(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.

TBA

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.

TBA

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

We see how smooth versions of peg problems give rise to constructions in symplectic topology. (No knowledge of symplectic topology is required).

The planar separator theorem by Lipton and Tarjan [FOCS ’77, SIAM Journal on Applied Mathematics ’79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan’s theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC ’90, Journal of the AMS ’90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest. This is a joint work with Édouard Bonnet, Tuukka Korhonen, Hung Le, and Jason Li.

We have a leisurely discussion of the Toeplitz Square peg problem and a little of its history.

I will discuss the sine–Gordon QFT, which lies in the same universality class as the Coulomb gas and the XY model, and exhibits the BKT phase transition (Nobel prize 2016). The sine–Gordon action admits infinitely many homotopy classes. Even though the action has no energy minimizers in homotopy classes with high topological charge, I will show that the Gibbs measure nevertheless concentrates and exhibits Ornstein–Uhlenbeck fluctuations around the multi-soliton manifold. I will then discuss the geometry of this multi-soliton manifold, including how solitons are arranged, such as their expected locations and gaps. Furthermore, I will explain the asymptotics of the partition function using Gaussian multiplicative chaos. Based on joint work with Hao Shen and Philippe Sosoe. https://arxiv.org/abs/2512.23957

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.

Boolean function analysis for the hypercube $\{ 0, 1 \}^n$ is a well-developed field and has many famous results such as the FKN Theorem or Nisan-Szegedy Theorem. One easy example is the classification of Boolean degree $1$ functions: If $f$ is a real, $n$-variate affine function which is Boolean on the $n$-dimensional hypercube (that is, $f(x) \in \{ 0, 1 \}$ for $x \in \{ 0, 1 \}^n$), then $f(x) = 0$, $f(x) = 1$, $f(x) = x_i$ or $f(x) = 1 – x_i$. The same classification (essentially) holds if we restrict $\{ 0, 1\}^n$ to elements with Hamming weight $k$ if $n-k, k \geq 2$. If we replace $k$-sets of $\{ 1, \ldots, n \}$ by $k$-spaces in $V(n, q)$, the $n$-dimensional vector space over the field with $q$ elements, then suddenly even the simple question of classifying Boolean degree $1$ functions, here traditionally known as Cameron-Liebler classes, becomes seemingly hard to solve. We will discuss some results on low-degree Boolean functions in the vector space setting. Most notably, we will discuss how vector space Ramsey numbers, so extremal combinatorics, can be utilized in this finite geometrical setting.

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

https://wj-kim.com/2026-mini-togeda-workshop-at-kaist/

위상수학 및 기하학을 데이터 분석에 활용하는 데에 관심있는 젊은 한국인 연구자들을 위한 교류의 장을 만들고자 합니다. 2026.01.07~2026.01.08 이틀동안 진행됩니다. 자세한 정보는 다음 웹페이지 참조해주세요. https://wj-kim.com/2026-mini-togeda-workshop-at-kaist/

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.

In [1], Fabianski et. al. developed a simple, yet surprisingly powerful algorithmic framework to develop efficient parameterized graph algorithms. Notably they derive a simple parameterized algorithm for the dominating set problem on a variety of graph classes, including powers of nowhere dense classes and biclique-free classes. These results encompass a wide range of previously known results and often improve the best known running times. Similar results follow for the distance-r variation of dominating set and for independent set. The running time of the algorithm is closely tied to model-theoretic properties, i.e. stability and the Helly property. We build upon these results and develop a similar algorithm which only relies on the strong Helly property and does not need stability. For the dominating set problem, we get a parameterized algorithm that works (additionally to biclique-free classes and powers of nowhere dense classes) weakly gamma-closed classes while being simpler and faster than previously known results. In this talk, we introduce the basic framework, results by Fabianski et. al and connections to other areas. We discuss our new insights and possible research directions. [1] Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, Szymon Torunczyk: Progressive Algorithms for Domination and Independence. STACS 2019

Since the development of the first randomized polynomial-time algorithm for volume computation by Dyer, Frieze, and Kannan in 1989, convex-body sampling has been a central problem at the intersection of algorithms, geometry, and probability. A major milestone came in 1997, when Kannan, Lovász, and Simonovits analyzed the Ball Walk and formulated the influential KLS conjecture. This was extended to log-concave distributions by Lovász and Vempala in 2006, and further accelerated by Cousins and Vempala in 2015 through warm-start generation techniques. In this talk, I will present a gentle introduction to these milestones and how they have been streamlined and improved in the past few years. The talk is based on joint work with Santosh Vempala.

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.

We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann-Plücker functions. The equivalence is motivated by Cayley’s identities expressing principal and almost-principal minors of a skew-symmetric matrix in terms of its pfaffians. As a corollary of the new cryptomorphism, we deduce that each component of the orthogonal Grassmannian is parameterized by certain part of the Plücker coordinates. This is joint work with Changxin Ding.

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.

Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Unlike previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate our algorithm’s performance by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering.

Given a prime power $q \equiv 1 \pmod 4$, the Paley graph of order $q$ is the graph defined over $\mathbb{F}_q$ (the finite field with $q$ elements), such that two vertices are adjacent if and only if their difference is a square in $\mathbb{F}_q$. In this talk, I will present some recent progress on the clique number of Paley graphs of non-square order, the characterization of maximum cliques in Paley graphs of square order, as well as their extensions to cyclotomic graphs. In particular, I will highlight a new proof of the Van Lint–MacWilliams’ conjecture using ideas from arithmetic combinatorics.

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.

We present a dynamic data structure that maintains a tree decomposition of width at most 9k+8 of a dynamic graph with treewidth at most k, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where n is the number of vertices. The data structure also supports maintaining any “dynamic programming scheme” on the tree decomposition, providing, for example, a dynamic version of Courcelle’s theorem with $O_k(\log n)$ amortized update time; the $O_k(⋅)$ notation hides factors that depend on k. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is “downwards well-linked”, which allows us to implement local rotations and analysis similar to those for splay trees. This talk is based on arXiv:2504.02790.

Let A be a random n×n matrix with independent entries, and suppose that the entries are “uniformly anticoncentrated” (for example, A could be a uniformly random n×n matrix with ±1 entries). We prove that the permanent of A is exponentially anticoncentrated, significantly improving previous bounds of Tao and Vu. Our proof also works for the determinant, giving an alternative proof of a classical theorem of Kahn, Komlós and Szemerédi. Joint work with Zach Hunter and Lisa Sauermann.

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

TBA

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.

Neural networks have become increasingly effective for approximating solutions to partial differential equations (PDEs). This talk presents three advances that improve both accuracy and computational efficiency. First, I introduce an augmented Lagrangian formulation of the physics-informed loss that strengthens constraint enforcement and improves accuracy near domain boundaries. Second, I develop efficient architectures based on hypernetworks and graph neural networks that learn PDE solution operators with markedly small model sizes. Finally, I describe Neural-Galerkin schemes with low rank approximations for operator learning, which achieves a favorable accuracy-efficiency trade-off.

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.