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.

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.

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.

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.

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.

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).

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.

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?

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)).