The phase-field (PF) model has been applied to a wide range of problems beyond its traditional scope in materials science. In this study, we reinterpret the Allen–Cahn (AC) equation, the governing equation of the PF model, as a mathematical framework for data classification. We develop an efficient numerical scheme to solve the AC equation with a fidelity term, employing an explicit-type approach based on the convex splitting method to ensure both energy stability and computational efficiency. Comparative experiments with conventional machine learning classifiers, such as support vector machines and artificial neural networks, demonstrate that our approach achieves competitive accuracy at significantly reduced computational cost. Moreover, the proposed PDE-informed framework exhibits superior performance on unbalanced datasets, where traditional classifiers often fail to generalize effectively.

In doing mathematics, we often encounter beautiful identities and proofs, shining like a full moon in the night sky. Some of them have combinatorial flavors. This talk is an introduction to combinatorial proof methods using bijections and weight-preserving-sign-reversing involutions, with examples including Franklin's bijective proof of the Euler pentagonal number theorem, a combinatorial proof of the Cayley-Hamilton theorem, the Robinson-Schensted correspondence and recent combinatorial proofs of some identities involving secant and tangent numbers.

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

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.

Khovanov homology is a knot homology theory, introduced by M. Khovanov in 2000 as a categorification of the Jones polynomial. Equivariant versions of Khovanov homology are also known, and they play an important role in understanding the Rasmussen invariant. In this talk, I will present the results established in my joint work with M. Khovanov in September 2025 (arXiv:2509.03785): (i) an order-two symmetry inherent in equivariant Khovanov homology, (ii) the existence of a signed Shumakovitch operator, and (iii) its relationship to the Rasmussen invariant.

We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds, finding the explicit asymptotics along an equidistance foliation. We prove that the divergent terms are completely expressed in terms of the data from the Weitzenböck geometry of hyperbolic ends and the conformal boundary. For this, it is essential to extend the framework of submanifolds in Riemannian geometry to Riemann-Cartan geometry, which addresses connections with torsion. This procedure naturally introduces a complex-valued geometric quantity consisting of mean curvature and torsion 2-form, which appears in the leading coefficient of the asymptotics. We also obtain several geometric results regarding the complex-valued quantity that generalize classical minimal surface theory.

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심사위원장: 백형렬, 심사위원: 박진성(KIAS), 김현규(KIAS), 박정환, 최서영.

I will discuss recent progress on the vanishing-viscosity limit of the two-dimensional Navier–Stokes equation. Our approach is Lagrangian and probabilistic: 1. We develop a stochastic counterpart of the DiPerna–Lions theory to construct and control stochastic Lagrangian flows for the viscous dynamics. 2. We also establish a large-deviation principle that quantifies convergence to the Euler dynamics. This talk is based on joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.

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.

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.

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 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.

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 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.