# Colloquium

The Colloquium is held on Mondays at 4pm on Zoom with meeting number 938 9042 5892.

### Spring Quarter, 2022

- April 18,
**Pengfei Guan** (McGill University)

**Location:** 208 University Hall

Curvature flows and Isoperimetric type inequalities
**Abstract**: The classical isoperimetric inequality states that: area of a bounded plane domain is controlled by square of its perimeter in optimal way; the equality is achieved only by round balls. There are many different proofs of this classical inequality, a proof by Gage-Hamilton using curve shorting flow is an interesting one. This inequality (its higher dimensional version) is equivalent to the Sobolev inequality which is fundamental in analysis. The inequality can be viewed a Lagrange multiplier problem in calculus variations: finding “minimum” of surface area functional under the constraint the volume functional is fixed. We seek a “good path” descending to the optimal solution. It leads to a mean curvature type flow. The key is the variational property of surface area functional. This idea can be adapted to solve isoperimetric type problems for geometric functionals. We will discuss why this type of problems naturally lead to some new curvature flows, and how to use them to establish new geometric inequalities. Through examples, we will see deep connection of nonlinear PDE and geometry.

- April 25

Film screening: Julia Robinson and Hilbert’s Tenth Problem
- May 2,
**Gregory G. Smith** (Queens University)

Sums of squares: a real projective story
**Abstract**: A multivariate real polynomial is nonnegative if its value at every real point is greater than or equal to zero. These special polynomials play a central role in many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems. However, it is very difficult, in general, to decide whether a given polynomial is nonnegative. In this talk, we will review some classic methods for certifying that a polynomial is nonnegative. We will then present novel certificates in some important cases. This talk is based on joint work with Grigoriy Blekherman, Rainer Sinn, and Mauricio Velasco.

- May 9,
**Julianna Tymoczko** (Smith College)

Combinatorial and algebraic subvarieties of the flag variety
**Abstract**: Flag varieties are manifolds at the nexus of combinatorics, algebra, and geometry. Each flag represents an ordered basis, equivalently a nested sequence of linear subspaces, or equivalently a coset of the group of invertible matrices.

In this talk, we describe a family of subvarieties of the flag variety called Hessenberg varieties. Hessenberg varieties are defined using two parameters, and varying these parameters independently or together gives a filtration on the flag variety with rich structure. We describe some of the important connections between Hessenberg varieties and representation theory, combinatorics, and knot theory, as well as open questions.

- May 23,
**Rafe Mazzeo** (Stanford)

**Location:** 208 University Hall

Noncompactness phenomena in low-dimensional gauge theories, and the relationship with special submanifolds
**Abstract**: A thread of research over the past few decades investigates noncompactness of moduli spaces of solutions for certain gauge-theoretic equations in dimensions 2, 3, 4 which are due to the noncompactness of the underlying group. This starts from some classical facts about Riemann surfaces, and generalizes to Taubes’ theory of Z_2 harmonic spinors and Donaldson’s multivalued harmonic functions. I will survey a number of recent results by various researchers, concluding with some new work by S. He on degenerations of calibrated submanifolds. This will be presented with focus on general geometric and analytic ideas, and meant to be broadly accessible.

### Winter Quarter, 2022

- January 28,
**Geordie Williamson** (U. Sydney)

Towards the combinatorial invariance conjecture
**Abstract**: The combinatorial invariance conjecture is a fascinating conjecture concerning Kazhdan-Lusztig polynomials. It was first formulated independently by Lusztig and Dyer in the 1980s. It says that Kazhdan-Lusztig polynomials (certain polynomials of central importance in geometric representation theory, are determined by a Bruhat graph). Some rather special cases are known, but the general case remains widely open. I will explain a new conjecture which sheds some light on this conjecture. Interestingly, we came across this conjecture whilst trying to understand certain machine learning models which had been trained to predict Kazhdan-Lusztig polynomials from the Bruhat graph.

- January 31,
**Sangwon (Justin) Hyun** (USC)

**Location** : 208 University Hall

Sparse Multivariate Mixture of Experts for Learning Environmental Drivers of Marine Microbial Systems
- February 7,
**Yuexia Zhang** (University of Toronto)

**Location** : 208 University Hall

Inverse Probability Weighting-based Mediation Analysis for Microbiome Data
**Abstract**: Mediation analysis is an important tool to study casual associations in biomedical and other scientific areas and has recently gained attention in microbiome studies. With a microbiome study of acute myeloid leukemia (AML) patients, we investigate whether the effect of induction chemotherapy intensity levels on the infection status is mediated by the microbial taxa abundance. The unique characteristics of the microbial mediators—high-dimensionality, zero-inflation, and dependence—call for new methodological developments in mediation analysis. The presence of an exposure-induced mediator-outcome confounder, antibiotics usage, further requires a delicate treatment in the analysis. To address these unique challenges brought by our motivating microbiome study, we propose a novel nonparametric identification formula for the interventional indirect effect (IIE), a measure recently developed for studying mediation effects. We develop the corresponding estimation algorithm and test the presence of mediation effects via constructing the nonparametric bias-corrected and accelerated bootstrap confidence intervals. Simulation studies show that the proposed method has good finite-sample performance in terms of the IIE estimation, and type-I error rate and power of the corresponding test. In the AML microbiome study, our findings suggest that the effect of induction chemotherapy intensity levels on infection is mainly mediated by patients’ gut microbiome.

- February 14,
**Kathryn Mann** (Cornell University)

Groups acting at infinity
**Abstract**: In the 1980s, Gromov introduced the notion of hyperbolic groups – a way to naturally think of some discrete groups as coarsely negatively curved metric spaces. Like more familiar hyperbolic spaces (imagine the Poincaré disc), these groups can be compactified by a boundary at infinity, and the group acts on its boundary by homeomorphisms. My talk will introduce you to the rich dynamics of these actions at infinity. I’ll discuss some of the history of the subject and explain a recent result, joint with Jason Manning, that proves a new dynamical property – groups acting on their boundaries exhibit remarkable stability under perturbation.

### Fall Quarter, 2021

- October 11,
**Daniil Rudenko** (U. Chicago)

Goncharov depth conjecture and volumes of orthoschemes
**Abstract**: Goncharov conjectured that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. In the first part of the talk I will explain how this conjecture fits into the general scheme of conjectures about polylogarithms. In the second part of the talk I will sketch the proof of the Goncharov conjecture. The proof is based on an explicit formula, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Surprisingly, almost the same formula gives a volume of a hyperbolic orthoscheme generalising the formula of Lobachevsky in dimension 3 to an arbitrary dimension.

- October 25,
**Laura Rider** (University of Georgia)

Examples of t-structures in Geometric Representation Theory
**Abstract**: In this talk, I’ll discuss the notion of `t-structure’. In the cases I’ll present, a t-structure is a way to relate two (a priori unrelated) abelian categories. When this happens, we hope to utilize better homological properties to gain traction on our problems. As time permits, I’ll spend some extra time on the examples of constructible sheaves, coherent sheaves, and Koszul duality.

- November 15,
**Ian Hambleton** (McMaster University)

Euler Characteristics and 4-manifolds
**Abstract**: The topology and total curvature of a Riemann surface is determined by a single integer, the Euler characteristic (Leonhard Euler, 1707-1783). In dimension four, the Euler characteristic gives an interesting invariant for finitely presented groups. The talk will survey some recent joint work with Alejandro Adem on this theme.

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