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.

# Past Functional Analysis Seminar

Quantum permutation groups, introduced by Wang, are a quantum analogue of permutation groups.

These quantum groups have a surprisingly rich structure, and they appear naturally in a variety of contexts,

including combinatorics, operator algebras, and free probability.

In this talk I will give an introduction to these quantum groups, starting with some background and basic definitions.

I will then present a computation of the K-groups of the C*-algebras associated with quantum permutation groups,

relying on methods from the Baum-Connes conjecture.

The Fourier algebra of a locally compact group was first defined by Eymard in 1964. Eymard showed that this algebra is in fact the space of all coefficient functions of the left regular representation equipped with pointwise operations. The Fourier algebra is a semi-simple commutative Banach algebra, and thus it admits no non-zero continuous derivation into itself. In this talk we study weak amenability, which is a weaker form of differentiability, for Fourier algebras. A commutative Banach algebra is called weakly amenable if it admits no non-zero continuous derivations into its dual space. The notion of weak amenability was first defined and studied for certain important examples by Bade, Curtis and Dales.

In 1994, Johnson constructed a non-zero continuous derivation from the Fourier algebra of the rotation group in 3 dimensions into its dual. Subsequently, using the structure theory of Lie groups and Lie algebras, this result was extended to any non-Abelian, compact, connected group. Using techniques of non-commutative harmonic analysis, we prove that semi-simple connected Lie groups and 1-connected non-Abelian nilpotent Lie groups are not weak amenable by reducing the problem to two special cases: the $ax+b$ group and the 3-dimensional Heisenberg group. These are the first examples of classes of locally compact groups with non-weak amenable Fourier algebras which do not contain closed copies of the rotation group in 3 dimensions.