(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

In this talk, we present Peppy, an AI-assisted workflow for discovering tight, analytic convergence proofs for optimization algorithms. While existing automated tools can generate numerical convergence bounds for first-order methods, translating these numerical certificates into analytic Lyapunov proofs independent of iteration number remains a complex manual task. Peppy bridges this gap by combining structured, verifiable blocks with AI agents. Case studies on first-order methods demonstrate its ability to support a rigorous, practical, and reproducible paradigm for AI-assisted theorem synthesis in optimization.

We will discuss the proof of stability for the Schwarzschild black hole as a vacuum solution to the spherically symmetric Einstein-Vlasov system. We will focus on the regime where the characteristic curves of the Vlasov equation remain bounded, enabling the phase mixing mechanism to operate effectively within dynamical action-angle coordinates. This framework allows us to establish the polynomial decay of the time derivative of energy-momentum tensor and metric coefficients in time, up to a lifespan of $T_B = \epsilon^{-1}(\log(1/\epsilon))^2$. In this talk, we will review the foundational ideas of phase mixing for the gravitational Vlasov-Poisson system [1], and then discuss the new challenges introduced by the curved spacetime setting, including gauge choice and the monotonicity of the period function. ([1] Chaturvedi, S., and Luk, J. Linear and nonlinear phase mixing for the gravitational Vlasov-Poisson system under an external Kepler potential. To appear in Arch. Rat. Mech. Anal., 2026.)

Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. Multi-dimensional coagulation equations have been used in recent years in order to include information about the system of particles which cannot be otherwise incorporated. Depending on the model, we can describe the evolution of the shape, chemical composition or position in space of clusters. In this talk, we focus on a model that is inhomogeneous in space and contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass-conserving solutions for a class of coagulation rates for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs. This is based on a joint work with B. Niethammer and J. J. L. Velázquez.

New results on boundary of fine curve graphs Joint with Katie Mann and Frédéric Le Roux

A celebrated theorem of Mark Green from the 1980s describes the syzygies of a curve embedded by a line bundle of sufficiently large degree: if $L$ is a line bundle on a smooth projective curve $C$ of genus $g$ with $\deg(L)\geq 2g+1+p$, then $L$ satisfies property $N_p$; equivalently, the minimal free resolution of its homogeneous ideal is linear for the first $p$ steps. Since then, many efforts have been made to find analogues of Green’s theorem in broader settings. Two major directions are the study of linear syzygies of adjoint line bundles on higher-dimensional varieties, as predicted by syzygetic Fujita conjectures, and the study of higher-weight syzygies, encoded by properties $N_{d,p}$. In joint work with Wenbo Niu, we address both directions by studying the syzygies of tautological line bundle on symmetric products of curves. Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$, and let $T_{k+1,L}$ be the tautological line bundle on the symmetric product $C_{k+1}$, obtained by descent from $L^{\boxtimes (k+1)}$ on $C^{k+1}$. We prove that for each integer $d$ with $0\leq d\leq k$, if $\deg(L)\geq dg+2g+1+p$, then $T_{k+1,L}$ satisfies property $N_{k+2-d,p}$. This yields a family of sharp results on higher syzygies of higher weight for symmetric products of curves in arbitrary dimension, and in particular recovers Green’s classical theorem for curves. We also prove a sharp numerical criterion for the nefness of certain divisors on $C_{k+1}$. Taken together, these results provide strong evidence for syzygetic Fujita conjecture on property $N_p$ adjoint linear series and may be viewed as a natural higher-dimensional analogue of Green’s theorem.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

Following the foundational work of Fargues--Scholze, the categorical local Langlands correspondence was recently proven by Hansen--Mann for many classical groups. A natural question is to understand how it interacts with global Langlands, namely the cohomology of Shimura varieties. The main geometric input for answering such questions is the Igusa stack, an object constructed by Zhang in 2023 for PEL type Shimura varieties and extended to all abelian type cases in joint work with Daniels, van Hoften, and Zhang. I will discuss how arguments of Koshikawa can be used to prove strong local-global compatibility statements.

