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.

In this talk, we will discuss a general result about the regularity of the associated graded ring of localizations at the prime ideals of a fixed ring. We will then apply this result to give a simple proof of a result of Ballard, Iyengar, Lank, Mukhopadhyay, and Pollitz on generation of the derived category in positive characteristic. This is based on joint work with De Stefani, KC, and Núñez-Betancourt.

Scott rank measures the descriptive complexity of a countable structure. It has been precisely defined in many non-equivalent ways over the past several decades. Montalbán gave a definition of Scott rank about 10 years ago that has become standard in the literature because it is equivalent to many interesting measurements coming from various areas of logic. Later, he also defined the so-called parameterized Scott rank, which is his equivalent to unparameterized Scott rank after adding a parameter. This notion is equally robust. A lighthearted debate emerged about which rank is better, usually with the underlying assumption that it does not really matter which notion is used. This talk challenges this underlying assumption. In particular, we demonstrate the asymmetry of these notions in Ehrenfeucht theories and discuss how a counterexample to Vaught's conjecture would behave quite differently depending on which notion is preferred. We also discuss how to use our analysis to produce models where the unparameterized and parameterized Scott rank differ, addressing a question of Alvir, Csima, and MacLean. This is joint work with Dino Rossegger and Dan Turetsky.

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).

We develop a new Fréchet sufficient dimension reduction (FdSDR) framework tailored for high-dimensional functional data with complex, metric-space-valued responses. Our main contribution is a low-rank distance covariance criterion that enables scalable, model-free identification of low-dimensional predictor structures while capturing nonlinear dependence. The proposed method is computationally efficient in high dimensions and avoids restrictive distributional assumptions. We establish theoretical guarantees and demonstrate its effectiveness through simulations and real data, providing a practical and flexible approach for modern functional data analysis.

The Shapley value, a fundamental concept in cooperative game theory, provides a fair allocation of cooperative gains or costs among players. However, computing Shapley values for a game with $d$ players requires evaluating all $2^d$ coalitions, which is computationally infeasible for large $d$. This difficulty is exacerbated in modern applications such as artificial intelligence, data science, and genomics, where evaluating the value of even a single coalition can be costly. To enable fast approximation and probabilistic inference of the Shapley value, we propose the Bayesian framework, where the Gaussian process is adopted to infer unobserved coalition values. The posterior distribution of the coalition values are then transformed into that of the Shapley values, allowing both point estimation and uncertainty quantification. We derive theoretical results showing that the computational complexity of posterior evaluation can be reduced from exponential to polynomial order. To further improve efficiency, we integrate experimental design principles to select coalitions that minimize posterior variances. Compared with existing approaches, the proposed method offers three main advantages: (i) support for statistical inference, (ii) accurate estimation of Shapley values using as few as $d^2-d+1$ coalition evaluations, and (iii) robustness across a wide range of cooperative games. Simulation studies and case analyses demonstrate that the proposed approach achieves higher accuracy and efficiency than existing methods under comparable evaluation cost.

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In this talk, we discuss the existence and nonexistence of certain birational contractions of \(\overline{\mathrm{M}}_{g,n}\). Somewhat surprisingly, this depends on the characteristic of the base field: many such contractions exist only in positive characteristic. We present a precise form of this phenomenon and discuss two examples that highlight the difference between characteristic zero and positive characteristic. The first is a simple and explicit contraction that exists only in positive characteristic, and the second is a modular interpretation of the morphisms associated with psi classes on \(\overline{\mathrm{M}}_{1,n}\). We also offer some speculation on why such characteristic-dependent phenomena arise.

The second boundary value problem for the Monge-Ampere equation is central to applications in illumination design, such as the construction of refractors and reflectors. While semi-discrete optimal transport methods have worst-case computational complexity of O(N^2) in dimensions 2 and 3, finite difference methods have linear complexity O(N) when used with a stencil of size independent of the number of mesh points N. This talk will present a complete theoretical foundation—covering existence, uniqueness, and convergence—for a linear-complexity finite-difference discretization based on a reformulation of the second boundary condition that prescribes the asymptotic cone of the epigraph of a convex extension of the solution.

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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.

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