In this talk I will discuss some definitions of exponential mixing and other rates of mixing and discuss some of its consequences.
We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.
Minimax optimization, especially in its general nonconvex formulation, has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs) and adversarial training. It brings a series of unique challenges in addition to those that already persist in nonconvex minimization problems. This talk will cover a set of new phenomena, open problems, and recent results in this emerging field.
The unique games conjecture gives a very strong PCP theorem, which, if true, leads to a clean understanding of a broad family of approximation problems. We will describe recent progress on the conjecture and how certain type of expansion and hypercontractivity of the Grassmannian complex plays a key role.
High dimensional expansion generalizes edge and spectral expansion in graphs to higher dimensional hypergraphs or simplicial complexes. Unlike for graphs, it is exceptionally rare for a high dimensional complex to be both sparse and expanding. The only known such expanders are number-theoretic or group-theoretic.