Over a polynomial ring, an ideal I is said to be linearly presented if its first syzygy module is generated by linear relations. More generally, the resolution of I is linear for p steps if this linearity continues through the first p steps. In this talk, we explore some fundamental properties of ideals with partially linear minimal free resolution, such as asymptotic behavior of powers, Castelnuovo-Mumford regularity, and lower bounds on the number of minimal generators. In particular, we prove the Eisenbud-Ulrich conjecture on powers of linearly presented ideals and establish a linear effective bound toward the more general Eisenbud-Huneke-Ulrich conjecture.

We consider the matrix random walk where the multiplicative increments are the identity plus a bi-invariant matrix, and the free random walk where the multiplicative increments are the identity plus a R-diagonal operator. In this talk, I will first speak about the result that the limit of the empirical eigenvalue distribution of the matrix random walk as the size of the matrix approaches infinity is the Brown measure of the free random walk. Then I will speak about the result that the limit of the Brown measure of the free random walk as the number of steps approaches infinity is the lima bean law. This is joint work with Driver, Hall, Kemp, Nemish, Nikitopoulos, Parraud.

The rank $n$ superspace is the algebra of polynomial-valued differential forms on affine $n$-space. This carries a natural action of the symmetric group, or in general, of any pseudo-reflection group. The superspace coinvariant ring is the quotient of superspace cut out by elements that are invariant under this action, and have a vanishing constant term. Due to their interesting combinatorial and representation theoretic features, these objects have received a lot of attention in recent years. In this talk (based on joint work with Brendon Rhoades), I will go over some recent progress in understanding superspace coinvariants for an infinite family of pseudo-reflection groups, and some natural questions that remain open.

Momentum is a ubiquitous and critical component of modern optimizers, yet its role remains poorly understood beyond restricted settings, particularly in the training of large-scale neural networks. Recent studies suggest that the highly non-convex loss landscapes of large language models exhibit a “river-valley” structure: a low-loss manifold (the river) bordered by sharp, high-loss directions (the valley). In such landscapes, long-run optimization progress is determined primarily by movement along the river. Motivated by this structure, we investigate the role of heavy-ball momentum in this setting. We analyze gradient descent with heavy-ball momentum and show that, compared with vanilla gradient descent, momentum accelerates progress along the river by permitting a substantially larger learning rate. Specifically, momentum acts as a stabilizer: it controls the oscillations induced by an aggressive learning rate, which vanilla gradient descent cannot tolerate. We validate these insights through experiments on synthetic functions and language model training, and we offer practical guidance for tuning the learning rate and momentum parameters.

The first part of this talk is about my organization, CCR La Jolla, and its connections with UCSD. After a brief interlude about current job openings, the second half describes the recent solution to a question from Serre about generic polynomials that came out of his 2001 lecture series at UCLA.

A branchvariety of a projective k-scheme X is a geometrically reduced scheme Y equipped with a finite map to X. Alexeev and Knutson showed the existence of a proper moduli space of branchvarieties with fixed Hilbert polynomial and degree sequence, but the projectivity of this space remained an open question. In this talk, we will discuss positivity results for some line bundles related to the determinant line bundle on the moduli space of equidimensional branchvarieties. As a consequence, we establish that this moduli space is projective.

A numerical semigroup is a subset of the natural numbers that is closed under addition. There is a family of polyhedral cones $C_m$, called Kunz cones, for which each numerical semigroup with smallest positive element $m$ corresponds to an integer point in $C_m$. It has been shown that if two numerical semigroups correspond to points in the same face of $C_m$, they share many important properties. In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure.  

In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups. No familiarity with numerical semigroups or polyhedral geometry will be assumed for this talk.

Using a new, non-K-theoretic approach involving embedding spaces in II1 factors with plenty of freely independent Haar unitaries, we prove that the reduced free group C*-algebras for different numbers of free generators are pairwise non-isomorphic. This recovers the seminal result of Pimsner and Voiculescu with a short new proof. This talk is based on joint work with Srivatsav Kunnawalkam Elayavalli.

Quantization is the art of converting neural network weights to low precision, say to just 1–4 bits per scalar, while keeping the network accuracy high. We study one natural and popular way of doing this, which consists of choosing a surrogate loss that is quadratic in the parameters, and then applying a simple integer optimization algorithm to this loss. In particular, we discuss how the second step (integer optimization algorithm) influences the first step (information-theoretically optimal choice of the surrogate loss).

We will discuss the proof that quasi-unipotent p-adic differential modules over the Robba ring with bounded rank and highest ramification break all admit models on a common annulus, with a uniform bound on the connection on each circle. The radius and the bound depend only on the rank, the break and residue characteristic p. We also show that a Frobenius structure on such a module extends to a model of the same size. Finally, we explain how this uniformity provides a local input for cut-by-curves criteria for overconvergence of F-isocrystals.

We consider sparsely observed functional data generated by a latent Gauss–Markov process modeled through a linear stochastic differential equation (SDE). Motivated by an application to tumor growth data, we allow for outcome-dependent dropout, where sampling terminates once the most recent observation exceeds a prescribed threshold. Such observation schemes arise naturally in longitudinal studies and violate the fundamental missing completely at random assumption commonly imposed in the analysis of partially observed functional data. We develop a likelihood-based estimation framework that, in contrast to existing moment-based methods, avoids the bias induced by outcome dependent dropout. We establish convergence rates for the proposed estimators and show that they are minimax optimal up to logarithmic factors, with the diffusion coefficient admitting a faster rate than the drift coefficients under a random initial condition for the SDE. An application to tumor growth data further demonstrates the advantages of a fully probabilistic functional data model for tasks beyond second-order inference, including prediction bands, first-passage times, and tumor-age estimation.