Over the past two decades, gradient flows in the optimal transport framework have found wide-ranging applications in PDEs, probability, and machine learning. However, the long-time behavior of Wasserstein gradient flows for genuinely nonconvex energies remains much less understood. In geometric analysis, Łojasiewicz-type inequalities provide a powerful framework for studying the asymptotic behavior of gradient flows near critical points. This theory, originating from Łojasiewicz’s work in the 1960s and Simon’s infinite-dimensional extension in 1983, also encompasses questions such as Thom’s gradient conjecture on the asymptotic direction of analytic gradient trajectories. In this talk, I will discuss a Łojasiewicz theory for nonconvex Wasserstein gradient flows. As applications, we study the long-time behavior of the Keller–Segel equation for chemotactic aggregation and simplified mean-field models of self-attention dynamics motivated by in-context learning. This is joint work in progress with Beomjun Choi (KAIST), Pei-Ken Hung (University of Illinois Urbana-Champaign), and Geuntaek Seo (POSTECH).

This talk surveys major developments in the stability analysis of nonlinear waves in fluid dynamics. We begin with classical energy methods, which provide robust global $L^2$-bounds and nonlinear stability but often do not capture detailed spatial behavior or sharp decay rates. In many problems on unbounded domains, the absence of a spectral gap also prevents the direct use of standard semigroup theory.Pointwise semigroup methods overcome this difficulty through detailed Green’s function estimates, revealing how perturbations propagate, spread, and decay. Although powerful and sharp, this approach is often technically demanding and requires substantial spectral and resolvent information.We then discuss compactified spectral methods, a more recent and unified framework in which spatial transformations, weighted spaces, and compactification convert the original problem into a spectral or coercivity problem on a bounded interval. This can reveal an effective spectral gap, or a comparable coercive structure, even when none is apparent in the original formulation.These three approaches will be illustrated through examples from fluid mechanics, with the aim of providing a conceptual roadmap for modern stability analysis in nonlinear PDE.

Cactus groups arise from the Deligne–Mumford compactification of the moduli space of stable real genus-zero curves with n+1 marked points: the pure cactus group is its fundamental group, while the full cactus group is naturally related to the corresponding quotient orbifold. They also appear as analogues of braid groups in the theory of coboundary categories. Recently, Genevois initiated a systematic study of cactus groups from the viewpoint of geometric group theory, showing that they admit geometric actions on CAT(0) cube complexes and are virtually cocompact special. In this talk, using the geometry of CAT(0) cube complexes, we will discuss the structure of subgroups of cactus groups. At first, we determine the possible maximal rank of a free abelian subgroup. Secondly, we classify which finite groups can be realized inside cactus groups. Both results resolve conjectures of Genevois. As an application of the first result, we show that cactus groups are not quasi-isometric to any right-angled Artin group. The first result is joint work with Jan Kim.

Abstract: A central challenge in constructing entropy solutions of hyperbolic conservation laws is passing to the limit in nonlinear approximation schemes. Uniform (L^\infty) bounds provide weak compactness, but this alone is insufficient to identify the limit of the nonlinear flux. This talk introduces compensated compactness as a method for overcoming this difficulty by exploiting the entropy structure of the equations. We explain how compactness of entropy production yields constraints on oscillations in approximate solutions and how, under suitable structural assumptions, these constraints lead to strong convergence. We then discuss how this approach can be used to establish the existence of entropy solutions through vanishing viscosity and relaxation approximations. References [1] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer, 2016. [2] R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), 383–420.

Treewidth is a graph parameter commonly used to quantify how “close” a graph is to a tree. Although it is a cornerstone of structural graph theory and algorithm design, it is nearly useless for algorithmic purposes in many dense graph classes. In this talk, we discuss the tree-independence number, a more versatile graph parameter that replaces the standard width measure with the stability number. We will present recent results aimed at characterizing the graph classes in which this parameter enables sub-exponential time algorithms for problems that are, in general, NP-hard.

Vertex algebras first appeared in physics as the algebraic structures of 2D CFT (conformal field theory) and were mathematically defined by Borcherds in 1986 during the resolution of the Moonshine conjecture. Since then, the theory has been developed from both mathematical and mathematical physical perspectives. In this talk, we introduce the structural characteristics and complexity of vertex algebras, as well as the interesting phenomena and research topics arising from them.

