Forthcoming events in this series


Tue, 03 Mar 2015
17:00
Taught Course Centre

Robustness of strong stability of semigroups with applications in control theory

Lassi Paunonen
(Tampere)
Abstract
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.
Tue, 02 Dec 2014
17:00
C1

TBA

Leonard Konrad
(Oxford)
Tue, 25 Nov 2014
17:00
C1

The structure of quantum permutation groups

Christian Voigt
(Glasgow)
Abstract

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.

Tue, 04 Nov 2014
17:00
C1

Weak amenability of Fourier algebras of Lie groups

Mahya Ghandehari
(Waterloo)
Abstract

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.

Tue, 27 May 2014

17:00 - 18:15
C6

The F. and M. Riesz theorem without connectivity

Steve Hofmann
(Missouri-Columbia)
Abstract

A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis.

In this work, we nonetheless establish versions of the F. and M. Riesz theorem, in higher dimensions, in which other properties of har¬monic functions substitute for the absolute continuity of harmonic measure. These substitute properties are natural “proxies” for har¬monic measure estimates, in the sense that, in more topologically be¬nign settings, they are actually equivalent to a scale-invariant quanti¬tative version of absolute continuity.

Tue, 13 May 2014

17:00 - 18:15
C6

Noncommutative dimension and tensor products

Aaron Tikuisis
(Aberdeen)
Abstract

Inspired largely by the fact that commutative C*-algebras correspond to

(locally compact Hausdorff) topological spaces, C*-algebras are often

viewed as noncommutative topological spaces. In particular, this

perspective has inspired notions of noncommutative dimension: numerical

isomorphism invariants for C*-algebras, whose value at C(X) is equal to

the dimension of X. This talk will focus on certain recent notions of

dimension, especially decomposition rank as defined by Kirchberg and Winter.

A particularly interesting part of the dimension theory of C*-algebras

is occurrences of dimension reduction, where the act of tensoring

certain canonical C*-algebras (e.g. UHF algebras, Cuntz' algebras O_2

and O_infinity) can have the effect of (drastically) lowering the

dimension. This is in sharp contrast to the commutative case, where

taking a tensor product always increases the dimension.

I will discuss some results of this nature, in particular comparing the

dimension of C(X,A) to the dimension of X, for various C*-algebras A. I

will explain a relationship between dimension reduction in C(X,A) and

the well-known topological fact that S^n is not a retract of D^{n+1}.

Tue, 04 Mar 2014

17:00 - 18:15
C6

The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ coefficients

Andrew Morris
(Oxford)
Abstract

We consider the layer potentials associated with operators $L=-\mathrm{div} A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent.  A ``Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi--Nash--Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.

Tue, 18 Feb 2014

17:00 - 18:15
C6

Contractions of Lie groups: an application of Physics in Pure Mathematics

Tony Dooley
(Bath)
Abstract

Contractions of Lie groups have been used by physicists to understand how classical physics is the limit ``as the speed of light tends to infinity" of relativistic physics. It turns out that a contraction can be understood as an approximate homomorphism between two Lie algebras or Lie groups, and we can use this to transfer harmonic analysis from a group to its ``limit", finding relationships which generalise the traditional results that the Fourier transform on $\R$ is the limit of Fourier series on $\TT$. We can transfer $L^p$ estimates, solutions of differential operators, etc. The interesting limiting relationship between the representation theory of the groups involved can be understood geometrically via the Kirillov orbit method.

Tue, 11 Feb 2014

17:00 - 18:15
C6

A functional calculus construction of layer potentials for elliptic equations

Andreas Rosen
(Chalmers University)
Abstract

We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A\in L_\infty, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.

Tue, 04 Feb 2014

17:00 - 18:15
C6

Honesty theory and stochastic completeness

Chin Pin Wong
(Oxford)
Abstract

An important aspect in the study of Kato's perturbation theorem for substochastic semi-

groups is the study of the honesty of the perturbed semigroup, i.e. the consistency between

the semigroup and the modelled system. In the study of Laplacians on graphs, there is a

corresponding notion of stochastic completeness. This talk will demonstrate how the two

notions coincide.