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.

(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.

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.

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.

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.

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.

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.

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.

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.

There are many conjectures in the theory of algebraic cycles. However, apart from the case of number fields the results are largely in the case of "modular varieties"—namely, varieties arising from the theory of automorphic forms. In this talk we will survey some of the results and discuss some ideas linking modular forms and higher Chow cycles. (This talk helps prepare the audience for the up-coming talks by Shouhei Ma.)

Inverse problems, broadly defined as the task of estimating unknown input parameters of mathematical models from observed data, arise across a wide range of scientific and engineering disciplines. This talk presents deep generative approaches to solving such problems within a Bayesian inference framework, covering two complementary settings distinguished by whether the likelihood function is tractable. In the first half, we address the tractable likelihood setting, where Markov chain Monte Carlo (MCMC) has long served as the standard inference tool but suffers from slow mixing and high computational cost. We propose replacing MCMC with normalizing flow-based variational inference, which leverages GPU computing for substantially faster posterior approximation. We show, however, that naïve application of normalizing flows is insufficient: accurate posterior representation requires careful architectural choices—including mixture-based distributions to handle multimodality and tail-adaptive transformations to capture heavy-tailed behavior—as well as principled training strategies such as weight-adjusted fine-tuning to mitigate the mode-seeking bias of reverse KL divergence. In the second half, we turn to the intractable likelihood setting, where complex, high-dimensional, or semi-continuous data structures (such as spatial fields with excessive zeros) preclude explicit likelihood evaluation. Here, we employ denoising diffusion probabilistic models (DDPM) as emulators of the computer model output, and combine them with approximate Bayesian computation (ABC) in which a Siamese network extracts discriminative features to compute data-adaptive acceptance probabilities. Together, these methods extend the reach of principled Bayesian calibration to a broader class of scientifically important models.

The celebrated theorem of Komlos (1967) establishes L^1-boundedness as a sufficient condition for a sequence of measurable functions on a probability space to contain a subsequence along which, and along whose every further subsequence (“hereditarily”), the Cesaro averages converge to a “randomized mean” in the spirit of the Strong law of Large Numbers. We provide conditions not only sufficient, but also necessary, for this result, as well as for the hereditary analogues of the Weak Law of Large Numbers, of the Hsu-Robbins-Erdos Law of Large Numbers, and of the Law of the Iterated Logarithm. Joint work with I. Berkes (Budapest) and W. Schachermayer (Vienna).

(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.

We develop a mathematical theory for finance based on the following “viability” principle: That it should not be possible to fund a non-trivial liability starting with arbitrarily small initial capital. In the context of continuous asset prices modeled by semimartingales, we show that proscribing such egregious forms of what is commonly called “arbitrage” (but allowing for the possibility that one portfolio might outperform another), turns out to be equivalent to any one of the following conditions: (i) a portfolio with the local martingale numeraire property exists, (ii) a growth-optimal portfolio exists, (iii) a portfolio with the log-optimality property exists, (iv) a local martingale deflator exists, (v) the market has locally finite maximal growth. We assign precise meaning to these terms, and show that the above equivalent conditions can be formulated entirely, in fact very simply, in terms of the local characteristics (the drifts and covariations) of the underlying asset prices. Full-fledged theories for hedging and for portfolio/consumption optimization can then be developed, as can the important notion of “market completeness”. Book with the same title with C. Kardaras (London).

We provide a detailed, probabilistic interpretation for the variational characterization of conservative diffusions as entropic flows of steepest descent. Jordan, Kinderlehrer, and Otto showed in 1998, via a numerical scheme, that for diffusions of Langevin-Smoluchowski type the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features; these are valid along almost every trajectory of the motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by “aggregating”, i.e., taking expectations. As a bonus we obtain a version of the HWI inequality of Otto and Villani, relating relative entropy, Fisher information, and Wasserstein distance. Joint work with W. Schachermayer, B. Tschiderer and J. Maas (Vienna); we report also on related work of L.Yeung and D. Kim.

