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.

In this talk, we discuss the paper “Machine learning methods trained on simple models can predict critical transitions in complex natural systems” by Smita Deb, Sahil Sidheekh, Christopher F. Clements, Narayanan C. Krishnan, and Partha S. Dutta, in Royal Society Open Science, (2022).

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.

We will discuss classical and recent results about phase transitions in random subgraphs of the hypercube and beyond. The focus will be on the giant component, long cycles, large matchings, and isoperimetric properties.

In 2014, Bourgain and Demeter proved almost sharp decoupling inequalities for the paraboloid and the light cone, leading to various applications to the Schrodinger and the wave equations. I will explain some subsequent developments, including important contributions by Guth, Maldague and Wang, my joint work with Shaoming Guo, Zane Li and Pavel Zorin-Kranich, and joint work with Andrew Hassell, Pierre Portal and Jan Rozendaal.

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.

Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this talk, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the “tensor distribution” limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of independence, which we term “tensor freeness”. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables. This is joint work with Ion Nechita.

(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.

A local certification of a graph property is a protocol in which nodes are given  “certificates of a graph property” that allow the nodes to check whether their network has this property while only communicating with their local network. The key property of a local certification is that if certificates are corrupted, some node in the network will be able to recognize this. Inspired by practical concerns, the aim in LOCAL certification is to minimize the maximum size of a certificate. In this talk we introduce local certification and open problems in the area and  present some recent joint work with Eunjung Kim and Tomáš Masařík, A Tight Meta-theorem for LOCAL Certification of MSO2 Properties within Bounded Treewidth Graphs. In this work, instead of considering a specific graph property and developing a local certification protocol tailor-made for this property, we aim for generic protocols that can certify any property expressible in a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $\Omega(n^2/\log n)$ on general $n$-vertex graphs (Göös, Suomela 2016). Hence, we impose additional structural restrictions on the graph. Inspired by their importance in centralized computing and Robertson-Seymour Graph Minor theory, we consider graphs of bounded treewidth. We provide a local certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a local certification protocol for certifying bounded treewidth. That is, for each integer $k$ and each MSO$_2$-expressible property $\Pi$, we give a local certification protocol to certify that a graph satisfies $\Pi$ and has treewidth at most $k$ using certificates of size $\mathcal{O}(\log n)$ (which is asymptotically optimal). Our result improves upon the works of Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024),  Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang (PODC 2025).

In physics, the phase transition between localized and delocalized phases in disordered systems, often called the Anderson transition, has attracted significant interest. Several intriguing models display this behavior, including random Schrödinger operators, random band matrices, and sparse random matrices. Heavy-tailed random matrices similarly capture this phase transition, making them a crucial class of models in understanding localization phenomena. In this talk, we will discuss the phase transition of the right singular vector associated with the smallest singular value of a rectangular random matrix. This work is in collaboration with Zhigang Bao (University of Hong Kong) and Xiaocong Xu (University of Southern California).

We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern–Simons–Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern–Simons–Schrödinger equation.This talk is based on a work with Théotime Girardot

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.

The notion of asymptotic dimension of metric spaces, introduced by Gromov, describes their large-scale behaviour. Asymptotic dimension of graph families has been recently studied, in particular, by Bonamy et al. who proved that the asymptotic dimension of proper minor-closed graph families is at most two. We will discuss nerve-type theorems for asymptotic dimension. In particular, we show that the asymptotic dimension of intersection graphs of balls and spheres in $\mathbb{R}^d$ is at most $d+1$. Based on joint work with Zdeněk Dvořák and with Chun-Hung Liu.