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for events the day of Tuesday, April 9, 2019.

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Tuesday, April 9, 2019

11:00 am in 345 Altgeld Hall,Tuesday, April 9, 2019

Classifying spectra of finite groups and chromatic homotopy theory

Nathan Stapleton (U Kentucky math)

Abstract: We will discuss a question about the functoriality of certain evaluation maps for classifying spectra of finite groups that arose when thinking about questions related to chromatic homotopy theory. I will describe a solution to this problem found in joint work with Reeh, Schlank.

12:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

Geometry Group Theory in Music AI

Haizi Yu (University of Illinois)

Abstract: Is it possible to invent an AI to learn important music concepts directly from sheet music? Is it possible to do this in a human-interpretable form that resembles known music theory and also suggests new theory? We apply our generally developed automatic concept learning model to the domain of music, so as to tackle the above questions and the like. Starting from a connection between existing music concepts and their group-theoretic interpretations, we propose a formal representation of music objects as well as their abstractions and probabilistic patterns. This proposed representation not only reveals internal music structures as mathematical symmetries, but more importantly, are also operational in computational models. As a result, this further yields a learning algorithm that couples knowledge from geometric group theory and statistical inference to automatically discover music concepts without human intervention. Lastly, we briefly demonstrate an ongoing project, called MUS-ROVER, which builds a real web application that delivers to people automatically discovered music rules and concepts, teaching us music composition in a designated style.

1:00 pm in 345 Altgeld Hall,Tuesday, April 9, 2019

Multiplication of weak equivalence classes

Anton Bernshteyn (Carnegie Mellon)

Abstract: The relations of weak containment and weak equivalence were introduced by Kechris in order to provide a convenient framework for describing global properties of p.m.p. actions of countable groups. Weak equivalence is a rather coarse relation, which makes it relatively well-behaved; in particular, the set of all weak equivalence classes of p.m.p. actions of a given countable group $\Gamma$ carries a natural compact metrizable topology. Nevertheless, a lot of useful information about an action (such as its cost, type, etc.) can be recovered from its weak equivalence class. In addition to the topology, the space of weak equivalence classes is equipped with a multiplication operation, induced by taking products of actions, and it is natural to wonder whether this multiplication operation is continuous. The answer is positive for amenable groups, as was shown by Burton, Kechris, and Tamuz. In this talk, we will explore what happens in the nonamenable case. Number theory will make an appearance.

1:00 pm in 347 Altgeld Hall,Tuesday, April 9, 2019

Convexity of Whitham's wave of extreme form

Bruno Vergara (ICMAT, Spain)

Abstract: In this talk I will discuss a conjecture of Ehrnström and Wahlén concerning travelling wave solutions of greatest height to Whitham's non-local model of water waves. We will see that there exists a cusped periodic solution whose profile is convex between consecutive peaks of $C^{1/2}$-regularity. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.

2:00 pm in 345 Altgeld Hall,Tuesday, April 9, 2019

Quantitative inequalities for the expected lifetime of the Brownian motion

Daesung Kim (Purdue University)

Abstract: The isoperimetric-type inequality for the expected lifetime of the Brownian motion state that the $L^p$ norm of the expected lifetime in a region is maximized when the region is a ball with the same volume. In particular, if $p=1$, it is called the Saint-Venant inequality and has a close relation to the classical Faber—Krahn inequality for the first eigenvalue. In this talk, we prove a quantitative improvement of the inequalities, which explains how a region is close to being a ball when equality almost holds in these inequalities. We also discuss some related open problems.

2:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

Equitable colorings of infinite graphs

Anton Bernshteyn (Carnegie Mellon Math)

Abstract: A proper $k$-coloring of a finite graph $G$ is called equitable if every two color classes differ in size at most by one. In particular, if $G$ has $n$ vertices and $k$ divides $n$, then in an equitable $k$-coloring of $G$ every color class has size exactly $n/k$. There is a natural way to extend this definition to infinite graphs on probability spaces. Namely, if $G$ is a graph whose vertex set $V(G)$ is a probability space, then a proper $k$-coloring of $G$ is equitable when every color class has measure $1/k$. In this talk I will discuss extensions of some classical results about equitable colorings to this setting, including an infinite version of the Hajnal-Szemerédi theorem on equitable $k$-colorings for $k \geq \Delta(G) + 1$, and an analog of the Kostochka-Nakprasit theorem on equitable $\Delta$-colorings of graphs with small average degree. This is joint work with Clinton Conley.

3:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

Quantization of algebraic exact Lagrangians in cotangent bundles

Christopher Dodd (UIUC Math)

Abstract: Exact Lagrangians play an important role in symplectic topology; in algebraic geometry they seem to be almost unstudied. In this talk I’ll explain some recent results about their structure and in particular I’ll show that, in the affine case, they admit certain canonical noncommutative deformations. Time permitting I’ll explain how this implies the vanishing of certain invariants in their de Rham cohomology.

4:00 pm in 314 Altgeld Hall,Tuesday, April 9, 2019

Recent progress on existence of minimal surfaces

André Neves (University of Chicago)

Abstract: The Tondeur Memorial Lectures will be given by Andre Neves (University of Chicago), April 9-11, 2019. Following this lecture, a reception will be held in 239 Altgeld Hall.

A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.