We introduce Orthogonal Möbius Inversion, a concept analogous to Möbius inversion on finite posets, applicable to order-preserving functions from a finite poset to the Grassmannian $\mathrm{Gr}(V)$ of an inner product space $V$. This notion relies critically on the inner product structure on $V$, enabling it to capture finer information than standard integer-valued persistence diagrams. Orthogonal inversion is a special case of the broader concept of orthomodular inversion, in which the target is an arbitrary orthomodular lattice (which we also identify). We apply orthogonal inversion to construct a “nonnegative” persistence diagram for any given multiparameter filtration $F$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$, by applying it to the birth–death spaces of $F$. Analogously to classical one-parameter persistence diagrams, these multiparameter Grassmannian persistence diagrams admit a straightforward interpretation. Specifically, for each segment $(b,d)\in \mathrm{Seg}(P)$: the Grassmannian persistence diagram canonically assigns a vector subspace of degree-$*$ cycles in $K$ that are born at $b$ and become boundaries at $d$, and this assignment is exhaustive at the homology level. This is joint work with Aziz Gülen and Zhengchao Wan.

Accurate segmentation of organoids in bright-field microscopy is essential for drug screening and personalized medicine, yet separating touching instances remains challenging. We present a training-free method that combines phase congruency and persistent homology to delineate touching instances without shape priors or learned representations. By utilizing maximally persistent H₁ cycles with their birth and death simplices, our method remains robust to common brightfield imaging artifacts while producing interpretable separation of contours that align with true organoid boundaries.

Ergodic theory emerged from the attempt to understand the long-term behavior of dynamical systems. Instead of tracking individual trajectories, the theory seeks to describe almost sure behavior by associating "invariant measures" with the system. This talk will provide a historical survey of research aimed at understanding these measures, with a particular focus on the fundamental question: how many invariant measures can a system admit?

Conformal Heat Flow is a pair of evolution equations of a map and a metric on the domain. This new type of flow can be understood as a harmonic map flow with metric evolution which is in conformal direction. In this talk, I will present basic idea of conformal heat flow of harmonic maps in 2-dimensional domain where the metric evolves almost proportional to the energy density. As its variant, I also introduce the system with Yamabe flow, which is in higher dimensions and the conformal factor satisfies a kind of Yamabe flow. This is a joint work with Hyo Seok Jang and Ki-Ahm Lee.

The Korteweg-de Vries-Burgers (KdVB) equation is a fundamental model capturing the interplay of nonlinearity, viscosity (dissipation), and dispersion, with broad physical relevance. It is well known that the KdVB equation admits traveling wave solutions, called viscous-dispersive shocks. These shock profiles are monotone in the viscosity-dominant regime, while they exhibit infinitely many oscillations when dispersion dominates. In this talk, we study the stability of such viscous-dispersive shocks, focusing on an L2 contraction property under arbitrarily large perturbations, up to a time-dependent shift. We begin with viscous shocks of the viscous Burgers equation (i.e., the KdVB equation without dispersion), then treat monotone viscous-dispersive shocks and finally address oscillatory shocks. We also present detailed structural properties of the oscillatory profiles. This is joint work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).

In this talk, for a finite group G, we consider G-metric spaces: metric spaces equipped with an isometric G-action. We introduce a G-equivariant Gromov–Hausdorff distance for compact G-metric spaces and derive lower bounds using equivariant persistent invariants and related constructions in equivariant topology. To analyze and compare these bounds, we further develop two complementary G-equivariant distances—the homotopy-type and interleaving distances—and establish stability relations linking them to the G-Gromov–Hausdorff distance. As applications: (1) we analyze how the G-actions descend to and enrich persistence modules and obtain lower bounds via the G-interleaving distance, comparing these to those induced by equivariant topology; (2) we prove equivariant rigidity and finiteness theorems; (3) we obtain sharp bounds on the Gromov–Hausdorff distance between spheres; and (4) we obtain a G-equivariant quantitative Borsuk–Ulam theorem. This is joint work with Sunhyuk Lim.

(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.

Generative models have made impressive progress across machine learning, yet we still lack a clear understanding of why some training methods are reliable while others fail. In this talk, I highlight several mathematical viewpoints—centered around optimal transport—that offer a unifying way to think about generative modeling and help relate major approaches such as diffusion models and GANs. I will then focus on a concrete issue that arises when we try to learn “transport maps” from data: popular methods can sometimes converge to misleading solutions, especially when the data have low-dimensional structure. I will explain the geometric reason for this phenomenon and discuss practical remedies that make training more stable and the learned maps more faithful, along with a few examples that illustrate the impact in modern generative modeling tasks.