Hausdorff-Kolloquium 2015

Date: April 29 - July 8, 2015

Venue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

 

Abstracts:

André Arroja Neves: Min-max theory and Geometry

Min-max methods were introduced in Geometry by Birkhoff in the 1910's.  Recently  the theory has been used to solve several long standing open questions in Geometry and Topology. I will survey these developments and mention some new results.

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Jan Hendrik Bruinier: Modular forms and moduli spaces

Generating series of interesting arithmetic functions are often related to modular forms. For instance, representation numbers of quadratic forms are given by coefficients of theta functions, and class numbers of imaginary quadratic fields by certain Eisenstein series. After introducing the basic notions, we discuss analogues of such results in the geometry and arithmetic of moduli spaces. For instance, a famous theorem of Gross, Kohnen, and Zagier states that the generating series of Heegner divisors on a modular curve is an elliptic modular form of weight 3/2 with values in the Picard group. This result can be viewed as an elegant description of the relations among Heegner divisors. More generally, Kudla conjectured that the generating series of codimension g special cycles on an orthogonal Shimura variety of dimension n is a Siegel modular form of genus g and weight 1+n/2 with coefficients in the Chow group of codimension g cycles. We report on recent progress on this and related conjectures.

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Gitta Kutyniok: Compactly Supported Shearlets: Theory and Applications

Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified  treatment of the continuum and digital realm, leading to faithful implementations. An additional advantage is the availability of stable compactly supported systems for high spatial localization.

In this talk, we will provide an introduction to the anisotropic representation system of shearlets, in particular, compactly supported shearlets, present the main theoretical results, and discuss applications to imaging science and adaptive numerical solution of partial differential equations. Finally, we will present a general framework for systems with optimal sparse approximation properties with respect to anisotropic features, coined parabolic molecules, which in particular includes compactly supported shearlet systems.

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Angela Stevens: Mathematical Modeling of the Cellular Cytoskeleton and Cell Motion

The ability of eukaryotic cells to actively move along different substrates plays a vital role in many biological processes. A key player in these processes is the cellular cytoskeleton. In this talk a free boundary problem for a hyperbolic-parabolic PDE-system is introduced which describes the dynamics of the actin filaments within the cytoskeleton. Emergence of Dirac measures in the densities of actin filament tips is shown, which can be interpreted as sharp polymerization fronts observed experimentally. These fronts lead to polarization and directed motion of a cell. (Joint work with Jan Fuhrmann)

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Andreas Thom: Entropy, Determinants, and L2-Torsion

This talk is about the relationship between the entropy of dynamical systems of algebraic origin and certain determinants defined on infinite-dimensional algebras of operators. All notions will be motivated and explained in detail. We give a proof of a conjecture by Christopher Deninger about the entropy of principal algebraic actions as well as a conjecture by Wolfgang Lück about the vanishing of L2-Torsion of amenable groups.

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Stefan Wenger: Plateau's problem, isoperimetric inequalities, and the geometry of metric spaces

The first part of the talk will describe a generalization of the classical problem of Plateau to the setting of metric spaces. It will be shown that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space admits a local quadratic isoperimetric inequality for curves then such a disc is locally Hölder continuous in the interior and continuous up to the boundary.

The second part of the talk will explain how the existence and regularity of area minimizers can be used to study the large scale geometry of Riemannian manifolds as well as the local structure of metric spaces with a quadratic isoperimetric inequality for curves. For example, it will be shown that metric spaces of non-positive curvature in the sense of Alexandrov are characterized by a Euclidean isoperimetric inequality. The talk is based on joint work with Alexander Lytchak.

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