Analysis Study Seminar -- Winter 2006
Thursdays 3:00-5:00 -- East Hall 2866

Speaker: Saara Lehto, University of Helsinki

Title: Dyadic structures in metric spaces by Michael Christ

Abstract: The dyadic structure is a useful tool in R^n. One can also construct dyadic structures in general metric spaces. We will discuss a construction by Michael Christ that can be found in his paper "A T(b) theorem with remarks on analytic capacity and the Cauchy integral".

Speaker: Marshall Williams, UofM.

Title: "Assouad Dimension: Antifractal Metrization, Porous Sets, and Homogenous Measures" by Jouni Luukkainen

Abstract: There are various notions of dimension in a metric space, including its topological, Hausdorff, box counting, packing, and Assouad dimensions, among others. In general, these do not coincide, but we will discuss a result by Luukkainen which says that a separable metric space can be remetrized, with a totally bounded metric, so that the Assouad dimension coincides with the topological dimension (and thus also with the other dimensions, which in the bounded case lie between the two). We'll also talk about some applications related to porous sets and homogeneous measures.

Speaker: Sergey Merenkov, UofM.

Title: "Removability theorems for Sobolev functions and quasiconformal maps", by Peter Jones and Stanislav Smirnov, Ark. Mat., 2000.

Abstract: From the abstract: "We establish several conditions, sufficient for a set to be (quasi)conformally removable... This is accomplished by proving removability theorems for Sobolev spaces..."

Speaker: Pietro Poggi-Corradini, Kansas State U./UofM.

Title: "The coarea formula for Sobolev mappings", by Maly, Swanson and Ziemer.

Abstract: From the abstract: "We extend Federer's coarea formula to mappings f belonging to the Sobolev class W^{1,p}(R^n;R^m), 1&le m < n, p > m."

Speaker: Jasun Gong, UofM.

Title: A Counter-Example for the Diffeomorphic Schoenflies Extension Problem

Abstract: The classical Schoenflies Problem asserts that, given a continuous embedding F of the unit (n-1)-sphere into Euclidean n-space R^n, does there exist an continuous embedding G of the unit n-ball into R^n which extends F? An affirmative answer and proof were given by M. Brown in 1960, and in the subsequent years the problem has been reformulated for various categories of homeomorphisms such as differentiable, smooth, piecewise-linear, and quasiconformal. In low dimensions (n < 4) diffeomorphic extensions do exist, but this fails in higher dimensions. In this talk we will demonstrate a diffeomorphism of the 6-sphere which has no diffeomorphic extension into the 7-ball. This will be possible by appealing to J. Milnor's construction of exotic 7-spheres.

Speaker: John Mackay, UofM.

Title: Quasiconformal groups

Abstract: A group of homeomorphisms of the n-sphere is called a quasiconformal (qc) group if each map is a K-qc map for some fixed K. We will discuss a result of Tukia which shows that if such a group acts cocompactly on the space of triples then the group is conjugate (by one fixed qc map) to a group of Mobius transformations.

Speaker: Marie Snipes, UofM.

Title: Flat chains in Banach spaces, Part II.

Abstract: Federer and Fleming developed the theory of flat chains in n-dimensional Euclidian space. I will discuss recent work of T. Adams that extends this theory to Banach spaces. The main result proves that certain sets of flat chains are compact.

Speaker: Marie Snipes, UofM.

Title: Flat chains in Banach spaces

Abstract: Federer and Fleming developed the theory of flat chains in n-dimensional Euclidian space. I will discuss recent work of T. Adams that extends this theory to Banach spaces. The main result proves that certain sets of flat chains are compact.

Speaker: Mario Bonk, UofM.

Title: Quasiregular self-maps of manifolds and word hyperbolic groups

Abstract: I'll discuss a recent paper by Bridson, Hinkkanen, and Martin. The authors show that a quasiregular self-map of a closed negatively curved n-manifold, n different from 4, does not have branching, i.e., it is quasiconformal.

Speaker: Pietro Poggi-Corradini, Kansas State U. and UofM.

Title: Conformal welding and Koebe's Theorem, after Chris Bishop

Abstract: We will discuss a recent paper of Chris Bishop which shows that every orientation preserving homeomorphism of the unit circle to itself is "almost" a conformal welding. The paper contains several results, so we might focus on a few well-chosen theorems.

Speaker: Juha Heinonen, UofM.

Title: Metric differentiability.

Abstract: A Lipschitz function from an interval to a Banach space need not be differentiable anywhere, but there is a recent simple notion of metric differentiability due to Kirchheim that allows for a generalization of the classical Rademacher theorem. I discuss this and another related result due to Ambrosio. The talk is elementary.

Speaker: Serban Costea, UofM.

Title: An algebraic characterization of quasi-Mobius homeomorphisms, part II.

Abstract: Let Z_1, Z_2 be compact metric spaces. Under some natural assumptions, we characterize quasi-Mobius homeomorphisms between Z_1 and Z_2 in terms of isomorphisms between function algebras on Z_i, which are defined by Besov norms.

Speaker: Serban Costea, UofM.

Title: An algebraic characterization of quasi-Mobius homeomorphisms.

Abstract: Let Z_1, Z_2 be compact metric spaces. Under some natural assumptions, we characterize quasi-Mobius homeomorphisms between Z_1 and Z_2 in terms of isomorphisms between function algebras on Z_i, which are defined by Besov norms.