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카이스트 동문 김동률 박사와 동명이인입니다.

This final talk illustrates the scope and utility of the auxiliary-space viewpoint through several important classes of numerical methods. The framework applies to a broad range of advanced iterative methods, including subspace correction methods, Hiptmair--Xu preconditioners, saddle point solvers, and iterative substructuring methods. Through these applications, we show how apparently different methods can be understood within a common structure: each may be interpreted as an elementary iteration on a suitably enlarged space. This perspective clarifies relationships among existing algorithms and suggests new ways to design efficient solvers. We conclude by discussing how this viewpoint may inform the development of numerical methods for problems arising in machine learning, where complexity is often shifted from the optimization procedure to the underlying representation.

This second talk develops the theoretical framework behind the interpretation of advanced iterative methods as elementary iterations on larger spaces. The main tool is an auxiliary-space framework, which recasts an iterative method for the original system as an equivalent, but more elementary, method for a suitably enlarged auxiliary system. Within this framework, many methods that appear sophisticated on the original problem can be understood as simple iterations applied to auxiliary variables. In particular, multigrid and domain decomposition methods can be viewed as Jacobi- or Gauss--Seidel-type iterations for appropriate expanded systems. This provides a rigorous way to connect advanced solvers with elementary iterative principles and clarifies the algebraic structure underlying these methods.

A central goal of scientific computing is to develop accurate and efficient solvers for scientific problems, and this goal is often pursued through sophisticated numerical methods. In modern machine learning, by contrast, the basic optimization procedure is often comparatively simple, typically gradient descent and its variants, while much of the complexity is shifted to larger models. This first talk introduces the main motivation of the talk series: to examine advanced iterative methods in scientific computing from a viewpoint inspired by this contrast. We begin with basic examples and preliminary concepts, including classical iterative methods, convergence of stationary iterations, and elementary schemes such as Jacobi, Gauss--Seidel, and Richardson iterations. These examples provide the foundation for the auxiliary-space perspective developed in the subsequent talks.

The classical Moser-Trudinger inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space H1, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints. Efforts have also been made to show similar inequalities in higher dimensions. We have improved Beckner’s inequality, the higher dimensional counterpart of Onofri’s inequality, for axially symmetric functions when the dimension n = 4, 6, 8. Numerical computations are exploited to provide rigorous proof. I will also present some new results on higher dimensional counterpart of Huber’s isoperimetric inequalities.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

Mirror Symmetry predicts that each Calabi-Yau variety X admits a mirror-dual Calabi-Yau variety Y, so that the symplectic Gromov-Witten (GW) invariants of X are computed from period integrals on Y. Mirror Symmetry inspired calculations of GW invariants have been achieved in several large families of cases yielding lots of data, yet we are still lacking a general approach. Comes in Intrinsic Mirror Symmetry (Gross-Siebert, 2022), which provides a general construction of Y from X. I will talk about an ongoing joint project with Siebert, where we show that the generating function of GW invariants of X equals the expansion of a canonical period integral on the Intrinsic Mirror Y of X. The project heavily relies on AI-assisted proofs, using the existing databasis of mirror-symmetric GW calculations.

Puncture-forgetting maps play an important role in the study of Teichmüller spaces, mapping class groups, and curves on surfaces. In my talk at KAIST last winter, I introduced the basic idea of puncture-forgetting maps for measured foliations and outlined the proof of the main result. In this talk, to keep the presentation self-contained, we will begin by reviewing the relevant definitions and the main theorems. We will then discuss recent developments in this theory and present a new application to the universal curve over Teichmüller space, viewed as the (infinite-volume) quotient of Teichmüller space by the point-pushing mapping class group. Then, we will also discuss several related questions and directions for future research. This is joint work with Jeremy Kahn.

In this talk, we discuss the kinetic theory of one-dimensional nonlinear oscillator chains, of which the most famous example is the Fermi-Pasta-Ulam-Tsingou equation.

