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 Hardy's inequality is a classical result in analysis, providing a relationship between the weighted norms of functions and their gradient norm, with several applications in Sobolev spaces, partial differential equations, and mathematical physics. In recent decades, interest has grown in fractional versions of this inequality, where the classical Laplacian is replaced by a non-local operator, such as the fractional Laplacian. In this talk, we introduce a weighted version of the classical Hardy inequality on bounded domains and extend it to the case of the fractional Laplacian. We also present some recent results related to this topic.

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 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 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.

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).

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.

In this talk, we introduce a two-dimensional semiflexible membrane model whose Hamiltonian is given by $H[\phi]=\sum_x |\nabla \phi_x|^2 +N^\lambda |\Delta \phi_x|^2$, interpolating between the discrete Gaussian free field (DGFF) and the membrane model (MM). We analyze its infinite-volume behavior and identify distinct regimes depending on the parameter $\lambda$. If $\lambda<0$, the covariance behaves similarly to that of the DGFF, while for $\lambda>2$, it resembles a rescaled MM. In the intermediate regime $0\le \lambda\le 2$, we observe a qualitatively different behavior: a nontrivial dependence on the distance between points. We further determine the leading order constant of the covariance. These results provide a unified description of the crossover from gradient-dominated to curvature-dominated behavior in this class of models.

We study the dynamics of a single vortex ring of small cross-section in the three-dimensional incompressible Euler equations. For a broad class of initial vorticities concentrated near a vortex ring, we prove that the solution remains sharply localized around a moving core for all times and propagates along its axis with the classical logarithmic speed predicted by the vortex filament conjecture. Moreover, we show that such vortex rings are dynamically unstable under arbitrarily small perturbations: suitable smooth perturbations lead to linear-in-time filamentation in the axial direction. These results provide a quantitative description of the coexistence of long-time coherence and instability mechanisms for vortex rings in inviscid flows.

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.

Separating hash families are useful combinatorial structures that generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of the universe for a $t$-perfect hash family of length $N$ over an alphabet of size ( q ). We show that $q^{2 – o(1)} < p_t(t, q) = o(q^2)$ for all  $t \ge 3$, thereby resolving an open problem raised by Blackburn et al. (2008) for certain parameter ranges. Previously, this result was known only for $t = 3$ and $t = 4$. Our approach establishes the existence of a large set of integers that avoids nontrivial solutions to a system of correlated linear equations. This is joint work with Xiande Zhang and Gennian Ge.

In this talk, we discuss the existence and stability of subsonic potential flow for the steady Euler--Poisson system in a 2-dimensional nozzle of finite length with prescribing some suitable boundary conditions. The purpose of this talk is to introduce the small-perturbation method for the steady Euler--Poisson system, which reduces to an elliptic system in our present situation. As a starting point of our discussion, we introduce the notion of background solution about which we may linearize the equations, and then point out the property of the linearized coefficients that turns out to be extremely crucial for the $$H^1$$-estimates. Next, we establish the iteration scheme and show the necessary estimates in a very brief manner, which may immediately lead us to the proof of the main existence and stability theorem. Finally, if time permits, we will take a glance at the general situation such as flows with nonzero vorticity. Main Reference: M. Bae, B. Duan, and C. Xie, Subsonic Flow for the Multidimensional Euler–Poisson System, Arch. Rational Mech. Anal. 220 (2016), 155–191.

Hirzebruch proved a beautiful inequality for complex line arrangements in CP^2, giving strong bounds on the their combinatorics. In the quest for a topological proof of this inequality, Paolo Aceto and I studied odd and even line arrangements (which I will define in the talk). We proved Hirzebruch-like inequalities for these arrangements, and drew some corollaries about configurations of lines. Time (and audience) permitting, I will also discuss some more speculative ideas and generalisations of our results.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

In this talk, we discuss the initial–boundary value problem for one-dimensional hyperbolic conservation laws on the half-line, focusing on linear systems and scalar conservation laws. We begin with a discussion of the theory of the Cauchy problem. We then turn to the half-line setting, where we introduce two formulations of boundary conditions: one based on the vanishing viscosity method and the other based on the Riemann problem. We show that these two formulations are equivalent for linear systems and scalar conservation laws. Finally, we present remarks on boundary conditions for general hyperbolic systems of conservation laws. Reference: Dubois, F., and LeFloch, P. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71, 1 (1988), 93–122.

