A variety is "unirational" if it admits a dominant rational map from projective space. For moduli spaces this amounts to an explicit “recipe” for writing down a general member of the universal family. In characteristic zero, tensor forms obstruct unirationality -- famously employed by Harris--Mumford (1982) to prove that M_g is not unirational for g > 22. In positive characteristic, unirationality behaves much wilder due to the existence of inseparable maps. Consequently, we know the (non)-unirationality of few moduli spaces in positive characteristic. I will exhibit new techniques to obstruct unirationality in positive characteristic inspired by methods used to prove hyperbolicity in complex geometry. As applications, I will present a counterexample to a 1977 conjecture of Shioda regarding the unirationality of general type surfaces and prove that many Hilbert modular varieties over positive characteristics are not unirational.

For the discretisation of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu–Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein–Gelfand–Gelfand framework of the finite element exterior calculus. We also construct hp-bounded projection operators satisfying a commuting diagram property and hp-stable Hodge decompositions. Numerical examples are provided.

We discuss Chernikov’s conjecture that the burden is sub-additive. As partial progress toward this conjecture, we show that if T has a stronger version of dependent dividing (where dividing is witnessed by a formula in an existentially NIP reduct T_0 of T), then the burden agrees with the dp-rank witnessed by NIP formulas in T_0 and is thus sub-additive.

It is well known that asset returns usually do not follow a normal distribution, rather, they have long and fat tails. This paper focuses on the quantile portfolio methodology, which considers the whole distribution of asset returns and employs expected loss as a risk measurement. In particular, we explore statistical properties of tau risk and propose related theories of quantile portfolio optimization. We also introduce portfolio performance terms for the quantile portfolio framework.

Ash, Grayson, and Green computed the action of Hecke operators on the cuspidal cohomology of congruence subgroups $\Gamma_0(3, p) \subseteq \mathrm{SL}(3, \mathbb{Z})$ for small $p$.  A natural question to ask is for what other congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$ can one perform analogous computations.  In this talk, we detail techniques for working with congruence subgroups that are Iwahori at $p$, providing a framework for understanding the action of Hecke operators on the corresponding cohomology.  If time permits, we will discuss some improvements for the $\Gamma_0(3, p)$ setting as well. $$$$ Note the room change!  

Given a polynomial, the Strong Monodromy Conjecture predicts a mysterious relationship between its p-adic zeta function and Bernstein-Sato polynomial. While the conjecture remains widely open in general, progress has been made for specific classes of polynomials. In 2009, Budur-Mustațǎ-Teitler introduced the n/d conjecture and showed that it would imply the Strong Monodromy Conjecture for all hyperplane arrangements. In this talk, I will present a solution of the n/d conjecture, based on our new theory of multivariate V-filtration and a wall crossing theory for mixed Hodge modules. The latter is inspired by the recent breakthrough on the unitary dual problem of real Lie groups, by Davis-Vilonen. The talk is based on the upcoming work, joint with Dougal Davis.

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Let T_P be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of T_P, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in T_P, we define an additive rank on all imaginaries, called the geometric rank. It takes values in ω*N+Z and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in T_P^eq. As a consequence, we derive an explicit criterion for determining forking independence.

Since its inception, Heegaard Floer homology has been an invaluable tool for the study of 3- and 4- manifolds. Diffeomorphisms of surfaces (up to isotopy) are inherent to its construction; however, more remains to be explored about how invariants in Heegaard Floer homology interact with techniques and structures in mapping class groups of surfaces. In this talk, I will discuss work with Santana Afton and Tye Lidman where we study the relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group and discover some surprising phenomena.

Probit models have been prominent tools to analyze binary/ordinal data, but the computational complexity of maximum likelihood functions presents challenges in their usage. Furthermore, the model identification necessitates the covariance matrix of the latent multivariate normal variables to be a correlation matrix, which brings a rigorous task to develop efficient Markov chain Monte Carlo (MCMC) sampling methods. Data augmentation has been inevitable explored for both identifiable univariate and multivariate probit models. Particularly, it is well-known that parameter-expanded data augmentation (PX-DA) based on non-identifiable models accelerates the convergence and improves the mixing of MCMC components. However, comprehensive investigation has seldom been undertaken, and various algorithms due to incorrectly constructed non-identifiable models further bring obstacles to develop efficient MCMC sampling methods. We tackle this issue by constructing correct non-identifiable models and develop PX-DA algorithms to estimate both univariate and multivariate probit models. Our investigation exhibits that the proposed PX-DA algorithms advance the performance of MCMC sampling considerably and illustrates the essentials of using PX-DA, especially for data with large sample sizes.

For hundreds of years mathematicians have been fascinated with slicing high-dimensional mathematical objects as a way to get knowledge and intuition of higher dimensions. There are many classical results and conjectures about slices with hyperplanes (e.g., Bourgain’s conjecture, recently a theorem, on the relation of volume and area of slices). This topic is central to geometry, analysis, and of course combinatorics!! Given a d-dimensional convex polytope P, what is the ``best’’ slice of P by a hyperplane? For a combinatorialists ``best’’ can mean for example, one with the largest number of vertices! Not only we investigate the above combinatorial optimization theorem but also, note that as we slice P with different hyperplanes, we create many combinatorially different (d-1)-slices, which are also polytopes of course. E.g., for a 3-dimensional regular cube there are 4 combinatorial types of slices (triangles, quadrilaterals, pentagons, hexagons). We investigate: How many combinatorially different slices are there for a polytope P? How can we count them all? Can we give lower/upper bounds on their number? What are extremal cases? I will explain a powerful new combinatorial model (a moduli space of slices) and an algorithmic framework that answers these problems (and many others) in polynomial time when dim(P) is fixed. Moreover, we show the problems have hard complexity otherwise. The results are joint work with Marie-Charlotte Brandenburg (U Bochum) and Chiara Meroni (ETH) and Antonio Torres and Gyivan López (UC Davis). This talk will have lots of pretty pictures and will be understandable by everyone. I will present lots of open questions for enthusiastic researchers in the audience.

In this expository talk, we will walk through some basics of the random graph theory, aiming to understand a high-level motivation for the Kahn--Kalai Conjecture (now the Park--Pham Theorem), which has been a central conjecture in the area of probabilistic combinatorics. Below is a more formal description of the work that we will discuss, but I will try to use concrete examples rather than formal language, and will not assume much prior knowledge other than undergraduate-level combinatorics and probability. More formal description: for a finite set $X$, a family $F$ of subsets of $X $is said to be increasing if any set $A$ that contains $B$ in $F$ is also in $F$. The $p$-biased product measure of $F$ increases as $p$ increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures, with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound $q(F)$ (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window.

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We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume. This is a joint work with Chris Connell and Shi Wang (arXiv:2410.19981).

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Modern decision-making systems, from online marketplaces to large language models (LLMs), increasingly rely on high-dimensional human feedback, where heterogeneous user preferences and massive feature spaces pose major challenges for statistical efficiency and alignment. In this talk, I will present low-rank reinforcement learning (RL) methods that exploit latent structures in human feedback to enable scalable and theoretically grounded learning. In the first part, we study the dynamic assortment problem in high-dimensional e-commerce and show how a low-rank structure in user–item interactions reduces the complexity of estimating personalized utilities and enables efficient exploration–exploitation strategies with provable regret guarantees. In the second part, we extend these ideas to reinforcement learning from human feedback (RLHF) in large-scale contextual environments, proposing a low-rank contextual framework that accommodates diverse user preferences and complex latent spaces in LLMs while providing theoretical guarantees on sample efficiency and robustness under distribution shifts.