학과 세미나 및 콜로퀴엄
(This is a reading seminar for two graduate students.) This talk studies the birational geometry of fibered surfaces, which are integral, projective, flat schemes of dimension 2 over a Dedekind scheme. In contrast to smooth projective curves, birational equivalence for surfaces does not imply isomorphism, which leads to the problem of understanding and selecting canonical representatives within a birational class.
We first introduce basic tools for birational surface theory, including blowing-ups, contraction, and desingularization. We then explain how intersection theory on regular surfaces is used to analyze these operations and to identify exceptional curves. This perspective naturally leads to minimal surfaces and to applications of contraction criteria in the construction of canonical models.
Asymptotically Locally Flat (ALF) Ricci-flat metrics are expected to model certain long-time singularities in four-dimensional Ricci flow, so understanding their stability is essential. In this talk, I will discuss that conformally Kähler, non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. Our work establishes three key tools in this setting: a Fredholm theory for the Laplacian on ALF metrics, the preservation of the ALF structure along the Ricci flow, and an extension of Perelman’s λ-functional to ALF metrics. This is joint work with Tristan Ozuch.
Puncture–forgetting maps have been studied for a variety of objects, including Teichmüller spaces, mapping class groups, and closed curves. In this talk, we discuss several ideas of forgetting punctures in measured foliations, which give rise to upper semi-continuous maps between spaces of measured foliations.
In the proof, we introduce complexes of pre-homotopic multicurves and show that they are hyperbolic CAT(0) cube complexes. We then study the action of point-pushing mapping classes on these complexes. This theory is motivated by applications to Teichmüller geodesics and the dynamics of post-critically finite rational maps. This is joint work with Jeremy Kahn.
A fundamental problem in low-dimensional topology is to
find the minimal genus of embedded surfaces in a 3-manifold or 4-manifold,
in a given homology class. Ni and Wu solved a 3-dimensional minimal
genus problem for rationally null-homologous knots. In this talk, we will
discuss an analogous 4-dimensional minimal genus problem for rationally
null-homologous knots. This is a joint work with Zhongtao Wu.
Interactive theorem provers (ITPs) are becoming crucial in mathematics, supporting the development of fully verified proofs and mathematically rigorous infrastructure.
Their importance comes not only from catching mistakes that humans overlook, but also from enabling large-scale formalization projects that would be impossible to manage manually.
In this talk, I will give an accessible overview of formalizing mathematics in Lean4 and outline how its underlying theory and tooling support modern proof development.
In advance, I will present several mathematical projects that showcase the strengths of ITPs.
Theorem proving with large language models has recently gained substantial attention as a direction for developing trustworthy and verifiable AI reasoning. This talk gives a brief introduction to the proof assistant Lean, which provides the verification layer for formal theorem proving, and reviews recent trends in LLM-based theorem proving. These methods rely increasingly on natural-language descriptions of mathematical arguments, which brings renewed attention to the relationship between informal mathematical language and formal representations. I will discuss the gap between natural-language mathematical expressions and the formal representations, and explain why the high level of abstraction in existing formal structures can be a limitation for natural-language–driven methods. This motivates the need for more human-oriented forms of formalization. As one approach, I will present a rule-based method for autoformalization. This talk is based on the work done at the 2025 KIAS winter school on Mathematics and AI.
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 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.
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.
심사위원장: 백형렬, 심사위원: 박진성(KIAS), 김현규(KIAS), 박정환, 최서영.
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.