In this talk, I will begin by introducing graphons, which are infinite-size limits of adjacency matrices of sequences of growing graphs. I will then define graph reaction-diffusion (RD) equations, which are systems of differential equations that are defined on the nodes of a graph. For a sequence of growing graphs that converges to a graphon, the solutions of the sequence of graph RD equations also converge. The limiting solution solves a nonlocal differential equation that we call a graphon RD equation. Furthermore, the graph RD equation is related to a stochastic birth-death process on graphs. I will show that this birth-death process converges to the graphon RD equation via a hydrodynamic limit.

Transport by fluid flow can provide one of the less understood regularization mechanisms in PDE. In this talk, I will focus on the 2D Keller-Segel equation for chemotaxis set on a general domain and coupled via buoyancy with the fluid obeying Darcy's law - a much studied model of the incompressible fluid flow in porous media. It is well known that solutions to the 2D Keller-Segel equation can form singularities in finite time if the mass of the initial data is larger than critical. It turns out that if the equation is coupled with fluid flow obeying Darcy's law via buoyancy, this completely regularizes the system, leading to globally regular solutions for arbitrarily large initial data. One of the key ingredients in the proof is a new generalized Nash inequality, which employs anisotropic norm that is natural in the context of the incompressible porous media flow. This talk is based on works joint with Kevin Hu, Naji Sarsam, and Yao Yao.

The main goal of this talk is to introduce twisted crossed products of Banach algebras by locally compact groups. 

Classical crossed products of Banach algebras have been extensively studied for different classes of representations, including contractive representations on L^p-spaces. In this talk, we will give a general formulation for Banach algebras associated with twisted dynamical systems. Recent developments in L^p-twisted crossed products have mostly focused on situations where either the algebra is the complex numbers or when the group is discrete (more generally for étale groupoids). We present a universal characterization of the twisted crossed product when the acting group is locally compact and the Banach algebra has a contractive approximate identity. 

As an application, we focus on the case when the representations are contractive ones acting on L^p spaces. We briefly discuss a reduced version for L^p-operator algebras and present conjectures regarding amenability and rigidity when p\neq 2. Time permitting, we will present a generalization of the so called Packer–Raeburn trick to the L^p-setting, by showing that the universal L^p twisted crossed product is ``stably'' isometrically isomorphic to an untwisted one. 

This is joint work with Carla Farsi and Judith Packer.

TBD

Benedict Gross has a conjecture relating the square roots of the central values of a certain L-function of a cuspidal eigenform $f$ to the Fourier coefficients of the lift of $f$ to the group $G_2$. We describe our methods to compute the central values of the L-function of $f$, twisted by a Dirichlet character associated to a Galois cubic field. We will provide evidence for this conjecture of Gross via comparison with Fourier coefficients on $G_2$ computed by Aaron Pollack. This is joint work with Maya Chang.

Benedict Gross has a conjecture relating the square roots of the central values of a certain L-function of a cuspidal eigenform $f$ to the Fourier coefficients of the lift of $f$ to the group $G_2$. We describe methods to compute the central values of the L-function of $f$, twisted by a Dirichlet character associated to a Galois cubic field. We will provide evidence for this conjecture of Gross via comparison with Fourier coefficients on $G_2$ computed by Aaron Pollack. This is joint work with Michael Hoffman.

We show that up to isomorphism there are exactly twenty pairs $(C, E)$, where $C$ is a genus-2 curve over the complex numbers, where $E$ is an elliptic curve over the complex numbers, and where for every integer $n > 1$ there is a map of degree $n$ from $C$ to $E$. For example, if $C$ is the curve $y^2 = x^5 + 5 x^3 + 5 x$, and if $E$ is an elliptic curve with CM by the order of discriminant -20, then $(C, E)$ is such a pair.

On the other hand, we produce some finite sets $S$ of integers such that if $C$ is a genus-2 curve in characteristic 0, and if for every $n$ in $S$ there is an elliptic curve $E_n$ and a degree-$n$ map $\phi_n$ from $C$ to $E_n$, then for at least one of these $n$, the map $\phi_n$ factors through a nontrivial isogeny.

