Have you ever looked at two matrices and thought to yourself ``Man, I wonder if there's a polynomial of the first matrix equal to a polynomial of the second matrix?'' If yes, then boy is this the perfect talk for your highly specific interests. For everyone else, I hope to convince you that asking such a question can be a surprisingly interesting and fun process.
Specifically, we're going to look at this question when our two matrices come from some graph G. When our matrices satisfy a certain relation, we'll be able to use this relation to translate from eigenvalues of one matrix to eigenvalues of the other, and using spectral graph theory we'll be able to conclude various properties about our original graph from this.

We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We will discuss a construction that leads to a family of II$_1$ factors whose Cartan subalgebras, up to unitary conjugacy, are not classifiable by countable structures. We do this via establishing a strong dichotomy, depending if the action is strongly ergodic or not, on the complexity of the space of homomorphisms from a given equivalence relation to $E_0$. We will start with some of the necessary preliminaries, and then outline the proofs of the aforementioned results.

A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\rm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q(\!(t)\!)\) is given by the so-called the characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel's lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?

In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving an answer to Lubotzky's question in this case.

The most fundamental objects in number theory are number fields, field extensions of the rational numbers that are finite dimensional as vector spaces over Q. Their arithmetic is governed heavily by certain invariants such as the discriminant, Artin conductors, and the class group; for example, the ring of integers inside a number field has unique prime factorization if and only if its class group is trivial. The behavior of these invariants is truly mysterious: it is not known how many number fields there are having a given discriminant or conductor, and it is an open conjecture dating back to Gauss as to how many quadratic fields have trivial class group.

Nonetheless, one may hope for statistical information regarding these invariants of number fields, the most basic such question being “How are such invariants distributed amongst number fields of degree d?” To obtain more refined asymptotics, one may fix the Galois structure of the number fields in question. There are many foundational conjectures that predict the statistical behavior of these invariants in such families; however, only a handful of unconditional results are known. In this talk, I will describe a combination of algebraic, analytic, and geometric methods to prove many new instances of these conjectures, including some joint results with Altug, Bhargava, Ho, Shankar, and Wilson.

A major open problem in the periodic homogenization theory of Hamilton-Jacobi equations is to understand ``deep'' properties of the effective equation, in particular, how the effective Hamiltonian depends on the original Hamiltonian. In this talk, I will present some recent progress in both the convex and non-convex settings.

The enumerative geometry of curves on K3 surfaces is governed by modular forms. I will discuss a parallel connection between the enumerative geometry of hyper-Kaehler varieties and Jacobi forms. The case of genus 1 curves is particularly interested and leads to the Igusa cusp form conjecture. In the last part I will explain recent work with Junliang Shen and Aaron Pixton which yields a proof of this conjecture.

Physics predicts that the Gromov-Witten theory of Calabi-Yau threefolds satisfies two fundamental properties: Finite generation and a holomorphic anomaly equation. I will explain a recent conjecture with Pixton that extends these conjectures to all elliptic fibrations, and indicate how to prove it in several basic cases. If time permits, we will also discuss holomorphic anomaly equations for hyper-Kaehler varieties.

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

In an attempt to generalize beyond solvable methods of analysis for last passage percolation, recently Eric Cator (Radboud University, Nijmegen) and I have started analyzing the piecewise linearity of the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model, our work in progress to use it to produce probabilistic information, and some connections to algebraic geometry.

I will discuss some recent ongoing work with Paul Skoufranis to create a non-microstates bi-free entropy. I will propose a definition of bi-free conjugate variables and bi-free Fisher information, which admit desirable properties such as additivity in the presence of bi-free independence and versions of Cramer-Rao and Stam inequalities. I will also discuss the analogue of the free difference quotient, and some of the quirks present in the bi-free setting.

Resolvent degree is an invariant of a branched cover which quantifies how ``hard'' is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover $\mathbb{P}^n/S_{n-1}\to \mathbb{P}^n/S_n$, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians.

The cell membrane is the first interface that separates the inside of a cell from its surrounding medium. It serves not only as a protective mechanical barrier, but also as a platform for cells to exchange material with their environment. In this talk we will illustrate both of these essential membrane functions through two examples where the biophysics of lipid bilayers determine the response of the cell membrane.
First, we will examine the out-of-equilibrium response of cell-sized lipid vesicles exposed to solute imbalance. Based on experimental observations that giant vesicles in hypotonic condition exhibit a non-intuitive pulsatile behavior characterized by swell-burst cycles, we will present a theoretical description of the system in the form of coupled stochastic differential equations. We will show how thermal fluctuations enable stochastic pore nucleation, leading to a dependence of the lytic strain on the load rates, and unravel scaling relationships between the pulsatile dynamics and the vesicles properties. We will then demonstrate how vesicles encapsulating polymer solutions - mimicking the crowded cytoplasm of a cell - undergo swell-burst cycles even in the absence of a concentration imbalance.

Then, we will investigate how membrane necks, a necessary step to produce trafficking membrane vesicles, are generated by curvature-inducing proteins. Based on an augmented Helfrich model for lipid bilayers to include membrane-protein interaction, we will show how the spontaneous curvature field induced by proteins can be computed based on the knowledge of the neck geometry. We will apply this methodology to catenoid-shaped necks, for which the shape equation reduces to a variable coefficient Helmholtz equation for spontaneous curvature, where the source term is proportional to the Gaussian curvature. We will finally present numerical results showing how boundary conditions and geometric asymmetries determine an energetic landscape constraining the geometry of catenoid-shaped membrane necks.

Donaldson and Chen introduced the J-functional in '99, and explained its importance in the existence problem for constant scalar curvature metrics on compact Kahler manifolds. An important open problem
is to find algebro-geometric conditions under which the J-functional has a critical point. The critical points of the J-functional are described by a fully-nonlinear PDE called the J-equation. I will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the J-equation.

Representation stability is an exciting new area that combines
ideas from commutative algebra and representation theory. The meta-idea
is to combine a sequence of objects together using some newly defined
algebraic structure, and then to translate abstract properties about
this structure to concrete properties about the original object of
study. Finite generation is a particularly important property, which
translates to the existence of bounds on algebraic invariants, or some
predictable behavior. I'll discuss some examples coming from topology
(configuration spaces) and algebraic geometry (secant varieties).