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.

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.

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.

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.