# Math Calendar

### Upcoming Events

## Talk Title TBA

Craig Kleski, Miami University in Ohio

Location: Stevenson 1432

## Introduction to the Discharging Method, Part II

Ryan Solava, Vanderbilt University

Location: Stevenson 1432

The discharging method is a tool with a number of applications, the most natural ones being coloring problems on sparse graphs (especially planar ones). Most famously, discharging was used by Appel and Haken to prove the Four Color Theorem. In this talk we will look at some simpler proofs in detail, with an eye toward learning how to apply the method ourselves.

## Special Colloquium: Stochastic Homogenization for Reaction-Diffusion Equations

Jessica Lin, University of Wisconsin-Madison

Location: Stevenson 5211

We study heterogeneous reaction-diffusion equations in stationary ergodic random media with both ignition and KPP-type nonlinearities. Under suitable hypotheses on the environment, we prove the existence of deterministic asymptotic speeds of propagation for solutions with both compactly supported and front-like initial data. We subsequently obtain a general stochastic homogenization result which shows that, in the large-scale-large-time limit, the behavior of typical solutions in such environments is governed by a simple deterministic Hamilton-Jacobi equation modeling front propagation. This talk is based on joint work with Andrej Zlatos. Tea at 3:30 pm in Stevenson 1425. (Contact Person: Ed Saff)

## Geometry of Surfaces

Krishnendu Khan, Vanderbilt University

Location: Stevenson 1206

What is a surface? How can we distinguish different surfaces from each other? What kind of formal structures do we need to do that?

## Well Partial Orderings, With Applications to Algebra

Stephen G. Simpson, Vanderbilt University

Location: Stevenson 1310

A partial ordering consists of a set P and a binary relation < on P which is transitive (x < y < z implies x < z) and irreflexive (x is never < x). Within P, a descending chain is a sequence a > b > c > …, and an antichain is a set of elements a, b, c, … which are pairwise incomparable (neither a < b nor b < a). A well partial ordering is a partial ordering which has no infinite descending chain and no infinite antichain. To each well partial ordering P one can associate an ordinal number o(P). For example, the natural numbers N with their usual ordering form a well ordering of order type omega and hence a well partial ordering with o(N) = omega. The class of well partial orderings is closed under finite sums, finite products, and certain other finitary operations. As noted in a 1972 paper by J. B. Kruskal, well partial ordering theory is a “frequently discovered concept” with many applications, especially in abstract algebra (G. Higman, I. Kaplansky, …). As a simple example, Dickson’s Lemma says that for each positive integer k the finite product N^k is a well partial ordering, and this is the key to a proof of the Hilbert Basis Theorem: for any field K and positive integer k, the polynomial ring K[x_1,...,x_k] has no infinite ascending chain of ideals. The ordinal number involved here is omega^omega. There are also generalizations involving larger ordinal numbers such as omega^{omega^omega}. There is a subclass of the well partial orderings, the better partial orderings, which has stronger closure properties. For example, if P is a better partial ordering, then the downwardly closed subsets of P form a better partial ordering under the subset relation. This fact from better partial ordering theory can be used to prove that for any field K, the group ring K[S] of the infinite symmetric group S (the direct limit of the finite symmetric groups S_n as n goes to infinity) has no infinite ascending chain of two-sided ideals and no infinite antichain of two-sided ideals. There seems to be an open question as to how far this theorem can be generalized from S to other locally finite groups. R. Laver has used better partial ordering theory to prove that the countable linear orderings form a well partial ordering (or rather, a well quasi-ordering) under the embeddability relation. N. Robertson and P. Seymour have proved a difficult theorem: the finite graphs form a well quasi-ordering under the minor embeddability relation. I. Kriz has proved that the Friedman trees are well quasi-ordered under the gap embeddability relation.

## Transversals to Horocycle Flow on the Moduli Space of Translation Surfaces

Grace Work, Vanderbilt University

Location: Stevenson 1310

Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group had multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in $\mathcal{H}(2)$.

