Forthcoming events in this series
We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.
C*-algebras constructed from topological groupoids allow us to study many interesting and a priori very different constructions
of C*-algebras in a common framework. Moreover, they are general enough to appear intrinsically in the theory. In particular, it was recently shown
by Xin Li that all C*-algebras falling within the scope of the classification program admit (twisted) groupoid models.
In this talk I will give a gentle introduction to this class of C*-algebras and discuss some of their structural properties, which appear in connection
with the classification program.
I will give an overview of semigroup C*-algebras, which are C*-algebras generated by left regular representations of semigroups. The main focus will be on examples from number theory and group theory.
Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.
We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.
I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities. Having its roots in a pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.
The well known Katznelson-Tzafriri theorem states that a power-bounded operator $T$ on a Banach space $X$ satisfies $\|T^n(I-T)\| \to 0$ as $n \to \infty$ if and only if the spectrum of $T$ touches the complex unit circle nowhere except possibly at the point $\{1\}$. As it turns out, the rate at which $\|T^n(I-T)\|$ goes to zero is largely determined by estimates on the resolvent of $T$ on the unit circle minus $\{1\}$ and not only is this interesting from a purely spectral and operator theoretic perspective, the applications of such quantified decay rates are myriad, ranging from the mean ergodic theorem to so-called alternating projections, from probability theory to continuous-in-time evolution equations. In this talk, we will trace the story of these so-called quantified Katznelson-Tzafriri theorems through previously known results up to the present, ending with a new result proved just a few weeks ago that largely completes the adventure.
ABSTRACT: Given a metric measure space $(X,d,\mu)$, its doubling constant is given by
$$
C_\mu=\sup_{x\in X, r>0} \frac{\mu(B(x,2r))}{\mu(B(x,r))},
$$
where $B(x,r)$ denotes the open ball of radius $r$ centered at $x$. Clearly, $C_\mu\geq1$, and in the case $X$ reduces to a singleton $C_\mu=1$. One might think that for a metric space with more than one point, the constant $C_\mu$ could be very close to one. However, we will show that in general $C_\mu\geq2$. The talk is based on a joint work with Javier Soria (Barcelona).
The theory of algebraic extensions of commutative Banach algebras is well established and have been used to solve many problems. In his thesis, Cole constructed algebraic extensions of a certain uniform algebra to give a counterexample to the peak point conjecture. Cole’s method for extending uniform algebras ensures that certain properties of the original algebra are preserved by the extension. In this talk, we discuss the general theory of uniform algebra extensions and a certain class of uniform algebra extensions which generalise Cole’s construction.
Consider a periodic function $f$, such that its restriction to the unit segment lies in the Banach space $L^2=L^2(0,1)$. Denote by $S$ the family of dilations $f(nx)$ for all $n$ positive integer. The purpose of this talk is to discuss the following question: When does $S$ form a Riesz basis of $L^2$?
In this talk, we will present a new \textit{mutli-term} criteria for determining Riesz basis properties of $S$ in $L^2$. This method was established in [L. Boulton, H. Melkonian; arXiv: 1708.08545 J. (2017), to appear at the Journal of Analysis and its Applications (ZAA)] and it relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients.
Beginning with a short introduction and a review of Hilbert space
techniques used in the admissibility analysis of dynamical systems,
we will focus on state-delayed systems.
Using the "lifting" method to reformulate the problem, we will firstly
analyse a retarded delay system assuming only contraction property
of the undelayed semigroup. Next, we will turn our attention to problems
where more can be said about the underlying semigroup.
In particular, we will investigate diagonal systems.
This talk will present results of a joint work with Jonathan Partington.
I will present the general strategy for classifying separable, nuclear, purely infinite C*-algebras
This is part of a meeting of the North British Functional Analysis Seminar
In this talk I will present some new $L_p$-$L_q$-Fourier multiplier theorems which hold for operator-valued symbols under geometric restrictions on the underlying Banach spaces such as (Fourier) (co)type. I will show how the multiplier theorems can be applied to obtain new stability results for semigroups arising in evolution equations. This is based on joint work with Jan Rozendaal (ANU, Canberra).
This is part of a meeting of the North British Functional Analysis Seminar.
In this talk I will present an overview on generalized square functions in Banach spaces and some of their recent uses in “Analysis in Banach Spaces”. I will introduce the notions of $R$-boundedness and $\gamma$-radonifying operators and discuss their origins and some of their applications to harmonic analysis, functional calculus, control theory, and stochastic analysis.
This talk is associated with the NBFAS meeting.
We discuss the quantitative asymptotic behaviour of operator semigroups. Batty and Duyckaerts obtained upper and lower bounds on the rate of decay of a semigroup given bounds on the resolvent growth of the semigroup generator. They conjectured that in the Hilbert space setting and for the special case of polynomial resolvent growth it is possible to improve the upper bound so as to yield the exact rate of decay up to constants. This conjecture was proved to be correct by Borichev and Tomilov, and the conclusion was extended by Batty, Chill and Tomilov to certain cases in which the resolvent growth is not exactly polynomial but almost. In this talk we extend their result by showing that one can improve the upper bound under a significantly milder assumption on the resolvent growth. This result is optimal in a certain sense. We also discuss how this improved result can be used to obtain sharper estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary. The talk is based on joint work with J. Rozendaal and R. Stahn.
This is part of a meeting of the North British Functional Aanlysis Seminar. There will be a tea break (15:30-16:00)
In a first talk, I shall recall the basic definitions and properties of ${\mathcal{H}}^\infty}$ functional calculus. I shall show how a second order problem can be reduced to a first order system and how to construct potential operators.
In a second talk, we will see how to use potential operators for the specific problem of the Stokes operator with the so-called “natural” boundary conditions in non smooth domains.
Most parts which will be presented are taken from a joint work with Alan McIntosh (to be published soon in the journal "Revista Matematica Iberoamericana”)
The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. It also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with rough features. I aim to give an overview of recent developments, with particular focus on the NP operator's action on the energy space of the domain. The energy space framework ties together Poincare’s efforts to solve the Dirichlet problem with the operator-theoretic symmetrisation theory of Krein. I will also indicate recent work for domains in 3D with conical points. In this situation, we have been able to describe the spectrum both for boundary data in $L^2$ and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval.
I plan to present several results linking the numerical range of a Hilbert space operator to the circle structure of its spectrum. I'll try to explain how the numerical ranges approach helps to unify, extend or supplement several results where the circular structure of the spectrum is crucial, e.g. Arveson's theorem on almost-wandering vectors of unitary actions and Hamdan's recent result on supports of Rajchman measures. Moreover, I'll give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations. If time permits, I'll also address the same or similar issues in a more general setting of operator tuples. This is joint work with V. M\" uller (Prague).