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.

The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations. In this talk, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished. This talk is based on a collaborative work with Daisuke Kawagoe and Chun-Hsiung Hsia.

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.

The syzygy scheme is the scheme defined by the quadric forms associated to the linear syzygies of certain order of a given scheme. It is natural to ask whether the syzygy scheme is equal to the scheme itself. In this talk, I will discuss about the classification of the second syzygy schemes for 4-gonal canonical curves of genus at least 6. This talk is based on the work by Aprodu-Bruno-Sernesi.

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.

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.

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.

In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. We will also discuss possible extensions to the viscous SQG equation in the context of Hamiltonian conservation and existence of weak solutions for a rough initial data.

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.

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.

The Lyapunov-Schmidt reduction is a powerful tool to solve PDEs. This method reduces the equations, which are essentially infinite-dimensional, to finite-dimensional ones. In this talk, we illustrate the reduction by showing the existence of a positive solution to the singularly perturbed problem in for positive smooth and appropriate . To show the existence, we first construct an -dimensional surface of approximate solutions. Then, we reduce the problem onto that surface by the Lyapunov-Schmidt reduction. The key to the reduction is proving the invertibility of a certain operator, which in turn, is proved by a certain uniqueness result. After the reduction, we end the proof by solving the equation on the -dimensional surface.

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 recent years, syzygies of projections of algebraic varieties have drawn a lot of attentions. It turns out that their Betti diagrams carry geometric information like the codimension of the projection and the position of the projection center, by the investigations of E. Park, S. Kwak and so on. In this talk, I will show that for a generic canonical curve $C$ in $\mathbb{P}^{g−1}$, its projection $C'$ away from a generic point into $\mathbb{P}^{g−2}$ is cut out by quadrics for $g \geq 9$. I will also give the predictions of the Betti diagrams with the help of Macaulay2.

We study support properties of solutions to stochastic heat equations $\partial_t u = \Delta u + \sigma(u) \xi$ where $\xi$ is Gaussian noise. For $\sigma(u) = u^\lambda$ with colored noise, we show the compact support property holds if and only if $\lambda \in (0, 1)$. Here, the compact support property refers to the property that if the initial function has compact support, then so does the solution for all time. For space-time white noise with general $\sigma$, we characterize when solutions maintain compact support versus become strictly positive. We also discuss how the initial function influences these support properties. This is based on joint work with Beom-Seok Han and Jaeyun Yi.

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.

We investigate compact minimal surfaces in the Einstein-Maxwell theory with both electric and magnetic charges and a negative cosmological constant. A two-sided, embedded and strictly stable minimal surface that maximizes the magnetically charged Hawking mass naturally corresponds to the event horizon of a black hole. Our main theorem shows that the geometry near such a surface is rigid: a neighborhood is isometric to the dyonic Reissner-Nordstrom-Anti-de Sitter space, the canonical model of a charged black hole in Anti-de Sitter spacetime. In addition, we provide an area estimate for the surface that depends only on its topology and the relevant physical parameters.

In this talk, we prove that the inviscid surface quasi-geostrophic (SQG) equation is strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^2(\mathbb{R}^2)$ without any solutions in $L^{\infty}_tH^2$. Then, we introduce similar ill-posedness results for $\alpha$-SQG and two-dimensional incompressible Euler equations. This talk is based on joint works with In-Jee Jeong(SNU), Young-Pil Choi(Yonsei Univ.), Jinwook Jung(Hanyang Univ.), and Min Jun Jo(Duke Univ.).

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation given noisy observation data on its solution at a set of spatial points. We consider the Gaussian prior probability for the log-slowness, which is expressed as a countable linear expansion of mutually independent normal random variables. The well-posedness of the inverse problem is established, using the variational formulation of the eikonal equation. We approximate the posterior by finitely truncating the expansion of the log-slowness, with an explicit error estimate in the Hellinger metric with respect to the truncation level. Solving the truncated eikonal equation by the Fast Matching Method, we obtain an approximation for the posterior in terms of the truncation level and the discrete grid size in the Fast Matching Method resolution. Using this result, we develop and justify the convergence of a Multilevel Markov Chain Monte Carlo (MLMCMC) method. In comparison to the case of a forward log-normal elliptic equation, proving error estimate for the MLMCMC method is technically more complicated, as the available result on the error of the Fast Matching Method only holds when the grid size is not more than a threshold, which is not uniform for all the realizations of the log-normal slowness. Using the heap sort procedure for the Fast Marching Method, our MLMCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity, which is equivalent to that of the forward solver. This reduces the computation complexity drastically, in comparison to the plain Monte Carlo method where a large number of realizations of the forward equation are solved with equal high accuracy. Numerical examples confirm the theoretical results on the convergence rate of the method and the optimal complexity. This is a joint work with Zhan Fei Yeo.