TBD

We say a graph $G$ is $(2,m)$-linked if, for every choice of $m+2$ distinct vertices $a_1,\ldots,a_m,b_1,b_2$ in $G$, there exist two vertex-disjoint connected subgraphs $A$ and $B$ of $G$ such that $\{a_1,\ldots,a_m\}\subseteq V(A)$ and $\{b_1,b_2\}\subseteq V(B)$. A related notion is $k$-linkedness: a graph is $k$-linked if, for any distinct vertices $s_1,\ldots,s_k,t_1,\ldots,t_k$, it contains $k$ pairwise vertex-disjoint paths joining $s_i$ to $t_i$ for $i=1,\ldots,k$. A fundamental result in graph theory is the characterization of $2$-linked graphs, obtained independently by Robertson and Chakravarti, Seymour, and Thomassen: $G$ does not contain disjoint paths joining $s_1$ to $t_1$ and $s_2$ to $t_2$ if and only if $G$ admits a certain “essentially planar” structure with $s_1,s_2,t_1,t_2$ appearing on the outer boundary in this cyclic order. For $k$-linkedness and $(2,m)$-linkedness with $k,m\geq 3$, comparable complete structural characterizations seem much more difficult. In this talk, I will present a full structural characterization of $(2,3)$-linked graphs. This is joint work with Robin Thomas, Shijie Xie, and Xingxing Yu.

Geometric group theory, as the name suggests, uses invariants of a geometric nature, e.g., Cayley graphs and growth, to study finitely generated groups. The study of nilpotency, especially the works of Gromov and Bass–Guivarc'h, revealed the importance of LCS ranks, numerical invariants computed from the lower central series. Unfortunately, LCS ranks are notoriously hard to compute explicitly, so they are usually replaced by more tractable approximations: the Chen ranks. Works by Papadima and Suciu showed that Chen ranks connect directly to algebro-geometric objects: Koszul modules and resonance loci. I will explain how Koszul modules and their resonance varieties can be used to prove Green's conjecture on the syzygies of canonical curves, via vanishing theorems for vector bundles.

TBA

In this second talk, we apply the channel of energy method to the nonlinear analysis of radial Type II blow-up for the three-dimensional focusing energy-critical wave equation. We discuss how radiation estimates interact with concentration and compactness to produce rigidity, and how this leads to the classification of possible blow-up profiles. Finally, we relate these ideas to the soliton resolution picture for bounded radial solutions. [References] 1. T. Duyckaerts, C. E. Kenig, F. Merle, Universality of blow-up profile for small radial Type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. 13 (2011), 533–599. 2. T. Duyckaerts, C. E. Kenig, F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge J. Math. 1 (2013), 75–144.

TBA

Let $B$ be a three-dimensional Brownian motion. We compare two regularizations of its self-intersection local time: a temporal cutoff excluding pairs of times separated by less than $\varepsilon$, and a spatial regularization obtained by replacing the Dirac mass with the heat kernel $p_\varepsilon$. We prove that the difference between these regularizations converges to a non-degenerate centered Gaussian random variable as $\varepsilon\to 0$, with convergence of the corresponding variances. The limiting variance is identified by an explicit $L^2$-calculation. We also briefly discuss how the comparison of regularization schemes arises naturally in the analysis of the three-dimensional Edwards model.

This talk explores the intricate interplay between the intrinsic geometry of complex projective varieties and the homological invariants of their defining ideals. We focus on the structure of syzygies—encoding higher order algebraic relations among defining equations—and the Castelnuovo–Mumford regularity, a key invariant governing the complexity of embedded varieties. Finally, we survey classic and recent developments regarding linear syzygies, the status of the Eisenbud–Goto conjecture, and the higher syzygies of secant varieties.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

Singularly perturbed differential equations arise in many problems in fluid mechanics and related areas, where small diffusivity or viscosity can generate sharp boundary, interior, and corner layers. These multiscale structures present significant challenges for both classical numerical methods and standard neural network approaches. We develop neural network methods that use asymptotic analysis to identify and incorporate the dominant structure of singular layers directly into the approximation. By explicitly incorporating the leading singular components into the neural network ansatz, the trainable component is used primarily to approximate the smoother part of the solution. We also consider conservative formulations based on finite volume residuals, which incorporate local flux balance into the learning framework. Numerical examples for a range of singularly perturbed problems demonstrate the accuracy and robustness of these approaches as the perturbation parameter tends to zero.

