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?

Differential privacy provides a principled framework for protecting sensitive data, yet existing private hypothesis testing methods often suffer from impracticality or significant power loss. We propose a general framework for differentially private permutation tests that extends classical non-private permutation tests while strictly preserving finite-sample validity. We characterize conditions for consistency and non-asymptotic uniform power, highlighting the role of the test statistic. As a concrete instantiation, we develop differentially private kernel tests for two-sample and independence testing. These methods are simple to implement, applicable to diverse data types, and achieve minimax-optimal power across privacy regimes.