[pre-talk at 3:00PM]

In this talk, we will discuss an ergodic proof of the Sumset Conjecture of Erdős, which asks if every set $A \subseteq \mathbb{N}$ with positive density contains $B + C$ for some $B,C \subseteq \mathbb{N}$ infinite. This result was originally proved by Moreira, Richter, and Robertson in 2019 using ultrafilters, however in this proof we will adapt the method of progressive measures recently developed by Kra, Moreira, Richter, and Robertson. We closely follow their proof, simplifying what we can along the way.

In this talk, I will present a joint work with W. Yang and C. Yuan on the weak decomposition of the Hardy and boundedness of the small Hankel operator on the Hardy space. It is well known that a function $f\in H^p(D)$, the Hardy space over the unit disc $D$ can be written as the product of functions $f_1$ and $ f_2 $ in $H^{2p}(D)$ with $\|f\|_{H^p}=\|f_1|_{2p} \|f_2\|_{H^2}$. This is no longer true for the Hardy space on the unit ball $B_n$ in ${\bf C}^n$ when $n>1$. A well known theorem was proved by Coiffman, Rochberg and Weiss (Ann. of Math., 1976) that there is a weak decomposition for $H^1(B^n)$ through atomic decomposition. Their theorem has been proved true for $H^p$ when $p\le 1$. In this talk,I will show that it remains true when $p>1$. As a consequence, we obtained boundedness of the small Hankel operator on Hardy spaces.

Let $A$ be a finite alphabet. The automorphism group $\mathrm{Aut}(A^\mathbb{Z})$ is the group of invertible sliding block codes from the full $A$-shift to itself. Suppose $N\trianglelefteq\mathrm{Aut}(A^\mathbb{Z})$. By emulating methods from Kim and Roush's embedding, we show that either $N\le\mathbb{Z}$ or the commutator subgroup $[\mathrm{Aut}(2^\mathbb{Z}),\mathrm{Aut}(2^\mathbb{Z})]$ embeds into $N$. It is known that the free group on $2$ generators embeds into this commutator subgroup.

TBD

This thesis studies a central theoretical challenge in online reinforcement learning from human feedback (RLHF): under Bradley--Terry preference feedback, comparing against a fixed weak reference can make learning exponentially inefficient in the reward scale because preference signals saturate. To isolate this issue, the thesis considers a simplified dueling-bandit setting and proposes Ladder-BAI, a self-updating baseline algorithm that repeatedly promotes the current best arm and identifies better arms through simple fixed-baseline comparisons. The main result shows that Ladder-BAI finds an $\epsilon$-optimal arm using $\tilde{O}(K R_{\max} + K/\epsilon^2)$ preference queries, achieving linear dependence on the reward scale $R_{\max}$. This improves substantially over prior exponential or higher-degree polynomial guarantees. The analysis is based on a reward-ladder argument: each epoch yields a constant reward improvement by keeping comparisons informative, and a final refinement step achieves $\epsilon$-accuracy. Synthetic experiments support the theory, confirming linear scaling in reward scale and number of arms, as well as the expected $1/\epsilon^2$ dependence on target accuracy.

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.

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.

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.

While Weierstrass’ Approximation Theorem guarantees that continuous functions can be uniformly approximated by polynomials, it provides no information about the rate of this convergence. Bernstein’s Lethargy Theorem (BLT) classically addresses this gap by proving that the error of best polynomial approximation can decay at an arbitrarily slow, prescribed rate. This talk explores the evolution of BLT from its roots in classical approximation theory to its broad applications in functional analysis. We will discuss extensions of BLT to abstract Banach spaces and Frechet spaces. Building on this framework, we will investigate the deep connections between lethargy phenomena and operator ideals, the influence of Banach space reflexivity on the existence of lethargic convergence, and the interplay between BLT and interpolation theory via the Peetre K-functional.