In this talk, I will begin by presenting some classic constructions of smooth non-orientable 4-manifolds arising from certain Brieskorn homology 3-spheres. I will then explain how to construct new examples, including infinitely many smooth fake copies of *RP4#*CP2. In addition, I will describe a method for generating a collection of Brieskorn homology 3-spheres that can be realized via integer surgery on knots in the 3-sphere. This is joint work with Jae Choon Cha and Oguz Savk.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

Hamiltonian dynamics is a fundamental mathematical framework for describing classical mechanics, and it can be formulated in terms of vector fields on manifolds. While studying the three-body problem, a central example in Hamiltonian dynamics, Poincaré highlighted the crucial role of periodic orbits. This theme remains central in modern symplectic geometry. In this talk, we introduce the relationship between Hamiltonian dynamics and symplectic geometry, and survey classical and modern approaches to the study of periodic orbits. We also explain how minimal period orbits can be understood from a symplectic-geometric perspective and present an approach to establishing the existence of Birkhoff sections of minimal area using these ideas.

Curves in the complex projective planes can be viewed as PL-submanifolds. Taking this perspective allows to deduce a number of interesting results about them. The goal of these lectures is two-fold: first, I will give a topological description of some algebro-geometric objects (singularities and Milnor fibres, curves, blow-ups), and then I will talk about some topological tools one can use to study complex curves. I will focus on rational cuspidal curves (those which are homeomorphic to spheres) and line arrangements (collections of lines).

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 first three talks, I will explain the backgrounds on Selmer groups and flat cohomology.

In this seminar, we study the Vlasov–Maxwell system, a fundamental collisionless kinetic model for plasmas, posed in a three-dimensional half-space with boundaries. We begin with a brief warm-up by revisiting the one-dimensional Vlasov–Poisson system in the absence of magnetic fields, focusing on Penrose’s classical 1960 spectral criterion for linear stability and instability. We then turn to the full Vlasov–Maxwell system and discuss the major analytical difficulties introduced by electromagnetic coupling, boundary effects, and nonlinear interactions. In particular, we highlight the role of an effective gravitational force directed toward the boundary and its interplay with boundary temperature conditions. This viewpoint naturally leads us to formulate a conjectural linear instability criterion associated with boundary-induced confinement effects. Within this framework, we construct global-in-time classical solutions to the nonlinear Vlasov–Maxwell system beyond the vacuum scattering regime. Our approach combines the construction of stationary boundary equilibria with a proof of their asymptotic stability in the $L^\infty$ setting under small perturbations. This work provides a new framework for analyzing long-time plasma dynamics in bounded domains with interacting magnetic fields. To our knowledge, it yields the first construction of asymptotically stable non-vacuum steady states for the full three-dimensional nonlinear Vlasov–Maxwell system. This is joint work with Chanwoo Kim.

Wenrui Hao Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning. The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification. To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity. Finally, we demonstrate the framework’s utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

Wenrui Hao Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning. The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification. To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity. Finally, we demonstrate the framework’s utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

Wenrui Hao Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning. The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification. To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity. Finally, we demonstrate the framework’s utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

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 ability of predictions remain critical for real-world applications, particularly in drug discovery. In this presentation, we introduce an 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 will show how generative modeling can facilitate the structural prediction of protein-ligand complexes and its expansion. Second, we focus on strategies that improve the 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. Third, large language models fine-tuned with drug-related data can be used to elucidating complex relationships between drugs, proteins, and diseases. Through case studies in drug design and broader molecular applications, we demonstrate how these generative modeling can help accelerate drug discovery, offering a pathway to more practical and innovative solutions across molecular discovery domains.

