit of the Abelian sandpile on infinite periodic graphs

Abstract: Joint work with Lionel Levine and Wesley Pegden. The Abelian sandpile is a deterministic diffusion process on graphs which, at least on periodic planar graphs, generates striking fractal configurations. The scaling limit on the two dimensional integer lattice can be described in terms of an Apollonian circle packing. General periodic graphs share some of this structure, but appear to have more complicated scaling limits in general. I will discuss both the square lattice and general cases.

### Octobor 2014

#### AWM Colloquium

Date: Friday, Octobor 3, 2014 (UNUSUAL DAY)

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: Sylvie Méléard (Ecole Polytechnique)

Title: Stochastic Modeling for Darwinian Evolution

Abstract: We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. We consider a multi-resources chemostat model, where the competition between bacteria results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states. An essential technical ingredient is the study of the long time behavior of a multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we give a characterization of the evolutionary branching points.

### May 2014

#### AWM Colloquium

Date: Wednesday, May 21, 2014

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: Lauren Williams (Berkeley)

Title: The positive Grassmannian

Abstract: The positive Grassmannian is a remarkable subset of the real Grassmannian which has recently arisen in diverse contexts such as integrable systems, scattering amplitudes, and free probability. I will provide an elementary introduction to the positive Grassmannian, focusing on a few intriguing aspects: what it looks like," and its connection to shallow water waves (via the KP hierarchy).

#### Department Colloquium

Date: Wednesday, May 7, 2014

Time: 2:00pm--3:00pm

Venue: Eckhart 206

Speaker: Alice Guionnet (MIT)

Title: Free probability and random matrices; from isomorphisms to universality

Abstract:

### March 2014

#### Department Colloquium

Date: Wednesday, March 19, 2014

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: William P. Minicozzi II (Johns Hopkins University)

Title: Uniqueness of blowups and Lojasiewicz inequalities

Abstract: Mean curvature flow starting from any smooth closed hypersurface is smooth for short time, but always becomes singular as the nonlinearities dominate. Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, by Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow (MCF), by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a MCF looked at under a microscope will resemble one blowup, but under higher magnification, it might microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is perhaps the most fundamental question about singularities. We will discuss the proof of this long standing open question for MCF at all generic singularities and for mean convex MCF at all singularities. This is joint work with Toby Colding.

#### Department Colloquium (Joint with CS Theory Seminar)

Date: Friday, March 7, 2014

Time: 1:30pm--2:30pm

Venue: Eckhart 133

Speaker: Assaf Naor (Courant Institute)

Title: Vertical versus horizontal Poincare inequalities

Abstract: A classical result of Pansu and Semmes asserts that the Heisenberg group does not admit a bi-Lipschitz embedding into any Euclidean space. Several alternative proofs of this fact have been subsequently found, yielding the non-embeddability of the Heisenberg group into a variety of spaces, including uniformly convex spaces and L_1. These proofs rely on metric differentiation methods, i.e., the use of a limiting procedure to show that it suffices to rule out certain more structured embeddings. For certain applications, which will be explained in this talk, it is important to get quantitative bounds, in which case turning the metric differentiation arguments into quantitative statements is quite challenging and yields sub-optimal bounds. This talk, which assumes no prerequisites, will start by explaining the approaches to the Heisenberg embeddability question based on metric differentiation, and then present a new and different approach based on Littlewood-Paley theory that yields asymptotically sharp distortion bounds for embeddings of balls in the Heisenberg group into uniformly convex spaces. We will end with a conjectural isoperimetric inequality that is motivated by the new Littlewood-Paley approach, and explain its implications to approximation algorithms.

#### Department Colloquium

Date: Wednesday, March 5, 2014

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: Bhargav Bhatt (Columbia)

Title: Hodge theory in the p-adic world

Abstract: Hodge theory, which is the study of "hidden" structures on the cohomology groups of complex manifolds, is a cornerstone of complex geometry. My talk will discuss the analogous theory in p-adic algebraic geometry. More precisely, after reviewing some relevant aspects of the complex story, I will explain the p-adic analog of de Rham's theorem (identifying de Rham and singular cohomology for manifolds), emphasizing the crucial role played by the arithmetic of p-adic fields. The presentation will be based on recent work of Beilinson and myself.

