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.

Classical variational approach of maximizing the kinetic energy with various constraints provides vortex stability in several special cases, but in general this approach fails when the vorticity is concentrated at several points in the fluid domain. This is simply because such configurations are not local kinetic energy maximizers, even when we restrict the admissible class using all the other coercive conserved quantities of fluid motion. In this talk, we present several results on the stability of multi-vortex solutions, obtained by combining classical variational approach with dynamical bootstrapping schemes. We focus on the case of multiple Lamb dipoles weakly interacting with each other. This is based on joint works with Ken Abe, Kyudong Choi, and Yao Yao.

Abstract: In this talk, we consider the second-order quasilinear degenerate elliptic equation whose dominant part has the form $(2x - au_x)u_{xx} + bu_{yy} - u_x = 0$, where $a$ and $b$ are positive constants. We first introduce the physical situation that motivates the present analysis in a very brief manner, and then discuss mathematical difficulties involved in the analysis of the problem. The main part of this talk focuses on methods to overcome those difficulties, such as vanishing viscosity approximation and parabolic scaling. - Reference: [1] Chen, G.-Q. and Feldman, M. (2010). Global solutions to shock reflection by large-angle wedges, Ann. of Math. 171: 1019–1134. *Main reference [2] Bae, M., Chen, G.-Q. and Feldman, M. (2009). Regularity of solutions to regular shock reflection for potential flow, Invent. Math. 175: 505–543.

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.

Generative modeling has emerged as a powerful tool for molecular design and structure prediction, offering the ability for molecular discovery. However, challenges such as synthetic feasibility, novelty, diversity of generated molecules, and generalization remain critical for real-world applications, particularly in drug discovery. In this presentation, we provide a comprehensive overview of state-of-the-art generative models, including graph-based methods, generative flow networks, and diffusion methods, all aimed at addressing these challenges. First, we focus on strategies that improve molecular structural optimzation using geometric deep learning methods. Second, we show how generative modeling can be applied to design novel molecules with desired properties such as drug potency, binding affinities to a specific target protein. Third, we will consider synthesizability of generated molecules by incorporating chemical reaction templates, enabling the generation of novel compounds that are not only drug-like but also synthetically accessible. Moreover, advanced sampling techniques and adaptive learning allow these models to explore diverse molecular structures, including those composed of previously unseen building blocks, while optimizing for key properties such as binding affinity and drug-likeness. Through case studies in drug design and broader molecular applications, we demonstrate how these generative modeling can help accelerate molecular discovery, offering a pathway to more practical and innovative solutions across diverse chemistry domains.

Reinforcement learning (RL) focuses on achieving efficient learning and optimal decision-making from available trials. Recent breakthroughs such as ChatGPT, robotics, autonomous driving, and recommendation systems owe much to advancements in reinforcement learning. Reinforcement learning is often framed as the ‘exploration vs. exploitation’ dilemma. In each trial, the learning agent must decide between ‘exploring’ to discover new possible outcomes or ‘exploiting’ by choosing familiar actions that yield reliable rewards. Effective exploration is crucial to enabling the agent to understand its environment with fewer trials, thereby saving trial opportunities for exploitation, which ultimately maximizes cumulative reward. In this talk, we will delve into a deeper understanding of efficient exploration through two RL variants: the bandit problem and best-arm identification. Throughout the series of new results, we will discuss how to address the two key aspects of exploration research: the design of experiments and the stopping condition for exploration.

In quantum many-body systems, complexity arises not from randomness alone, but from the rich interplay of interactions and entanglement. These systems often exhibit emergent behavior, where global coherence emerges in ways that are absent in single- or few-body descriptions. While most many-body systems are governed by short-range interactions, we explore how strong correlations can arise even between spatially distant degrees of freedom by introducing the concept of multifractality in wave functions. In this talk, we present new perspectives on how quantum many-body systems can exhibit long-range and effectively all-to-all coupling, despite being governed by local Hamiltonians. We highlight key examples where multifractal wave functions naturally appear, such as in quasiperiodic systems, systems with mobility edges, and near localization–delocalization transitions. These critical states possess spatially inhomogeneous amplitude distributions that mediate strong, non-local entanglement and random long-distance couplings, offering a novel route toward engineering globally connected quantum systems.