## Talk Title TBA

Hung-Chang Liao, Penn State University

Location: Stevenson 1432

## The Trouble with Voting

Zach Gaslowitz, Vanderbilt University

Location: Stevenson 1206

We will explore some of the surprising mathematical challenges one runs into when trying to turn a pile of ballots into a single winner. How do we decide who should win, and how does this question influence our democracy as a whole?

## Metric Approximations of Wreath Products

Ben Hayes, Vanderbilt University

Location: Stevenson 1310

I will discuss joint work with Andrew Sale. In it, we investigate metric approximations of wreath products. A metric approximation of a group is a family of asymptotic homomorphisms into a class of groups so that the image of any nonidentity element is bounded away from zero. Metric approximations have received much recent interest and are related to several interesting conjectures, including Kaplansky’s direct finiteness, Gottschalk’s surjenctivity conjecture and the Connes embedding problem. Our results say the following: suppose that H is a sofic group. Then G wreath H is sofic (resp. linear sofic, resp. hyperlinear) if G is sofic (resp. linear sofic, resp. hyperlinear). No knowledge of sofic, linear sofic, or hyperlinear groups will be assumed.

## Periodicity in Homotopy Theory

Haynes Miller, MIT

Location: Stevenson 5211

In 1950 Jean-Pierre Serre calculated the rational homotopy groups of spheres. This work proved to be the vanguard of an extensive campaign of understanding homotopy theory through its localizations, resulting in the “chromatic” perspective on homotopy theory. I’ll review some of that history, and describe some recent work on analogous calculations in “motivic” homotopy theory. Tea at 3:30 pm in SC 1425. (Contact Person: Anna Marie Bohmann)

## Talk Title TBA

Elaine Cozzi, Oregon State University

Location: Stevenson 1307

## Talk Title TBA

Daniel Drimbe, UC San Diego

Location: Stevenson 1432

## On Weyl’s Spectral Decomposition Theorem

Nigel Higson, Penn State

Location: Stevenson 5211

Early in his career, Hermann Weyl examined and solved the problem of decomposing a function on a half-line as a continuous combination of the eigenfunctions of a Sturm-Liouville operator with asymptotically constant coefficients. Weyl’s theorem served as inspiration for Harish-Chandra in his pursuit of the Plancherel formula for semisimple groups, and for this and other reasons it continues to be of interest. I’ll try to explain how Weyl’s theorem arises again in efforts to view Harish-Chandra’s work from the perspective of noncommutative geometry, and I’ll describe a new, geometric, proof of Weyl’s theorem that seems to fit better with representation theory. This is joint work with Tyrone Crisp and Qijun Tan. Tea at 3:30 pm in SC 1425. (Contact Person: Rudy Rodsphon)

## New Examples of Gradient Expanding Kahler-Ricci Solitons

Ronan J. Conlon, Florida International University

Location: Stevenson 1310

A complete Kahler metric g on a Kahler manifold M is a gradient expanding Kahler-Ricci soliton if there exists a smooth real-valued function f:M->R with ∇^{g}f holomorphic such that Ric(g)-Hess(f)+g=0. I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint work with Alix Deruelle (Universite Paris-Sud).

## Talk Title TBA

Jing Tao, University of Oklahoma

Location: Stevenson 1310

## Talk Title TBA

Jing Tao, University of Oklahoma

Location: Stevenson 5211

Tea at 3:30 pm in SC 1425. (Contact Person: Ed Saff)

## Talk Title TBA

Danny Calegari, University of Chicago

Location: Stevenson 5211

Tea at 3:30 pm in SC 1425. (Contact Person: Mark Sapir)

## Talk Title TBA

Dan Margalit, Georgia Tech

Location: Stevenson 5211

Tea at 3:30 pm in SC 1425. (Contact Person: Spencer Dowdall)

## Talk Title TBA

Dimitri Bilyk, University of Minnesota

Location: Stevenson 5211

Tea at 3:30 pm in SC 1425. (Contact Person: Ed Saff)

## Talk Title TBA

Kate Juschenko, Northwestern University

Location: Stevenson 5211

Tea at 3:30 pm in SC 1425. (Contact Person: Denis Osin)

## Talk Title TBA

Miklos Maroti, University of Szeged

Location: Stevenson 5211

Tea at 3:30 pm in SC 1425. (Contact Person: Ralph McKenzie)

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