The Geometric Function Theory Seminar -- Winter 2008

Wednesdays 3:10-4:00 -- East Hall 4096

PAST SEMINARS: Fall 08  

To schedule a talk, or for more information, please contact Pekka Pankka (pankka 'at' umich.edu).

Wednesday, March 11.

Speaker: Jani Onninen (Syracuse Univerisity)
Title: Dynamics of Quasiconformal Fields

Abstract: This talk will focus on a uniqueness theorem for autonomous systems of ODEs, $\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. The talk is based on joint work with Tadeusz Iwaniec and Leonid V. Kovalev.

Monday, March 9. 3pm. EH 4088. Note the special time and place!

Speaker: Markus Nieß (Kath. Universität Eichstätt-Ingolstadt)
Title: Universal functions and power series.

Abstract: Roughly speaking, universality means „existence of a dense orbit“. Thus, in some sense, universal functions are „uncontrolled“. More precisely, we are considering functions g having the property that on any „suitable“ set, any „suitable“ function can be approximated by a subsequence of either translations, derivatives or partial sums of the power series expansion of g. The only known explicit functions having such properties are the Riemann Zeta-function and its relatives. This has been discovered by Voronin in 1975. A newer type of universality is linked with famous theorems of Jentzsch from 1917. Given a power series f with radius of convergence 1. He showed that every point on the boundary of the disk of convergence is a so-called limit point of zeros of the partial sums of f.

Wednesday, February 18.

Speaker: Marshall Williams (UM).
Title: Metric currents and differentiable structures.

Abstract: In 2000 Ambrosio and Kirchheim extended the Federer-Fleming theory of currents to general metric spaces. Around the same time Cheeger introduced the notion of a differentiable structure in similar generality.
Using this concept he proved a version of Rademacher's theorem on differentiability of Lipschitz functions. I will discuss the compatibility of Cheeger's differentiable structures with the Ambrosio-Kirchheim theory.

Wednesday, February 11.

Speaker: Laurent Moonens (visiting UM).
Title: Charges in Euclidean space.

Abstract: A m-dimensional flat cochain in the n-dimensional Euclidean space is a linear functional on the space of m-dimensional polyhedral chains, that is continuous with respect to the flat norm topology. On the other hand, a flat m-form is a bounded (measurable) m-differential form whose weak exterior derivative is also bounded. Wolfe's theorem shows that any m-dimensional flat cochain A is associated to an m-flat form w by an integral representation formula. Relaxing the continuity condition, we call m-charge any linear functional on the space of m-dimensional polyhedral chains whose values asymptotically vanishes on chains whose flat norms go to zero and they are all supported in a fixed compact set and remain bounded in perimeter. As we will show, any m-charge can be represented by a pair (w,z) of continuous m- and (m-1)-forms. This yields in particular a new integral representation of m-dimensional flat cochains.
This is a joint work with T. De Pauw and W.F. Pfeffer.

Thursday, January 22. 4pm, EH 3866. Note the special time and place!

Speaker: Raanan Schul (UCLA).
Title: Bi-Lipschitz decomposition of Lipschitz functions into a metric space.

Abstract: We will outline the proof of a quantitative version of the following statement. Given a Lipschitz function $f$ from the $k-$dimensional unit cube into a general metric space, one can decompose $f$ into a finite number of Bi-Lipschitz functions $f|_{F_i}$ so that the $k-$Hausdorff content of $f([0, 1]^k \smallsetminus \cup F_i$) is small. The case where the metric space is $\mathbb{R}^d$ is a theorem of P. Jones (1988). This positively answers problem 11.13 in ``Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from ``Thirty-three yes or no questions about mappings, measures, and metrics" by J. Heinonen and S. Semmes.
Bi-Lipschitz decomposition of Lipschitz functions into a metric space.

Wednesday, January 21.

Speaker: Jinho Baik (UM).
Title: Directed last passage site percolation

Abstract: Consider the points in the plane with integer coordinates. At each such point, an independent (positive) random variable is attached. Given a path, the passage time along the path is the sum of the random variables at the sites on the path. We are interested in the last passage time from (1,1) to (N,k) which is the maximum of the passage time along an admissible path. The last passage time is directly related to other statistical models such as a random growth model in 2D and other interacting particle systems. A surprise is its relation to random matrix theory and also combinatorics, as discovered during last decade. We will review some aspects of last passage percolation problem.