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.

In this talk, we present the global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension $d=2$ for large initial data in $H^s,s>0$. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, in particular the inverse theorems for Gowers uniformity norms by Green-Tao-Ziegler. In addition, we construct an approximate periodic solution showing ill-behavior of the flow map at the $L^2$ regularity. This is based on joint works with Sebastian Herr.

(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.