Tuesday, March 6, 2018
03/06/2018 - 4:00pm to 5:00pm
Motivated by a simple model of sequential decision making, we study small random perturbations of a dynamical system in aneighborhood of a heteroclinic network, that is, a collection of hyperbolic equilibrium points and corresponding connecting trajectories. Basedon a detailed study of the exit problem from a neighborhood of a hyperbolic equilibrium, we show that the probability of tracing any particular path in the network decays at most polynomially in the size of the noise and establish sharp asymptotics on the time required to complete these journeys. Using these results, we describe the metastable hierarchy that emerges on polynomially long timescales.
03/06/2018 - 4:30pm to 5:30pm
Positive representations of quantum groups were defined by I. Frenkel and I. Ip as certain q-deformations of principal series representations of universal enveloping algebras. It was conjectured by the authors that these representations are closed under tensor products. This conjecture happens to be closely related to the modular functor conjecture by V. Fock and A. Goncharov. I will speak about joint works with Gus Schrader (some of which are works in progress) where weprove the above conjectures.