(This is a reading seminar talk by a graduate student, Mr. Jaehong Kim.) This talk is a reading seminar about basic intersection theory, following chapter 1 to 6 of the book of William Fulton. The main objects to be dealt with are Chow groups, pullback/pushforward, pseudo-divisors, divisor intersection, Chern/Segre classes, deformation to the normal cone and intersection products.

Lattice field theories provide a discrete, probabilistic framework for approximating continuum quantum field theories. These models, originally motivated by statistical mechanics, are central to constructive approaches in mathematical physics. A fundamental challenge is to rigorously establish the continuum limit as the lattice spacing tends to zero, yielding singular but physically meaningful Gibbs measures on function spaces. Beyond this small scale (ultraviolet) limit, another major theme, especially from the viewpoint of statistical mechanics, is the analysis of large scale (infrared) behavior in the infinite volume limit. This involves understanding how thermodynamic properties, phase structure, and fluctuation phenomena emerge as the size of the physical system increases. In this three part minicourse, we will explore both aspects of this limiting procedure through the lens of probabilistic methods and stochastic quantization. While the Euclidean Φ^4 quantum field theory will serve as our primary example, the broader goal is to illustrate how continuum quantum field theories can be constructed as scaling limits of lattice models, unifying perspectives from statistical mechanics, field theory, PDEs, and probability.

Lattice field theories provide a discrete, probabilistic framework for approximating continuum quantum field theories. These models, originally motivated by statistical mechanics, are central to constructive approaches in mathematical physics. A fundamental challenge is to rigorously establish the continuum limit as the lattice spacing tends to zero, yielding singular but physically meaningful Gibbs measures on function spaces. Beyond this small scale (ultraviolet) limit, another major theme, especially from the viewpoint of statistical mechanics, is the analysis of large scale (infrared) behavior in the infinite volume limit. This involves understanding how thermodynamic properties, phase structure, and fluctuation phenomena emerge as the size of the physical system increases. In this three part minicourse, we will explore both aspects of this limiting procedure through the lens of probabilistic methods and stochastic quantization. While the Euclidean Φ^4 quantum field theory will serve as our primary example, the broader goal is to illustrate how continuum quantum field theories can be constructed as scaling limits of lattice models, unifying perspectives from statistical mechanics, field theory, PDEs, and probability.

(This is a reading seminar talk by a graduate student, Mr. Jaehong Kim.) This talk is a reading seminar about basic intersection theory, following chapter 1 to 6 of the book of William Fulton. The main objects to be dealt with are Chow groups, pullback/pushforward, pseudo-divisors, divisor intersection, Chern/Segre classes, deformation to the normal cone and intersection products.

Lattice field theories provide a discrete, probabilistic framework for approximating continuum quantum field theories. These models, originally motivated by statistical mechanics, are central to constructive approaches in mathematical physics. A fundamental challenge is to rigorously establish the continuum limit as the lattice spacing tends to zero, yielding singular but physically meaningful Gibbs measures on function spaces. Beyond this small scale (ultraviolet) limit, another major theme, especially from the viewpoint of statistical mechanics, is the analysis of large scale (infrared) behavior in the infinite volume limit. This involves understanding how thermodynamic properties, phase structure, and fluctuation phenomena emerge as the size of the physical system increases. In this three part minicourse, we will explore both aspects of this limiting procedure through the lens of probabilistic methods and stochastic quantization. While the Euclidean Φ^4 quantum field theory will serve as our primary example, the broader goal is to illustrate how continuum quantum field theories can be constructed as scaling limits of lattice models, unifying perspectives from statistical mechanics, field theory, PDEs, and probability.

(This is a reading seminar talk by a graduate student, Mr. Jaehong Kim.) This talk is a reading seminar about basic intersection theory, following chapter 1 to 6 of the book of William Fulton. The main objects to be dealt with are Chow groups, pullback/pushforward, pseudo-divisors, divisor intersection, Chern/Segre classes, deformation to the normal cone and intersection products.

Lecture 1: Artem Pulemotov (University of Queensland), 4:15-5:15PM
Title: The prescribed Ricci curvature problem on homogeneous spaces
Abstract: We will discuss the problem of recovering the ``shape" of a Riemannian manifold $M$ from its Ricci curvature. After reviewing the relevant background material and the history of the subject, we will focus on the case where $M$ is a homogeneous space for a compact Lie group. Based on joint work with Wolfgang Ziller (The University of Pennsylvania).

Lecture 2: Mikhail Feldman (University of Wisconsin-Madison), 5:30-6:30PM
Title: Self-similar solutions to two-dimensional Riemann problems with transonic shocks
Abstract: Multidimensional conservation laws is an active research area with open questions about existence, uniqueness, and stability of properly defined weak solutions, even for fundamental models such as the compressible Euler system. Understanding particular classes of weak solutions, such as Riemann problems, is crucial in this context. This talk focuses on self-similar solutions to two-dimensional Riemann problems involving transonic shocks for compressible Euler systems. Examples include regular shock reflection, Prandtl reflection, and four-shocks Riemann problem. We first review the results on existence, regularity, geometric properties and uniqueness of global self-similar solutions of regular reflection structure in the framework of potential flow equation. A significant open problem is to extend these results to compressible Euler system, i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions of regular reflection structure have low regularity. We further discuss existence, uniqueness and stability of renormalized solutions to the transport equation for vorticity in this low regularity setting.

***Tea Time 3:45PM-4:15PM in Room 1410***

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***Tea Time 3:45PM-4:15PM in Room 1410***

Confidence sequence provides ways to characterize uncertainty in stochastic environments, which is a widely-used tool for interactive machine learning algorithms and statistical problems including A/B testing, Bayesian optimization, reinforcement learning, and offline evaluation/learning.In these problems, constructing confidence sequences that are tight and correct is crucial since it has a significant impact on the performance of downstream tasks. In this talk, I will first show how to derive one of the tightest empirical Bernstein-style confidence bounds, both theoretically and numerically. This derivation is done via the existence of regret bounds in online learning, inspired by the seminal work of Raklin& Sridharan (2017). Then, I will discuss how our confidence bound extends to unbounded nonnegative random variables with provable tightness. In offline contextual bandits, this leads to the best-known second-order bound in the literature with promising preliminary empirical results. Finally, I will turn to the $[0,1]$-valued regression problem and show how the intuition from our confidence bounds extends to a novel betting-based loss function that exhibits variance-adaptivity. I will conclude with future work including some recent LLM-related topics.