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.

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).

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).

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.

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.

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.

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?

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.

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)).