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

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