In this talk, we present a unified learning framework for inverse problems governed by wave and elliptic partial differential equations (PDEs), where the forward operator is unknown and no ground-truth interior data is available. The key idea is to embed a physics-based forward solver directly into the training loop, enabling learning from boundary measurement data alone. This removes the need for supervised training pairs and allows simultaneous recovery of unknown quantities. The framework is applied to three representative problems: (1) a nonlinear photoacoustic model where the sound speed depends on the unknown initial pressure, (2) a wave inverse problem with spatially varying unknown sound speed, connected to Calderón-type structures, (3) an elliptic inverse problem based on the Dirichlet-to-Neumann map, where theoretical uniqueness is available. Numerical results demonstrate robustness under noise. This work suggests a general paradigm for solving PDE inverse problems via physics-informed self-supervised learning.

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(세미나 ZOOM 링크: https://cau.zoom.us/j/88050404196 // 회의 ID: 880 5040 4196)

(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 will look at an analogue theorem of the classical Erdős-Pósa Theorem. We prove a $GF(q)$-representable matroid analogue of Robertson and Seymour’s theorem that planar graphs have an Erdős-Pósa property. Given a matroid $N$, we prove that for every matroid $M$ with bounded branch width, $M$ either contains $r$ skew copies of $N$, or there is a small perturbation of $M$ that doesn’t contain $N$ as a minor. This is joint work with James Davies and Meike Hatzel.

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).

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.

(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.

This talk deals with induced minor obstructions to treewidth. The natural setup for this problem is to consider the class of graphs excluding some planar graph, and some complete bipartite graph as induced minors, and some complete graph as a subgraph. Unfortunately, such classes still contain graphs of arbitrarily large treewidth. Moreover, a result of Alecu, Bonnet, Bureo Villafana and Trotignon and its extensions suggests that there is no elegant characterization of families of bounded treewidth in terms of induced obstructions. On the other hand, it is conjectured that graphs in the classes as above have treewidth bounded by a poly-logarithmic function of their number of vertices. If true, this will imply the existence of quasi-polynomial time algorithms for a host of problems on such  class that are NP-complete in the general setting. While this conjecture remains open, in joint work with Julien Codsi, David Fischer and Daniel Lokshtanov, we were able to prove the existence of a sub-polynomial bound on treewidth in terms of the number of vertices. This in terms leads to sub-exponential algorithmic behavior. In this talk we will discuss some ideas of the proof, and, if time permits, some results in the more general setting when the bound on the clique size is removed.

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.

We study a problem related to submodular function optimization and the exact matching problem for which we show a rather peculiar status: its natural LP-relaxation can have fractional optimal vertices, but there is always also an optimal integral vertex, which we can also compute in polynomial time. More specifically, we consider the multiplicative assignment problem with upgrades in which we are given a set of customers and suppliers and we seek to assign each customer to a different supplier. Each customer has a demand and each supplier has a regular and an upgraded cost for each unit demand provided to the respective assigned client. Our goal is to upgrade at most k suppliers and to compute an assignment in order to minimize the total resulting cost. This can be cast as the problem to compute an optimal matching in a bipartite graph with the additional constraint that we must select k edges from a certain group of edges, similar to selecting k red edges in the exact matching problem. Also, selecting the suppliers to be upgraded corresponds to maximizing a submodular set function under a cardinality constraint. Our result yields an efficient LP-based algorithm to solve our problem optimally. In addition, we provide also a purely strongly polynomial-time algorithm for it. As an application, we obtain exact algorithms for the upgrading variant of the problem to schedule jobs on identical or uniformly related machines in order to minimize their sum of completion times, i.e., where we may upgrade up to k jobs to reduce their respective processing times. This is joint work with Alexander Armbruster, Lars Rohwedder, Andreas Wiese, and Ruilong Zhang.

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.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. We now explain the statement of the equivariant BSD conjecture.

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 and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets (which we rephrase as modules), and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts. This is joint work with Mathilde Bouvel, Valentin Féray, and Florent Koechlin.

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.

In four-dimensional topology, the smooth and topological categories have significant differences, called exotica. For instance, there are many smooth manifolds that are homeomorphic but not diffeomorphic. Moreover, there are many smoothly embedded surfaces in a 4-manifold that are isotopic topologically, but not smoothly. In this talk, we explore how exotic phenomena can be constructed and detected via knots.

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?

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the next two--three talks, I will explain the backgrounds on K_1 and relative K_0 of group rings for finite groups over local/global fields of characteristic 0 and their orders.

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.

We give an induced counterpart of the Forest Minor theorem: for any t ≥ 2, the $K_{t,t}$-subgraph-free H-induced-minor-free graphs have bounded pathwidth if and only if H belongs to a class F of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs H for which every weakly sparse H-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series. This is a joint work with É. Bonnet and R. Hickingbotham.

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.

In this talk we discuss recent work to that establishes that the bounds of the Vital Linkage Function is single-exponential. This has immediate impacts on the complexity of the k-Disjoint Paths Problem, Minor Checking, and more generally, the Folio-Problem. We in fact prove something even stronger: It turns out that it is not in fact the number of terminals (or more generally vertices) that matters in these problems, but rather their structure within the graph. Concretely, we show that the Vital Linkage Function is single-exponential only in the bidimensionality of the terminals, whilst the number of terminals contributes only polynomially. A direct consequence of this is an algorithm for the k-Disjoint Paths Problem running in $f(k)n^2$-time, where f(k) is singly exponential in k and doubly exponential in the bidimensionality of k. This derives directly from an algorithm for the Folio-problem we give that has an analogous runtime. Notably these are the first algorithms for these problems in which the function f is explicit. In particular, we give the first explicit bounds for the Vital Linkage Function. Joint work with Dario Cavallaro, Stephan Kreutzer, Dimitrios Thilikos, and Sebastian Wiederrecht.