Calculating the coefficients of equivariant generalized cohomology theories has been a fundamental question for equivariant homotopy theory. In this talk, I will talk about some calculations when the group is nonabelian. Examples include $RO(G)$-graded Eilenberg-MacLane cohomology of a point with constant coefficient when $G$ is a dihedral group of order $2p$ or the quaternion group $Q_8$, and coefficient ring of $\Sigma_3$-equivariant complex cobordism. I will discuss techniques in such computations: isotropy separation, cellular structures and dualities. This is joint work with Po Hu and Igor Kriz.

I will present a recent result showing that the direct product group $\Gamma = \mathbb F_2 \times \mathbb F_2$ is not Hilbert-Schmidt stable. Specifically, $\Gamma$ admits a sequence of asymptotic homomorphisms (with respect to the normalized Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms. As we will explain, while this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. We will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$ (in the normalized Hilbert-Schmidt norm), but there are no matrices $A’,B’$ close to $A,B$ such that $A’$ commutes with $B’$ and $B’^*$.

We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large scale dynamics of solutions to these equations is almost surely governed by a homogeneous Hamilton-Jacobi equation. While this was previously shown in one dimension as well as for radially symmetric reactions in several dimensions, we prove this phenomenon in the general non-isotropic multidimensional setting. We also show that the rate of convergence of solutions to the Hamilton-Jacobi dynamics is at least algebraic in the relevant space-time scales when the initial data is close to an indicator function of a convex set. This talk is based on joint work with Andrej Zlato\v{s}.

For computations with many variables in optimization or solving large systems in numerical linear algebra, developing efficient methods is highly desirable. This talk introduces an approach for large-scale optimization with sparse linear equality constraints that exploits computationally efficient orthogonal projections. For approximately solving large linear systems, (randomized) sketching methods are becoming increasingly popular. By recursively augmenting a deterministic sketching matrix, we develop a method with a finite termination property that compares favorably to randomized methods. Moreover, we describe the construction of logical linear systems that can be used in e.g., COVID-19 pooling tests, and a nonlinear least-squares method that addresses large data sizes in machine learning.

We define a Fibonacci walk to be any sequence of positive integers satisfying the recurrence $w_{k+2}=w_{k+2}=w_{k+1}+w_k$, and we say that a sequence is an $n$-Fibonacci walk if $w_k=n$ for some $k$. Note that every $n$ has a number of (boring) $n$-Fibonacci walks, e.g. the sequence starting $n,n,2n,\ldots$. To make things interesting, we consider $n$-Fibonacci walks which have $w_k=n$ with $k$ as large as possible, and we call this an $n$-slow Fibonacci walk. For example, the two 6-slow Fibonacci walks start 2, 2, 4, 6 and 4, 1, 5, 6. In this talk we discuss a number of properties about $n$-slow Fibonacci walks, such as the number of slow walks a given $n$ can have, as well as how many $n$ have a given number of walks. We also discuss slow walks that follow more general recurrence relations. This is joint work with Fan Chung and Ron Graham.

Have you ever thought to yourself ``Homological Algebra is great, but I wish there were more technicalities and operations to keep track of"? Do you ever worry that the chain complex feels left out after you take cohomology? Does it keep you up at night that elements whose products are 0 get less of a say in the cohomology ring's structure? Has your topologist friend ever ignored nullhomotopic maps, making them feel excluded and unheard? In this talk, Max will be explaining two closely related constructions, in homological algebra and in homotopy, that arise only when other operations give a trivial output. These so called ``secondary operations" play a large role in computational algebraic topology, and can be an indispensable tool for proving results both highly abstract (see Tyler Lawson's ``BP is not $E_\infty$") and very concrete (see Massey's ``Higher Order Linking Numbers").

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. This is a joint work with Mark Shimozono.

Let $\Gamma$ be a countably infinite group. Seward and Tucker-Drob proved that every free Borel action of $\Gamma$ on a Polish space $X$ admits a Borel equivariant map $\pi$ to the free part of the Bernoulli shift $k^\Gamma$, for any $k \geq 2$. Our goal in this talk is to generalize this result by putting extra restrictions on the image of $\pi$. For instance, can we ensure that $\pi(x)$ is a proper coloring of the Cayley graph of $\Gamma$ for all $x \in X$? More generally, can we guarantee that the image of $\pi$ is contained in a given subshift of finite type? The main result of this talk is a positive answer to this question in a very broad (and, in some sense, optimal) setting. The main tool used in the proof of our result is a probabilistic technique for constructing continuous functions with desirable properties, namely a continuous version of the Lov\'{a}sz Local Lemma.

This talk is expository. It describes the history of an exciting connection made by physicists between an unsolved problem in combinatorial design theory- the existence of maximal sets of $d^2$ complex equiangular lines in ${\mathbb C}^d$- rephrased as a problem in quantum information theory, and topics in algebraic number theory involving class fields of real quadratic fields. Work of my former student Gene Kopp recently uncovered a surprising, deep (unproved!) connection with the Stark conjectures. For infinitely many dimensions $d$ he predicts the existence of maximal equiangular sets, constructible by a specific recipe starting from suitable Stark units, in the rank one case. Numerically computing special values at $s=0$ of suitable L-functions then permits recovering the units numerically to high precision, then reconstructing them exactly, then testing they satisfy suitable extra algebraic identities to yield a construction of the set of equiangular lines. It has been carried out for $d=5, 11, 17$ and $23$.

The Mondrian process in machine learning is a Markov partition process that recursively divides space with random axis-aligned cuts. This process is used to build random forests for regression and classification as well as Laplace kernel approximations. The construction allows for efficient online algorithms, but the restriction to axis-aligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT processes approximate a large class of stationary kernels and STIT random forests achieve minimax rates for Lipschitz functions (forests and trees) and $C^2$ functions (forests only). This work opens many new questions at the intersection of stochastic geometry and statistical learning theory. Based on joint work with Ngoc Mai Tran.

For a smooth projective surface X, natural objects of study are its moduli spaces of (semi-) stable coherent sheaves. In rank one, their structural invariants are well-understood, starting with G\"{o}ttsche's famous formula for the Betti numbers of the Hilbert schemes of points of X in terms of the Betti numbers of X itself. Even for rank two