Graded Ehrhart theory is a new q-analogue of Ehrhart theory introduced by Reiner and Rhoades. Roughly, it is the study of how a canonical, graded lattice point count of a polytope behaves under dilations. The grading in this count is constructed via the orbit harmonics method. In this talk, I will discuss the graded Ehrhart theory of unimodular zonotopes and its connections to matroid theory. In particular, I will explain why graded lattice point counts of unimodular zonotopes are q-integer evaluations of Tutte polynomials and how arrangement Schubert varieties can be used to study the harmonic algebras of unimodular zonotopes. This talk is based on joint work with Colin Crowley.

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.