A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are subdivided into disjoint blocks of variables $X_1,\dots,X_c$, there is a natural multi-grading on $R$, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal $I$ of $R$. However, the full strength of the Bayer-Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals $I$ such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of $R/I$ in terms of almost regular sequences of the multi-graded generic initial ideal of $I$ restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.

We prove the geometric Breuil--M\'ezard conjecture for two-dimensional semi-stable representations of $G_{\mathbf{Q}_{p^2}}$ with parallel Hodge--Tate weights $(0,r)$, where $r$ is odd and lies in the Fontaine--Laffaille range. Our strategy is to prove the numerical Breuil--M\'ezard conjecture at a maximally non-split mod-$p$ representation on each irreducible component of the special fiber of the Emerton--Gee stack. Building on previous work on strongly divisible modules, we obtain explicit descriptions of the relevant semi-stable deformation rings at these points and use them to establish the numerical Breuil--M\'ezard conjecture. We then explain how these local results lead to the geometric Breuil--M\'ezard conjecture in this setting. This is joint work with Seongjae Han and Hyeonmin Kim.

Following our spring seminar exploring the foundational connections between mathematics and pharmaceutical sciences, this second lecture—“Mathematical path illuminating pharmacy: 2nd story”—dives deeper into advanced quantitative frameworks and systems-level modeling. While the first part focused on core receptor occupancy theories and basic enzyme kinetics, this presentation addresses the remaining critical challenges in modern pharmacokinetics and pharmacodynamics (PK/PD). Specifically, we examine the limitations of conventional Michaelis-Menten approximations and introduce modified modeling approaches for accurate in vitro-in vivo extrapolation (IVIVE). Furthermore, we discuss advanced Physiologically Based Pharmacokinetic (PBPK) models, the challenges of handling correlated covariates using Full Random Modeling (FREM), and the implementation of machine learning into PK/PD via Mixed Effect State Space (MESS) models. By bridging mathematical systems theory with clinical pharmacology, this session aims to provide a comprehensive perspective on precision medicine and data-driven drug development.

We introduce Morse handles on effective oriented smooth four-dimensional orbifolds and explain how invariant splittings determine their indices and ordered handle structures. Even for a fixed isotropy representation and Morse index, different splittings may give nonisomorphic ordered handles. We classify these handles up to orientation-preserving isomorphism by computing normalizer orbits of invariant subspaces for every finite subgroup of SO(4). We then define surgery on oriented 3-orbifolds by orbifold handle attachment and determine the attaching regions, belt regions, and boundary gluing maps for every ordered handle class of index 1,2, or 3. These calculations yield an orbifold version of the Lickorish–Wallace theorem. Every closed connected effective oriented smooth 3-orbifold can be transformed into the 3-sphere by a finite sequence of these surgeries. Unlike the manifold case, handles of index 2 alone do not always suffice. We prove that they suffice exactly when there are no tetrahedral, octahedral, or icosahedral vertices. Finally, we determine the minimum total number of handles of indices 1 and 3 from the numbers of these three vertex types.

Most multi-armed bandit and reinforcement learning algorithms typically rely on full optimism, using confidence bonuses large enough to upper-bound unknown values with high probability. This talk presentsquasi-optimism, a new exploration principle that permits a controlled degree of underestimation while retaining provably optimal regret guarantees. We first introduce the principle in tabular reinforcement learning, where a simple inverse-count bonus achieves minimax-optimal regret without empirical-variance estimation, and then discuss its extension to distributional and instance-dependent regret guarantees in bandits and reinforcement learning. Finally, we extend quasi-optimism to linear contextual bandits through a quadratic–linear bonus. Despite not being pointwise optimistic, the new linear bandit algorithm simultaneously achieves minimax and margin-dependent regret guarantees under a single gap-agnostic tuning, while substantially reducing unnecessary exploration in empirical evaluations. Together, these results suggest that full optimism is not necessary for exploration to be both theoretically optimal and practically effective. This talk is based on the following recent works: Oh & Lee,Linear Contextual Bandits with Quasi-Optimism, 2026. Lee & Oh, Unified Framework of Distributional Regret in Multi-Armed Bandits and Reinforcement Learning, COLT 2026.Lee & Oh,Minimax Optimal Reinforcement Learning with Quasi-Optimism, ICLR 2025.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.