Wall's stabilization principle suggests that exotic phenomena in dimension four in the orientable category disappear after taking connected sums with sufficiently many S2xS2. Since most known exotic pairs of closed 4-manifolds become diffeomorphic after one stabilization, a natural question was: is a single S2xS2 enough? Recently, Jianfeng Lin constructed an exotic diffeomorphism on a closed 4-manifold-a diffeomorphism topologically isotopic to the identity but not smoothly isotopic-that survives one stabilization. In this talk, we provide a relative exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations. This gives the first exotic phenomenon in the orientable category that survives two stabilizations. The obstruction to stabilization comes from equivariant Seiberg–Witten theory, together with a version of lattice homology. I will also survey some background and recent developments in equivariant gauge theory. This is joint work with Sungkyung Kang and JungHwan Park.

Using an operator-theoretic approach, we provide a unified framework for Optimal Transport (OT) between Gaussian measures on separable Hilbert spaces. This formulation allows us to fully characterize the Monge and Kantorovich problems without imposing any regularity or non-degeneracy conditions on the covariance operators. We then develop the dynamic picture, explicitly characterizing 2-Wasserstein geodesics and particle dynamics in this general setting. Extending these results to Entropic OT, we show that the optimal entropic coupling operates as a precise spectral shrinkage of the correlation operator. Time permitting, I will discuss the algorithmic advantages of this spectral perspective and present complementary viewpoints connecting these transport problems.

Functional principal component analysis (FPCA) is a fundamental tool for exploring variation in samples of random curves or surfaces. We propose a new approach to FPCA for functional data observed irregularly and sparsely over their domains, based on smoothing directly at the level of the eigenfunctions. Our formulation leads to an efficient optimization-based procedure whose computational and storage costs are comparable to those of standard multivariate PCA for regularly observed data. The method is flexible with respect to domain geometry and model class, accommodates structural constraints and penalties, and facilitates uncertainty quantification via resampling and asymptotic theory.

I will discuss how Gibbs measures concentrate and exhibit two distinct central limit theorems around multi-vortex manifold in QFT, especially in comparison with point vortices in incompressible fluids/Coulomb gases. Joint work with Martin Hairer.

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. This talk is based on the joint work with Kunlun Qi.

XGBoost is one of the most successful machine learning methods in practice, yet its theoretical foundations remain poorly understood. In particular, despite its widespread use, there is currently no rigorous characterization of the function class that XGBoost is capable of learning. In this talk, I will present a theoretical framework that addresses this question. I will introduce an infinite-dimensional function class that extends finite ensembles of bounded-depth regression trees, together with a complexity measure that generalizes the regularization penalty used by XGBoost. I will show that every minimizer of the XGBoost objective is a minimizer of an equivalent penalized regression problem over this larger function class, thereby revealing the function class that XGBoost implicitly targets. I will also discuss a smoothness-based characterization of this function class, connecting XGBoost to classical smoothness-based methods in nonparametric regression. Finally, I will present statistical guarantees showing that least squares estimation over this class achieves nearly minimax-optimal rates of convergence without suffering from the curse of dimensionality. These results provide a theoretical explanation for why XGBoost performs well in practice.

(This is a seminar talk given by an undergraduate student, Mr. Rayhyun Kim, after his individual reading course studies.) In this seminar, I want to discuss about some basic notions of homological algebra that appears in category of sheaves. Many useful functors which are not exact, and the failure of exactness often contains important information. In the sheaves category this point of view naturally leads to sheaf cohomology. Going one step further, I will also discuss how sheaf cohomology is related to higher direct images. From this perspective, I would like to explain how inverse and direct image functors, their adjunction, and the derived functors of left exact functors fit together in a natural example. TBA

(This is a seminar talk given by Mr. Joon Song, an undergraduate student, after his individual reading studies.) In many situations one meets the same phenomenon: a short exact sequence gives rise to a long exact sequence in cohomology. However, the construction of such long exact sequences differs from case to case. In particular, the construction of the long exact sequence in sheaf theory is different from that of complexes. Is there a single framework where all long exact sequence arise in the same way? This leads to distinguished triangles which generalize the short exact sequence. In this presentation, I will introduce abelian categories, resolutions, and the derived category with their basic properties, and show how we can use the derived category briefly, through derived functor.