Stochastic Volterra equations (SVEs for short) are useful to model dynamics with hereditary properties, memory effects and roughness of the path, which cannot be described by standard SDEs. However, the analysis of SVEs is much more difficult than the SDEs case since the solutions are no longer Markovian or semimartingales in general. In this talk, we introduce an infinite dimensional framework which captures Markov and semimartingale structures behind SVEs. We show that an SVE can be “lifted” to an infinite dimensional stochastic evolution equation (SEE for short) and that the solution of the SEE becomes a Markov process on a Hilbert space. Furthermore, we establish asymptotic properties and well-posedness results for lifted SEEs, and then apply them to the original SVEs.

We consider a class of linear estimates for evolution PDEs on the Euclidean space, called Strichartz estimate. Strichartz estimates are well-established for fundamental linear PDEs, such as heat and wave equations. As a simple model of such, we consider the Schrödinger example, introducing classical Strichartz estimates with proofs. Reference Terence Tao, Nonlinear dispersive equations: local and global analysis, Chapter 2.3

Macrophages play an essential role in wound healing due to their dynamic nature and functional plasticity, exhibiting highly heterogeneous morpho-kinetic behaviors depending on their activation states. However, quantitative analysis of macrophage behavior in in vivo settings remains limited, largely due to the complexity of their diverse morphologies and motility patterns over time. In this study, we present an analytic workflow to investigate macrophage dynamics in zebrafish. By computing a comprehensive set of morpho-kinetic features, we reveal the clear distinctions between M1 (pro-inflammatory) and M2 (anti-inflammatory) macrophages in terms of shape elongation, directional movement, and random-like motion. Based on these features, we classify macrophages in the transition period into M1-like and M2-like groups. We compare and analyze their behaviors, which allows us to estimate the timing of the phenotypic switch. In addition, we analyze the behavior of macrophages that do not express Tumor Necrosis Factor (TNF) and are not stimulated by wound signaling. In summary, this study provides a quantitative analysis of macrophage behavior during wound healing and suggests distinct behavioral landscapes across different macrophage activation states.

Given any smooth 4-manifold bounding a Seifert manifold, the Seifert action on its boundary can be used to define their boundary Dehn twists. If the given 4-manifold is simply-connected, this Dehn twist is always topologically isotopic to the identity, but usually not smoothly isotopic, making it a very nice potential example of exotic diffeomorphisms. In this talk, we prove that for any Brieskorn homology sphere bounding a positive-definite 4-manifold, their boundary Dehn twists are always infinite-order exotic. This is a joint work with JungHwan Park and Masaki Taniguchi.

In this talk I will discuss front propagation in the KPP type reaction-diffusion equations with spatially periodic coefficients. Since the pioneering work of Kolmogorov--Petrovsky--Piscounov and Fisher in 1937, front propagations in KPP type reaction-diffusion equations have been studied extensively. Starting around 1950's, KPP type equations have played an important role in mathematical ecology, in particular, in the study of biological invasions in a given habitat. What is particularly important is to estimate the speed of propagating fronts. In the spatially homogeneous case, there is a simple formula for the speed, which was given in the work of KPP and Fisher in 1937. However, if the coefficients are spatially periodic, estimating the front speed is much more difficult, and it involves the principal eigenvalue of a certain operator that is not self-adjoint. In this talk, I will mainly focus on the one-dimensional problem and give an overview of the past research on this theme starting around 1980's. I will also present a work of mine on KPP type equations in 2D in periodically stratified media.

In recent years, the behavior of solution fronts of reaction-diffusion equations in the presence of obstacles has attracted attention among many researchers. Of particular interest is the case where the equation has a bistable nonlinearity. In this talk, I will consider the case where the obstacle is a wall of infinite span with many holes and discuss whether the front can pass through the wall and continue to propagate (“propagation”) or is blocked by the wall (“blocking”). The answer depends largely on the size and the geometric configuration of the holes. This problem has led to a variety of interesting mathematical questions that are far richer than we had originally anticipated. Many questions still remain open. This is joint work with Henri Berestycki and François Hamel.