### February 2014

#### Department Colloquium/Joint Seminar

Date: Wednesday, Friday, February 26, 28, 2014

Time: 2:30pm--4:00pm

Venue: Eckhart 206

Speaker: Frank Calegari (Northwestern)

Title: The stable homology of congruence subgroups

Abstract: Let $F$ be a number field. A stability result due to Charney and Maazen says that the homology groups $H_d(\mathrm{SL}_N(\mathcal{O} _F),\mathbf{Z})$ (for $d$ fixed) are independent of $N$ for $N$ sufficiently large. The resulting stable cohomology groups are intimately related to the algebraic $K$-theory of $\mathcal{O}_F$. In these talks, we shall explore the homology of the $p$-power congruence subgroups of $\mathrm{SL}_N(\mathcal{O}_F)$ in fixed degree $d$ as $N$ becomes large. We show that the resulting homology groups consist of two parts: an unstable'' part which depends only on local behavior concerning how the prime $p$ splits in $F$, and a stable'' part which contains global information concerning $p$-adic regulator maps. Our argument consists of two parts. The first part (which is joint work with Matthew Emerton) explains how to modify the homology of congruence subgroups in a suitable way (using completed homology) to obtain groups which are literally stable for large $N$. We prove this by combining recent results in representation stability with the representation theory of $p$-adic groups. The second part consists of relating these stable homology groups to $K$-theory using ideas from algebraic topology. We give some applications of our results. The first concerns the computation of $H_2$ of the $p$-congruence subgroup of $\SL_N(\Z)$ for sufficiently large $N$. The second concerns describing the extent to which the stable classes in characteristic zero constructed by Borel become more and more divisible by $p$ when one passes to higher and higher $p$-power level.

#### Department Colloquium

Date: Wednesday, February 12, 2014

Time: 3:00pm--4:00pm

Venue: Eckhart 206

Speaker: Theodore A. Slaman (Berkeley)

Title: On Normal Numbers

Abstract: A real number is simply normal in base b if in its base-b expansion each digit appears with asymptotic frequency 1/b. It is normal in base b if it is simply normal in all powers of b, and absolutely normal if it is simply normal in every integer base. By a theorem of E. Borel, almost every real number is absolutely normal. We will present three main results. We will give an efficient algorithm, which runs in nearly quadratic time, to compute the binary expansion of an absolutely normal number. We will demonstrate the full logical independence between normality in one base and another. We will give a necessary and sufficient condition on a set of natural numbers M for there to exist a real number X such that X is simply normal to base b if and only if b is an element of M. If time allows, we will discuss normality for Liouville numbers.

### November 2013

#### Department Colloquium

Date: Wednesday, November 20, 2013

Time: 3 - 4 PM

Venue: Eckhart 206

Speaker: Nick Nozenblyum (Northwestern)

Title: Derived Geometry and Quantum Invariants

Abstract: Over the last several decades, quantum field theory has been a source of many new ideas in mathematics and particularly in geometry. The Feynman path integral has been used to define many interesting invariants of manifolds and algebraic varieties. For instance, the so called B-model suggests that there should be natural operations on the Hodge cohomology of a projective Calabi-Yau manifold parametrized by Riemann surfaces with boundary. I will describe how derived algebraic geometry, which is roughly the study of solutions of equations up to homotopy, provides a natural setting for studying these types of invariants. From this point of view, the Feynman integral can be interpreted as an integral over a derived space. I will describe how the B-model, as well as a plethora of other interesting invariants such as the Todd genus and the Witten genus, can be constructed using these ideas.

#### Special Representation Theory Seminar

Date: Tuesday, November 12, 2013

Time: 3 - 4 PM

Venue: Eckhart 308

Speaker: Steven Sam (University of California, Berkeley)

Title: Twisted commutative algebras and representation stability

Abstract: Twisted commutative algebras are commutative rings with an action of the infinite general linear group and are connected with recent studies of "representation stability". The simplest nontrivial example of a twisted commutative algebra is the infinite polynomial ring C[x_1, x_2, ...] with the natural action of GL(infinity). Its module category is equivalent to the category of (complex) FI-modules, which was recently introduced by Church-Ellenberg-Farb to study cohomology of configuration spaces and many other examples. In this talk I will explain joint work with Andrew Snowden on the structural properties of this category and the connection to FI-modules. Time permitting, I will explain what we know about more complicated twisted commutative algebras and their connections with representation stability.

This colloquium is jointly sponsored by the Association for Women in Mathematics:

Date: Tuesday, November 5, 2013

Time: 4:30PM -- 5:30 PM

Venue: Eckhart 206

Speaker: Wei Ho (Columbia University)

Title: Arithmetic invariant theory and applications

Abstract: The origins of "arithmetic invariant theory" come from the work of Gauss, who used integer binary quadratic forms to study ideal class groups of quadratic fields. The underlying philosophy---parametrizing arithmetic and geometric objects by orbits of group representations---has now been used to study higher degree number fields, curves, and higher-dimensional varieties. We will discuss some of these constructions and highlight the applications to topics such as bounding ranks of elliptic curves and dynamics on K3 surfaces.

This talk is intended for a general mathematical audience.

### May 2013

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).

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.

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 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.