Sampling from a high-dimensional distribution is a core challenge spanning statistics, operations research, applied mathematics and computer science. Diffusion-based methods have led to dramatic advances: in generative AI, they produce strikingly realistic images, video, and text, and they are finding growing use in areas including biology, drug design, materials science, and robotics.

This talk gives an overview of recent progress toward understanding the ``unreasonable effectiveness'' of diffusion-based samplers. We introduce geometric measures of the path complexity involved in traversing from initial noise to final samples. These measures adapt naturally to low-dimensional structure in the underlying distribution, and can be substantially smaller than what ambient dimension alone would suggest. Moreover, path complexity can be estimated from data, providing a way to calibrate and improve the performance of diffusion samplers.

While the talk focuses primarily on Gaussian diffusion, we also

discuss analogues for discrete diffusion, including masking and

noising-based samplers.

Based on:

M. J. Wainwright, Denoising growth complexity: Data geometry and certified schedules for diffusion sampling, arXiv:2607.26285 (2026).

M. J. Wainwright, The information geometry of product-reference discrete diffusion: Interaction growth complexity and optimal

scheduling, arXiv:2608.28949 (2026).

A wine reception in the Central Core will follow the lecture.