Recently, Balmer—Gallauer deduced the tensor-triangular geometry of the so-called "derived category of permutation modules," which controls both the usual modular representation theory of a finite group as well as that of its "p-local" subgroups. Their construction of "permutation twisted cohomology" plays a key role in their deduction in the case of elementary abelian $p$-groups; here the authors deduce far stronger geometric results. In this talk, after reviewing some basics about tensor-triangular geometry and permutation modules, we'll describe how one can utilize endotrivial complexes, the invertible objects of this category, to extend Balmer—Gallauer's results for elementary abelian $p$-groups to all $p$-groups.

The impressive performance of modern machine learning methods seems to arise through different mechanisms from those of classical statistical learning theory, mathematical statistics, and optimization theory. Simple gradient methods find excellent solutions to non-convex optimization problems, and without any explicit effort to control model complexity they exhibit excellent prediction performance in practice. This talk will describe recent progress in statistical learning theory and optimization theory that demonstrates the optimization benefits of step-sizes that are too large to allow gradient methods to be viewed as an accurate time discretization of a gradient flow differential equation, that characterizes the solutions that are favored by gradient optimization methods, and that illustrates when those solutions can overfit training data but still provide good predictive accuracy.

On a modular curve of arbitrary congruence level, we introduce the notion of a "Makdisi symbol", a device that simultaneously gives a moduli-friendly coordinate system while also having an elegant Hecke theory. Concretely, these are cuspidal projections of products of two weight 1 Eisenstein series, and their study was originally pioneered by the work of Kamal Khuri-Makdisi. We show that a certain subclass of symbols, the "invertible Makdisi symbols", yield precisely the eigenforms of rank zero; combining this with the moduli interpretation, we obtain a systematic and relatively efficient algorithm to determine the rational points on a modular curve, so long as the curve has a rank zero eigenform. The algorithm is $p$-adic in nature, based on the method of Chabauty-Coleman, and does not require finding any equations for the curve.

Motivated by privacy regulations and the need to mitigate the effects of harmful data, machine unlearning seeks to modify trained models so that they effectively ``forget'' designated data. A key challenge in verifying unlearning is forging—adversarially crafting data that mimics the gradient of a target point, thereby creating the appearance of unlearning without actually removing information. To capture this phenomenon, we consider the collection of data points whose gradients approximate a target gradient within tolerance $\epsilon$ ---which we call an $\epsilon$-forging set--- and develop a framework for its analysis. For linear regression and one-layer neural networks, we show that the Lebesgue measure of this set is small. It scales on the order of $\epsilon$, and when $\epsilon$ is small enough, $\epsilon^d$. More generally, under mild regularity assumptions, we prove that the forging set measure decays as $\epsilon^{(d-r)/2}$, where $d$ is the data dimension and $r < d$ is the dimension of vector space of right singular vectors corresponding to ``small" singular values of a variation matrix defined by the model gradients. Extensions to batch SGD and almost-everywhere smooth loss functions yield the same asymptotic scaling. In addition, we establish probability bounds showing that, under non-degenerate data distributions, the likelihood of randomly sampling a forging point is vanishingly small. These results provide evidence that adversarial forging is fundamentally limited and that false unlearning claims can, in principle, be detected.

Representation stability asks whether an approximate representation of a group can be approximated by an actual representation.  There are many technical variations of this basic question: I will focus mainly on approximate representations into finite-dimensional unitary groups.

I’ll introduce the two properties in the title - the LLP of Kirchberg and property FD of Lubotzky-Shalom - via group C*-algebras and explain how they imply some fairly weak representation stability results.  I’ll then explain some refinements you can get using K-theory (without assuming any background knowledge of K-theory).  Finally, I'll discuss the known range of validity of the LLP and property FD (I’ll also mention some related properties like Kechris’ property MD).

The non K-theoretic parts are based on joint work with Francesco Fournier-Facio.

Determining rational points on modular curves is an important problem in arithmetic geometry; those curves which have Jacobian rank at least equal to the genus remain the frontier. While quadratic Chabauty can be an effective p-adic tool for computing rational points on certain modular curves where the rank of the Jacobian equals the genus, many of the underlying computations, such as computing a basis of de Rham cohomology, as well as the local height computations, become computationally prohibitive for higher genus non-split Cartan modular curves.  We will discuss joint work in progress with Steffen Mueller and Jan Vonk to carry out quadratic Chabauty on the genus 8 non split Cartan modular curve $X_{ns}^+(19)$  and what remains to be done to complete the quadratic Chabauty computation.

In a classic paper from 1990, Beilison, Lusztig and MacPherson gave a geometric realization of the quantized enveloping algebra of gl_n by defining a convolution product on the space of invariant functions over the variety of pairs of n-step partial flags over a finite field. This construction was generalized by Rosso to the mirabolic setting by modifying the points on the variety to include the additional data of a vector. A presentation for this "mirabolic quantum group" in terms of generators and relations was recently given by Fan, Zhang and Ma. I will describe this construction of the mirabolic quantum group and discuss its representation theory. Time permitting, I will also discuss a mirabolic quantum Schur-Weyl duality that this algebra satisfies with a mirabolic version of the Hecke algebra of Type A.

This Ph.D. thesis concerns optimal transport and effective resistance on finite weighted graphs. We investigate a number of directions, including applications of these topics to geometric graph theory and combinatorial optimization, as well as extensions of them to graphs with matrix-valued edge weights. We conclude with a number of results elucidating their connections.

It was observed by Kollár that the moduli functor of stable varieties in characteristic p>0 is no longer proper when one considers varieties of dimension ≥ 3. The key point is the existence of families of plt good minimal models of general type for which taking the relative canonical model does not commute with base change. I am going to illustrate an example showing that this kind of pathological behavior is not limited to the relative canonical model, but can indeed occur for any step of the relative MMP.