Clustering is one of the fundamental problems in statistics and machine learning. Classical generative models such as the Stochastic Block Model (SBM) and Gaussian Mixture Model (GMM) are widely used for synthetic data generation and theoretical evaluation, but much of the literature assumes linearly separable clusters---an assumption that can fail in the presence of nonlinear geometry. We study a nonlinear multi-manifold model in which disjoint manifolds represent different clusters and the observations are corrupted by high-dimensional noise. We propose a kernel-based spectral embedding algorithm, based on the Random Geometric Graph (RGG) constructed from the data. Following the framework established by Ding and Ma (2023), we show that the embedding converges to its noiseless counterpart when the signal-to-noise ratio is sufficiently large. For downstream tasks, the embedding can be used for community detection problems. When different manifolds are sufficiently separated, the procedure recovers the community structure with vanishing error. Based on joint work with Xiucai Ding.

This dissertation consists of four papers that develop computational and structural results in equivariant stable homotopy theory. The results include the computation of the reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems, the construction of the first family of $C_{p^n}$-equivariant ``$v_1$''-self maps, the computation of the $C_{p^n}$-equivariant Mahowald invariants of all elements in the Burnside ring, extending the classical computations of Bredon--Landweber and Iriye, and the computation of the spoke-graded $C_3$-equivariant stable stems.

Understanding the loss-landscape geometry near a minimum is key to explaining the implicit bias of gradient-based methods in non-convex optimization problems such as deep neural network training and deep matrix factorization. A central quantity to characterize this geometry is the maximum eigenvalue of the Hessian of the loss. Its precise role has been obfuscated because no exact expressions for this sharpness measure are known in general settings. In this talk, I will present an analysis to derive a closed-form expression for the maximum eigenvalue of the Hessian matrix of an overparameterized deep matrix factorization problem with squared-error loss. I will show that this expression reveals fundamental properties of the loss landscape in deep matrix factorization. For instance, flat minima correspond to spectral-norm balanced minima in depth-2 matrix factorization. Furthermore, I will discuss the implications of this analysis. Beyond this, I will further discuss how l2 regularization reshapes the loss landscape and the set of minimizers of the overparameterized deep matrix factorization problem.

This is a continuation of Hui Tan's talk on joint work studying the structure and classification of q-Araki-Woods factors. I will focus on the proofs of the main results: the dichotomy for subalgebras in continuous cores underlying strong solidity, and the failure of biexactness for q-Araki-Woods factors with infinite-dimensional representations via norm estimates from Nou and Hiai.

Let $\mathrm{Mat}_{n\times m}(\mathbb{C})$ be the affine space of $n\times m$ complex matrices, and let $\mathcal{Z}_{n,m,r}$ (resp. $\mathcal{UZ}_{n,m,r}$) be the locus in $\mathrm{Mat}_{n\times m}(\mathbb{C})$ corresponding to rook placements with exactly (resp. at least) $r$ rooks. The orbit harmonics method yields two quotient rings $R(\mathcal{Z}_{n,m,r})$ and $R(\mathcal{UZ}_{n,m,r})$, where both rings have the additional structures of $\mathfrak{S}_n\times\mathfrak{S}_m$-modules. We find the generators of their defining ideals and compute their graded Frobenius image. Furthermore, we give a nontrivial generalization of Viennot's shadow line avatar of the Schensted correspondence to rook placements in $\mathcal{UZ}_{n,m,r}$. This generalization is used to determine the standard monomial basis of $R(\mathcal{UZ}_{n,m,r})$ with respect to a diagonal term order. Joint with Jasper (Moxuan) Liu.

Vaughan Jones introduced an index for inclusions of certain von Neumann algebra in the 1980's and proved that it is surpisingly rigid. This rigidity is due to a rich combinatorial structure that is inherent to the representation theory of a subfactor with finite index. Subfactor representations reveal interesting unitary tensor categories, or quantum symmetries, whose algebras of intertwiners always contain the Temperley-Lieb algebras and, if an intermediate subfactor is present, the Fuss-Catalan algebras of Jones and myself. The case of two intermediate subfactors is much more involved and not much progress had been made since the late 1990's.

I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a subfactor when two intermediate subfactors occur, and the four algebras form a cocommuting square. These new symmetries turn out to be related to partition algebras and Bell numbers.

The Wasserstein metric over a metric space X is an optimal-transport based distance on the set of probability measures on X. Metric spaces for which the optimal transport problem is "easiest" to solve are trees,  in the sense that the Wasserstein metric on trees isometrically embeds into $L^1$. Property A is a coarse invariant of metric spaces introduced by Yu as an approach to solving the coarse Baum-Connes conjecture. We prove a new characterization of bounded degree graphs X with Property A as precisely those that are coarsely equivalent to another space Y whose Wasserstein metric admits a biLipschitz embedding into $L^1$. Applications to group actions on Banach spaces will be discussed. Based on joint work with Tianyi Zheng and Ignacio Vergara.

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.

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.

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.

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. By emulating methods from Kim and Roush's embedding, we show that the commutator subgroup $[\mathrm{Aut}(2^\mathbb{Z}),\mathrm{Aut}(2^\mathbb{Z})]$ embeds into $\mathrm{Aut}(A^\mathbb{Z})$ for any $A$. 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.