Past Functional Analysis Seminar

27 May 2014
17:00
to
18:15
Steve Hofmann
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.
  • Functional Analysis Seminar
13 May 2014
17:00
to
18:15
Aaron Tikuisis
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}.
  • Functional Analysis Seminar
4 March 2014
17:00
to
18:15
Andrew Morris
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.

  • Functional Analysis Seminar
18 February 2014
17:00
to
18:15
Tony Dooley
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.
  • Functional Analysis Seminar
11 February 2014
17:00
to
18:15
Andreas Rosen
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.
  • Functional Analysis Seminar
4 February 2014
17:00
to
18:15
Chin Pin Wong
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.
  • Functional Analysis Seminar

Pages