I will review recent constructions of oligomorphic tensor categories generalizing Deligne's Rep(S_t). Then, I will lean into the model theoretic part of the question. Specifically, I will explain where there are no continuous families like the original Rep(S_t) and where you should look for n-parameter families, i.e., depending on n free variables. Ultimately, these questions are closely related to classes of structures in model theory.

The magical ability of a spinning top to stay upright fascinated people for many thousands of years and in many civilizations. Clay spinning tops dated to about 6,000 years ago were excavated in Iraq.  And for all these millenia another similarly counterintuitive phenomenon went undiscovered until 1908, when Stephenson observed that an upside-down pendulum becomes stable if its pivot is subjected to vertical vibrations (there is no feedback involved). Later other counterintuitive effects were discovered – and used in physical experiments. Traditionally, this subject has been treated by formal computations, sometimes quite long. This is an effective practical tool, but it does not explain what is really going on.  As an alternative to this approach, I will give a geometrical explanation of the seemingly mysterious effect in which an inverted multiple pendulum stands upright when its pivot undergoes vertical vibrations.  I will also describe the recently discovered “ponderomotive Lorentz force”: a point mass in a rapidly oscillating potential force field behaves as if it were electrically charged and in the presence of magnetic field. This is a purely mathematical effect: there is no electricity or magnetism involved; superficially this looks like the Faraday effect in which a changing electric field generates a magnetic field.

My talk will be about two open questions and a (perhaps surprising) link between them:

(1) Connes' rigidity conjecture, that in particular predicts that the von Neumann algebras of PSL_n(Z) are not isomorphic for different values of n. Ancient works with Vincent Lafforgue and Tim de Laat suggest a possible approach to it: does the non-commutative Lp space of the von Neumann of SL(n,Z) have the completely bounded approximation property for some non-trivial p?

(2) Kakeya conjecture : every subset of R^d containing a unit segment in every direction has dimension d.Both questions are open for large values of the parameters (n>2 and >3). I will explain why (1) is difficult: it implies some form of (2) for d<=(n+1)/2.

In this work, we consider an inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture.  Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult.   Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs.

Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement.  To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms with L1 regularization.  We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method.  Numerical experiments illustrate the effectiveness and robustness of the algorithm.

This is a joint work with Samy W. Fung (Colorado School of Mines), Siting Liu (UCR), Levon Nurbekyan (Emory University), and Stanley J. Osher (UCLA).