Abstract: I will explain how one can use ideas from analytic number theory and Manin's conjecture to understand the geometry of moduli spaces of curves on smooth low degree hypersurfaces. In particular, when e is large compared to g, the moduli space of degree e maps from smooth genus g curves to an arbitrary smooth hypersurface of low degree has at worst terminal singularities. Using a spreading-out argument together with a result of Mustata, we reduce the problem to counting points over finite fields on the jet schemes of these moduli spaces. We solve this counting problem by developing a suitable geometric interpretation of the circle method. This is joint work with Jakob Glas.
Abstract: Two-stage stochastic programs (SPs) with polynomial loss functions serve as a powerful framework for modeling decision-making problems under uncertainty. In this talk, we introduce a two-phase approach to find global optimal solutions for two-stage SPs with continuous decision variables and nonconvex recourse functions. Our method does not only generate global lower bounds for the nonconvex stochastic program, but also yields an explicit polynomial approximation for the recourse function. It is particularly suitable for the case where the random vector follows a continuous distribution or when dealing with a large number of scenarios.
Abstract: Under the operation of connected sum, the set of three-manifolds form a monoid. Modulo an equivalence relation called homology cobordism, this monoid (of homology spheres) becomes a group. What is the structure of this group? What families of three-manifolds generate (or don’t generate) this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with (various subsets of) I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.
Abstract: Machine learning has revolutionized computational science and engineering with impressive breakthroughs, e.g., making the efficient solution of high-dimensional computational tasks feasible and advancing domain knowledge via scientific data mining. This leads to an emerging field called scientific machine learning. In this talk, we introduce a new method for a symbolic approach to solving scientific machine learning problems. This method seeks interpretable learning outcomes via combinatorial optimization in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can efficiently learn high-dimensional complex functions. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for learning the solution of high-dimensional PDEs and learning the governing equations of raw data.
Abstract: Classical reaction-diffusion-advection models assume that all individuals in a species or subspecies move in the same way and only produce offspring of the same subspecies. There are at least three ways that populations can violate those assumptions. Individuals can change behavior, for example switching between search and resource exploitation. Populations can be stage structured and individuals at different stages can have different movement patterns. Individuals of one subspecies can produce offspring of another subspecies by genetic mutation or recombination. Models for populations whose members can switch between different movement modes can have properties that are different from those for populations where all individuals move in the same way. They may not satisfy the reduction principle, which says that slower dispersal is advantageous, and holds for many types of models with a single movement mode. They typically lead to systems that are cooperative at low densities but competitive at high densities. This talk will describe some models of this type and some of their properties.
Abstract: There is an array of long-standing open problems in the theory of countable Borel equivalence relations (CBER), all of which state that the class of hyperfinite CBERs is nice in some way. For instance, the unresolved Union Problem asks whether the class of hyperfinite CBERs is closed under increasing unions, and in a different direction, it is also open whether the hyperfinite CBERs form a $\mathbf{\Pi}^1_1$ set, which would be nicer than the naive complexity of $\mathbf{\Sigma}^1_2$. There are many other such problems, and it is widely believed that if one of them is false, then most of the others will be false as well, although there is no formal statement to this effect. To this end, we show an implication between the two aforementioned problems: precisely, we show that if the Union Problem has a negative answer, then the Borel complexity of the class of hyperfinite CBERs is as high as possible, namely $\mathbf{\Sigma}^1_2$-complete. This is joint with Joshua Frisch and Zoltan Vidnyanszky.
Abstract: Maxwell's equations govern the propagation of light and electromagnetic radiation. Computing the electromagnetic field is important in designing communications and sensing devices. To understand the challenges in approximating the solution of these equations I will consider a simple model scattering problem. Starting with the classical curl conforming edge element space, I will explain how standard software solves the Maxwell system. Then l will move on to the evolution of discontinuous Galerkin methods concentrating on Hybridizable Discontinuous Galerkin (HDG) methods. Finally, I will discuss Trefftz discontinuous Galerkin methods where simple exact solutions of the Maxwell system are used element by element to approximate a general solution. This method has proved successful in solving large problems such as scattering by an aircraft but still faces issues that need to be addressed. Several numerical examples will illustrate the performance of the method.
Abstract: Suppose that particles are randomly distributed in $R^d$, and that they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. The goal is to estimate characteristics of the particle displacement process from the count data. Such estimation problems arise in various application areas, e.g., in biology (studies of particles/cells motility), trasportation science (traffic estimation), etc.
We discuss probabilistic properties of the Smoluchowski processes and consider related statistical problems for two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In these settings we develop estimators with provable accuracy guarantees.
Abstract: In this session, we will break out into subgroups to work through the mathematics in the paper "Hidden-State Proofs of Quantumness" (https://arxiv.org/abs/2410.06368).
Each group will have at least one person with familiarity in cryptography familiarity to guide the process.
Participants should read the paper before the session, but are not expected to have grasped all of its concepts.
Abstract: Let $f:X\to S$ be a projective morphism of normal varieties. Assume $U$ is an open subset of $S$ and $L_U$ is a divisor on $X_U:=X\times_S U$ such that $L_U\equiv_U 0$. We explore when it is possible to extend $L_U$ to a global $\mathbb{Q}$-divisor $L$ on $X$ such that $L\equiv_S 0$. In particular, we can show that such $L$ always exists after a (weak) semi-stable reduction when $S$ is a curve.
On the other hand, we give an example showing that $L$ may not exist (after any reasonable modification of $f$) if $S$ has dimension $\ge2$, which also gives an $f_U$-nef divisor $M_U$ that cannot extend to an $f$-nef $\mathbb{Q}$-divisor $M$ for any compactification of $f|_U$, even after replacing $X_U$ with any higher birational model.
Abstract: This talk presents the global existence of entropy weak solutions for scalar balance laws with nonlocal singular sources, along with a partial uniqueness result. A detailed analysis of the solution structure is provided for a general class of initial data, particularly in neighborhoods where two shocks interact. Additionally, some open questions will be discussed.
Abstract: Stochastic approximation (SA) is a powerful and scalable computational method for iteratively estimating the solution of optimization problems in the presence of randomness, particularly well-suited for large-scale and streaming data settings. In this talk, we propose a theoretical framework for stochastic approximation (SA) applied to non-parametric least squares in reproducing kernel Hilbert spaces (RKHS), enabling online statistical inference in non-parametric regression models. Our approach combines an online multiplier bootstrap with functional stochastic gradient descent (SGD) in RKHS to achieve two key inferential advances: (1) scalable online confidence intervals/bands—constructing asymptotically valid pointwise confidence intervals for local inference and simultaneous confidence bands for global inference of nonlinear regression functions; and (2) minimax online hypothesis testing—building optimal Wald-type test statistics for nonparametric regression models. The main theoretical contributions consist of a unified framework for characterizing the non-asymptotic behavior of the functional SGD estimator and demonstrating the consistency of the multiplier bootstrap method. And the theory specifically establishes the interplay between the online learning rate and the minimax estimation/power performance in uncertainty quantification.