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Forthcoming events in this series
17:00
14:00
Algebraic characterization of autonomy of PDEs
Abstract
Given an ideal I in the polynomial ring C[x1,...,xn],
the variety V(I) of I is the set of common zeros in C^n
of all the polynomials belonging to I. In algebraic geometry,
one tries to link geometric properties of V(I) with algebraic properties of I.
Analogously, given a system of linear, constant coefficient
partial differential equations, one can consider its zeros, that is,
its solutions in various function and distribution spaces.
One could then hope to link analytic properties of the
set of solutions with algebraic properties of the polynomials
which describe the PDEs.
In this talk, we will focus on one such analytic property,
called autonomy, and we will provide an algebraic characterization
for it.
17:00
Robustness of strong stability of semigroups with applications in control theory
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.
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Fredholm Theory for Singular Integral Operators of Calderon-Zygmund Type in Uniformly Rectifiable Domains
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The structure of quantum permutation groups
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.
17:00
Dynamics of curve flows related to vortex filaments and ferromagnetism
17:00
Weak amenability of Fourier algebras of Lie groups
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.
17:00
The F. and M. Riesz theorem without connectivity
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.
Jacobi Matrices examples of the spectral phase transition phenomenon and the Green's Matrix estimates
Noncommutative dimension and tensor products
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}.
Maximal regularity for non-autonomous evolution equations: recent progress
The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ coefficients
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.
Contractions of Lie groups: an application of Physics in Pure Mathematics
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.
A functional calculus construction of layer potentials for elliptic equations
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.
Honesty theory and stochastic completeness
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.