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.

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.

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).

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.

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?

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.

Organoids are miniaturized representations of human organs and play an important role in precision medicine. They are widely used in applications such as drug discovery and personalized treatment development, making accurate instance segmentation essential. In this work, we present a method for organoid instance separation in bright-field images using phase congruency and persistent homology. Phase congruency provides illumination-invariant edge responses, which define a filtration over which persistent homology is computed. Focusing on H1(loop) features, maximally persistent cycles correspond to closed contours that align with organoid boundaries. These cycles are mapped back to the image domain to separate instance contours, enabling training-free delineation of touching organoids without reliance on shape priors or supervised learning. This approach demonstrates how persistent cycles can serve as a robust structural signal for resolving ambiguous boundaries under varying imaging conditions.

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.

(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).

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 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.

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.