Taught Course Centre

This talk will be by videolink from Warsaw.  The starting-time will be a little after 17:00 due to a TCC lecture and time needed to establish video connections.

Abstract: The Haagerup approximation property for finite von Neumann algebras  (i.e.von Neumann algebras with a tracial faithful normal state) has been studied for more than 30 years. The original motivation to study this property came from the case of group von Neumann algebras of discrete groups, where it corresponds to the geometric Haagerup property of the underlying group. Last few years brought a lot of interest in the Haagerup property for discrete and general locally compact quantum groups. If the discrete quantum group in question is not unimodular, the associated (quantum) group von Neumann algebra cannot be finite, so we need a broader framework for the operator algebraic property. In this talk, I will present recent developments regarding the Haagerup approximation property for arbitrary von Neumann algebras and will also discuss some questions relating it to the issues related to the classical Schoenberg correspondence. (Mainly based on joint work with Martijn Caspers.)

Stochastic homogenization shows that solutions to an elliptic problem 

with rapidly oscillating, ergodic random coefficients can be effectively 

described by an elliptic problem with homogeneous, deterministic 

coefficients. The definition of the latter is based on the construction 

of a "corrector" and invokes an elliptic operator that acts on the 

probability space of admissible coefficient fields. While qualitative 

homogenization is well understood and classical, quantitative results 

(e.g. estimates on the homogenization error and approximations to the 

homogenized coefficients) have only been obtained recently.  In the talk 

we discuss an optimal estimate on the associated semigroup (usually 

called the "random walk in the random environment") and show that it 

decays with an algebraic rate. The result relies on a link between a 

Spectral Gap of a Glauber dynamics on the space of coefficient fields (a 

notion that we borrow from statistical mechanics) and heat kernel 

estimates. As applications we obtain moment bounds on the corrector and 

an optimal convergence rate for the approximation of the homogenized 

coefficients via periodic representative volume elements.

We begin by reviewing different stability types for abstract differential equations and strongly continuous semigroups on Hilbert spaces. We concentrate on exponential stability, polynomial stability, and strong stability with a finite number of singularities on the imaginary axis. We illustrate each stability type with examples from partial differential equations and control theory. 



In the second part of the talk we study the preservation of strong and polynomial stabilities of a semigroup under bounded perturbations of its generator. As the main results we present conditions for preservation of these two stability types under finite rank and trace class perturbations. In particular, the conditions require that certain graph norms of the perturbing operators are sufficiently small.



In the final part of the talk we consider robust output tracking for linear systems, and explain how this control problem motivates the study of preservation of polynomial stability of semigroups. In particular, the solution of this problem requires determining which uncertainties in the parameters of the controlled system preserve the stability of the closed-loop system consisting of the system and the dynamic controller. We show that if the reference signal to be tracked is a nonsmooth periodic function, it is impossible to stabilize the closed-loop system exponentially, but polynomial stability is achievable under suitable assumptions. Subsequently, the uncertainties in the parameters of the system can be represented as a bounded perturbation to the system operator of the polynomially stable closed-loop system.