Flagella-driven motility is one of the most prevalent locomotion strategies employed by microorganisms. Numerous species of flagellated microorganism exhibit remarkable proficiency in navigating aqueous environments while simultaneously interacting with the physical and chemical properties of their microenvironments to facilitate various biological functions. Deciphering the underlying mechanisms of locomotion presents a significant challenge, necessitating a multidisciplinary framework that integrates principles from biology, physics, engineering, and applied mathematics. In this presentation, I will introduce a suite of mathematical models developed to investigate the propulsion mechanisms of microswimmers, with a focus on the hydrodynamic complexities associated with species such as Escherichia coli, Pseudomonas putida, Campylobacter jejuni and green algae Chlamydomonas reinhardttii. These computational simulations not only demonstrate strong agreement with experimental data but also provide novel insights into the biophysical principles governing swimming motility.

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight (Sym^4, det^-1) with at most pole of order 1, and that this construction is functorial with respect to degeneration, namely the K-theory elevator for the cycle corresponds to the Siegel operator for the modular form.

Topological Data Analysis provides tools for studying the global structure of data: clusters, loops, holes, branching patterns, and other geometric features that are often difficult to capture using classical statistical methods. In this lecture, we will present, in an informal and intuitive way, examples of tools such as persistent homology, Mapper, Ball Mapper, and Euler characteristic curves and profiles.

(This is a reading seminar given by the PhD Student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.

For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory.

This talk presents recent progress in differentially private hypothesis testing, focusing on the interplay between privacy, validity, and statistical efficiency. I will discuss a framework for private permutation testing that preserves finite-sample validity and extends naturally to kernel-based procedures. These ideas yield private testing methods with strong theoretical guarantees, including optimality properties in several regimes. I will then turn to minimax results for two-sample testing under central differential privacy, which reveal a rich structure in the privacy–power trade-off. The overall message is that rigorous privacy protection can be incorporated into modern hypothesis testing without sacrificing principled statistical guarantees.

Trees generalize in (at least) three different ways, CAT(0) cube complexes which is a fine metric notion, hyperbolic spaces which is a coarse metric notion and non-Hausdorff trees which is a topological notion that arises naturally when studying Anosov flows on closed three manifolds. I will discuss analogies between the three contexts with focus on recent joint work with Barthelm’e, Mann and Paulet where we build a counterpart of Hagen’s contact graph for bifoliated planes and use it to derive several genericity results for groups acting on bifoliated planes by foliation-preserving homeomorphisms.

(This is a seminar talk given by an undergraduate student, Mr. Dohyun Kwon, reporting on his reading course studies.) This talk aims to provide a geometric analysis of hyperelliptic curves within the framework of Riemann surface theory. In the beginning, the fundamental tools in Riemann surface theory, such as the Riemann-Roch theorem, Serre duality and the Hurwitz formula will be introduced briefly. With these tools, we will first compute the genus of hyperelliptic curves and provide an explicit basis for the space of holomorphic 1-forms. Then, we will focus on the relation between the canonical map and hyperelliptic curves. The main goal is to examine the canonical map of the compact Riemann surface for cases of genus 2 or greater, and understand why it characterizes the hyperelliptic case when the canonical map fails to be an embedding. In particular, we will explicitly observe the canonical map in genus 2 and 3 cases.

Inverse scattering problems aim to identify the geometric and material properties of scatterers from measured data. Despite their wide range of applications, these problems are inherently nonlinear and ill-posed. In this talk, we introduce the basics of inverse scattering problems, with a particular focus on acoustic obstacle scattering governed by the Helmholtz equation. After a brief overview of inverse problems, we discuss several types of inverse scattering problems and the main challenges arising in inverse obstacle scattering. We then study some commonly used reconstruction methods and approaches for these problems. In particular, we present layer potential theory, which serves as a fundamental tool in the analytical study of inverse problems.