Diffusion is a macroscopic phenomenon arising from the random movement of particles at the microscopic level. Fick’s law predicts uniform spreading of particles over time, while fractionation is often observed in heterogeneous environments, as in the Soret effect and Darken’s experiment. In this talk, we show that such heterogeneous diffusion can be described by a two-coefficient diffusion equation derived from particle dynamics. In particular, for persistent random walks, fractionation occurs only when both heterogeneity and anisotropy are present. We formally derive the limiting diffusion equation and present a methodology to rigorously establish convergence from a persistent discrete kinetic equation to the macroscopic diffusion equation.

This talk provides an overview of Photoacoustic Tomography (PAT) from both the imaging and mathematical perspectives, and then develops a unified integral-transform viewpoint via a generalized spherical mean operator. In PAT, a short optical pulse induces an initial acoustic pressure distribution \(f(\mathbf x)\), which evolves according to a wave equation. The measured time-dependent acoustic data on an acquisition surface \(\Gamma\) form the forward map, and the central inverse problem is to reconstruct \(f\) from boundary observations. Key mathematical issues include uniqueness, and explicit reconstruction formulas, all of which depend sensitively on the measurement geometry and observation time.

his 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 first two or three talks, I will explain the backgrounds on Selmer groups and flat cohomology.

Strong flip-flatness appears to be the analogue of uniform almost-wideness in the setting of dense classes of graphs. Almost-wideness is a notion that was central in different characterisations of nowhere dense classes of graphs, and in particular the game-theoretic one. In this talk I will present the flip-flatness notions and conjectures about the characterization of strongly flip-flat graph classes. Then, I will present a proof that strongly flip-flat classes of graphs that are weakly sparse are indeed uniformly almost-wide, making a step towards their characterisation. A consequence is a characterization of strongly flip-flat graph classes with low rank-depth colourings. This is a joint work with F. Ghasemi, J. Grange and F. Madelaine.

In this talk, we study the non-cutoff Boltzmann equation with moderately soft potentials, a classical kinetic model. The uniqueness of large weak solutions is challenging due to the nonlinearity and limited regularity. To overcome these difficulties, we utilize dilated dyadic decompositions in phase space $(v,\xi,\eta)$ to capture hypoellipticity and reduce the fractional derivative structure $(-\Delta_v)^s$ of the Boltzmann collision operator to a zeroth-order form. Within this framework, we establish the uniqueness of large-data weak solutions under the assumption of finite $L^2$--$L^r$ energy, namely that $\|\mu^{-\frac{1}{2}}(F-\mu)\|_{L^\infty_t L^{r}_{x,v}}+\|\mu^{-\frac{1}{2}}(F-\mu)\|_{L^\infty_t L^2_{x,v}}$ is bounded for some sufficiently large $r>0$. The challenges arising from large solutions are handled via a negative-order hypoelliptic estimate, which yields additional integrability in $(t,x)$.

Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation.

Any reasonable exotic phenomena in simply-connected 4-manifolds are unstable. It is an open question if there is an universal upper bound to the number of stabilizations needed. The case of 1 stabilization was proven in works of Lin and Guth-K., but whether we need more than two stabilizations has been open because it is significantly harder. In this talk, we discuss my recent proof with Park and Taniguchi that two stabilizations are indeed not enough for exotic diffeomorphisms.

A freshman can calculate that the probability of picking $k$ blue balls after sampling $n$ balls from a bin of $K$ blue balls and $N-K$ red balls is $$\frac{\dbinom{n}{k} \dbinom{N-n}{K-k}}{\dbinom{N}{K}}$$ if one samples without replacement, while it is $$\frac{\dbinom{n}{k} (\frac{K}{N})^k(\frac{N-K}{N})^{n-k}$$ if one samples with replacement. We demonstrate that comparing probabilities of sampling with replacement vs. without replacement leads to De Finetti's Theorem, the Aldous-Hoover Theorem, and even a weak form of Szemeredi's Regularity Lemma which plays a crucial role in the study of graphons. This comparison also leads to a strong version of a representation for DAG-exchangeable arrays (Jung, Lee, Staton, Yang (2021)) which generalize Aldous-Hoover arrays as well as Hierarchical Exchangeable arrays (Austin-Panchenko (2014)).