For many variant kinetic equations, we choose an appropriate approx- imatation equations. Also, this approximation equations are solvable more easier than the original equations and it retains the expected a priori bounds. Then, we use the variant compactness theorem to pass to the limit in the sense of distributions in the approximation equations. In the kinetic theory, this compactness theorem is called the averaing lemma, that is, the averaging in velocity improves regularity in the space and time variables. For this PDE seminar, we study the basic averaging lemma. In other words, we investigate the basic properties of the free transport operator ∂t + v · ∇x.

Associated to a group action on a bifoliated plane, satisfying some reasonable conditions, one can associate a combinatorial object known as a veering triangulation. Since their introduction by Agol (in a very different setting), these triangulations have recently played an interesting role in studying pseudo-Anosov flows, the structure of fibered 3-manifolds, algorithmic properties of mapping class groups, and fixed points of surface homeomorphisms, to name just a few (from my own biased perspective). This talk will be an overview of these applications, starting with the most basic properties from the initial bifoliated plane.

This is a reading seminar of a graduate student, following the Fields medal work of Daniel Quillen on the foundation of the higher algebraic K-theory.

In this talk, I will present the local existence theory for quasilinear symmetric hyperbolic systems, based on Sections 1.3 and 2.1 of [1]. I will begin by reviewing the framework of symmetric systems and then explain how it is applied to establish local-in-time existence of classical solutions. The main focus will be on the iteration scheme, energy estimates, and convergence arguments. We aim to understand how regularity and a priori bounds are used to construct solutions from smooth initial data.

This talk explores the relationship between 3-dimensional lens spaces and smooth 4-manifolds that bound them under various topological constraints—topics that connect to several central conjectures in low-dimensional topology. After reviewing the classifications of Lisca, Greene, and Aceto–McCoy–JH Park, I will present recent joint work with Wookhyeok Jo and Jongil Park investigating which lens spaces can bound smooth 4-manifolds with second Betti number one. In particular, we exhibit infinite families of lens spaces that bound simply connected 4-manifolds with b₂ = 1, yet do not bound 4-manifolds consisting of a single 0-handle and 2-handle. Moreover, we construct infinite families of lens spaces that bound 4-manifolds with b₁ = 0 and b₂ = 1, but do not bound simply connected 4-manifolds with b₂ = 1. These constructions are motivated by the study of rational homology projective planes with cyclic quotient singularities.

Geometric evolution equations describe how geometric objects such as curves, surfaces, or metrics evolve toward more symmetric or optimal shapes. Among the most fundamental examples are the mean curvature flow and the Ricci flow, which have played central roles in modern differential geometry and topology. In this talk, I will give an introduction to these flows, explaining how curvature acts as a driving mechanism that smooths and reshapes geometry. I will also outline the key ideas behind Perelman’s proof of the Poincaré conjecture, focusing on the role of singularity formation and the classification of canonical neighborhoods. Finally, I will discuss the problem of classifying singularity models arising under geometric flows and present some recent progress on the classification of ancient oval solutions, together with possible further directions.

We briefly introduce the restriction theory in harmonic analysis and its connections with PDEs through Strichartz estimate. We then discuss the Kakeya and multilinear Kakeya estimates, which naturally arise from restriction theory. The main part of the talk will focus on Larry Guth’s proof of the multilinear Kakeya estimate via the induction on scales method.

A (positive definite and integral) quadratic form $f$ is called irrecoverable (from its subforms) if there is a quadratic form $F$ that represents all proper subforms except for $f$ itself, and such a quadratic form $F$ is called an isolation of $f$. In this talk, we present recent advances on irrecoverable quadratic forms and discuss their possible generalizations.

(The is a PhD student reading seminar to be given by Mr. Jaehong Kim.)

TBD

In this talk, I will try to explain how the essence of the Weierstrass representation formula and the Bjorling representation formula for minimal surfaces in $E^3$ can be suitably applied to zero/constant mean curvature surfaces in the three-dimensional spaceforms in the Lorentz-Minkowski four-space.

In this talk, we introduce the concept of t-core partitions. We discuss the generating function and modularity, along with some results and applications of t-core partitions. Recent results on simultaneous core partitions will also be presented. Toward the end of the talk, we introduce numerical semigroups and explore connections between numerical semigroups (or numerical sets) and partitions. Additionally, we present some open problems related to these topics.

We study the partial dimensional semi-classical Weyl’s laws, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive lowest free energy states for the subband models describing non-interacting 2DEGs.