This talk presents recent progress in differentially private hypothesis testing, focusing on the interplay between privacy, validity, and statistical efficiency. I will discuss a framework for private permutation testing that preserves finite-sample validity and extends naturally to kernel-based procedures. These ideas yield private testing methods with strong theoretical guarantees, including optimality properties in several regimes. I will then turn to minimax results for two-sample testing under central differential privacy, which reveal a rich structure in the privacy–power trade-off. The overall message is that rigorous privacy protection can be incorporated into modern hypothesis testing without sacrificing principled statistical guarantees.

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?

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.