2024-2025 Mathematics Colloquium

Upcoming Talks:

Title: The Connections Between Discrete Geometric Mechanics, Information Geometry, Accelerated Optimization and Machine Learning
Abstract: Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.

Title: Real and complex structure, and beyond
Abstract: Real structure occurs naturally and crucially in very many areas of mathematics. Together with collaborators we have recently developed the theory of real operator spaces and (possibly nonselfadjoint) real operator algebras to a somewhat mature level. We begin by describing this theory and how standard constructions interact with the complexification. We characterize real structure in complex operator spaces, and characterize some of the most important objects in the subject. There are some hard questions, some of which we have solved very recently with Neal, Peralta and Su. Others involve duality. Generalizing further, joint work with Mehrdad Kalantar gives a novel framework that contains the operator space complexification, as well as the less-studied quaternification, as special cases. It also may be viewed as the appropriate variant of Frobenius and Mackey's induced representations for the category of operator spaces.

Click here to see a list of past colloquium talks.