28 October 2014

17:00

Pascal Auscher

28 October 2014

17:00

Pascal Auscher

14 October 2014

17:00

Felix Geyer

1 September 2014

14:00

to

15:15

3 June 2014

17:00

to

18:20

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.

20 May 2014

17:00

to

18:15

Sergey Naboko

Abstract

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}.

15 April 2014

17:00

to

18:20

El Maati Ouhabaz

Abstract

11 March 2014

17:00

to

18:15

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.