Geometric Analysis meets Geometric Topology

• Carla Cederbaum (University of Tübingen)

  On special hypersurfaces of the Schwarzschild spacetime and related uniqueness theorems (Tuesday 15h30)

  The famous Schwarzschild spacetime is well-known to possess a unique so-called “photon sphere” – a cylindrical, timelike hypersurface P such that any null geodesic initially tangent to P remains tangent to P – in all dimensions. After an introduction into the relevant notions from Lorentzian Geometry and General Relativity, we will show that the Schwarzschild spacetime also possesses a rich family of spatially spherically symmetric “photon surfaces” – general timelike hypersurfaces P such that any null geodesic initially tangent to P remains tangent to P. This generalizes a result of Foertsch, Hasse, and Perlick from 2+1 to higher dimensions.

  We will then explain why the Schwarzschild spacetime is indeed the only static, vacuum, asymptotically flat spacetime possessing static black hole horizons, photon spheres, and/or what Gregory J. Galloway and I call “equipotential” photon surfaces, again in all dimensions.

  The above results are joint work with Gregory J. Galloway. They build on a rigidity result of mine for asymptotically flat Riemannian manifolds of non-negative scalar curvature with special umbilic, CMC, constant scalar curvature inner boundary components that globally support a harmonic function satisfying an overdetermined system of Dirichlet and Neumann conditions on the inner boundary. The proof of this rigidity result generalizes and extends Riemannian and Conformal Geometry arguments first introduced by Bunting and Masood-ul-Alam in their proof of static black hole uniqueness, a higher dimensional analog by Gibbons, Ida, and Shiromizu, and previous joint work with Gregory J. Galloway on the uniqueness of photon spheres. It relies on Schoen and Yau’s higher dimensional positive mass theorem as well as on a result by McFerron and Szekelyhidi.

• Anna Dall'Acqua (Ulm University)

  On minimal elastic networks (Thursday 9h30)

  We consider planar networks of three curves that meet at two junctions with prescribed equal angles, so called Theta-Networks. Considering this as a model for elastic materials we associate to it an energy given by the combination of elastic energy and length. The questions we address are: does a minimizer exists? if so, is it a Theta-network or there is some change in topology? More precisely, the minimal configuration does still consist of three curves?

  This is a joint work with M. Novaga and A. Pluda. Accepted for publication in Indiana University Mathematical Journal.

• Viveka Erlandsson (University of Bristol)

  Determining the shape of a billiard table from its bounces (Monday 14h00)

  Consider a billiard table shaped as a Euclidean polygon with labeled sides. A ball moving around on the table determines a bi-infinite “bounce sequence” by recording the labels of the sides it bounces off. We call the set of all possible bounce sequences the “bounce spectrum” of the table. In this talk I will explain why the bounce spectrum essentially determines the shape of the table: with the exception of a very small family (right-angled tables), if two tables have the same bounce spectrum, then they have to be related by a Euclidean similarity. The main ingredient in proving this fact is a technical result about non-singular geodesics on surfaces equipped with flat cone metrics. This is joint work with Moon Duchin, Chris Leininger, and Chandrika Sadanand.

• Alessandra Iozzi (ETH)

  Mostow rigidity and bounded cohomology (Tuesday 14h00)

  We define the volume of a representation of a cocompact hyperbolic lattice and show a rigidity theorem for representations of maximal volume. As a consequence we obtain a proof of Mostow rigidity theorem. This is based on work in collaboration with Michelle Bucher and Marc Burger.

• Anna Lenzhen (Université de Rennes 1)

  Limit sets of disks and rays in Thurston compactification of Teichmüller space (Wednesday 15h30)

  Consider a Teichmüller disk, i.e, an isometrically embedded copy of a Poincaré disk in the Teichmüller space equipped with Teichmüller metric. We would like to know the set of accumulation points of the disk in the Thurston boundary. One could start by looking at accumulation points of rays in this disk. H. Masur showed in the early 80's that almost every Teichmüller ray converges to a unique point in the Thurston boundary. It is also known since a while that there are rays that have more than one accumulation point in the boundary. Moreover, there are rays whose limit set is a d-dimensional simplex. I will give an overview of what is understood so far about the limit sets, mentioning some recent progress.

• Katarzyna Mazowiecka (Université catholique de Louvain)

  On the size of the singular set of minimizing harmonic maps (Monday 15h30)

  Minimizing harmonic maps (i.e. minimizers of the Dirichlet integral) with prescribed boudary conditions may have singularities. At the beginning of this talk, I will consider minimizing harmonic maps from 3-dimensional domains into the two dimensional sphere and present an extension of Almgren and Lieb’s linear law on the bound of the singular set as well as Hardt and Lin’s stability theorem for singularities. Next, I will discuss new higher dimensional counterparts of those theorems. This is joint work with Michał Miśkiewicz and Armin Schikorra.

• Melanie Rupflin (University of Oxford)

  Flowing to minimal surfaces (Monday 11h00)

  In this talk I will discuss the construction and properties of a geometric flow, the Teichmueller harmonic map flow, that is designed to change surfaces into minimal surfaces. As I will explain, this flow, which is a natural gradient flow of the Dirichlet energy, succeeds in decomposing any closed surface in any compact target manifold into minimal surfaces.

• Anna Schilling (University of Heidelberg)

  Horofunction and Satake compactifications of symmetric spaces (Tuesday 9h30)

  There are various ways to compactify a Riemannian symmetric space X = G/K of non-compact type. The horofunction compactification embeds X into the space of real valued function using the metric on the space and takes the closure there. The generalized Satake compactification is associated to a faitthful projective representation of G. In this talk I will explain these two compactifications and show that any generalized Satake compactification can be realized as a horofunction compactification with respect to a special Finsler norm on X. This ist joint work with Thomas Haettel, Cormac Walsh and Anna Wienhard.

• Petra Schwer (University Magdeburg)

  Kostant Convexity in the affine flag variety and affine grassmannian (Wednesday 11h00)

  In this talk I will explain the classical Kostant convexity theorem and will show how one can obtain natural analogs for the affine flag variety and affine grassmannian using combinatorial and geometric methods.

• Anna Siffert (Max-Planck-Institut Bonn)

  Large genus minimal surfaces in positive Ricci curvature (Tuesday 11h00)

  We use Colding–Minicozzi lamination theory to study the systole of large genus minimal surfaces in an ambient three-manifold of positive Ricci curvature. This is joint work with Henrik Matthiesen.

• Gabriela Weitze-Schmithüsen (Saarland University)

  Systols on Origami Translation Surfaces (Thursday 11h00)

  Translation surfaces are obtained by a charming concrete construction: Take finitely many polygons in the Euclidean plane and glue pairs of their edges via translations such that you obtain a connected surface. The study of these surfaces leads to such multifaceted topics as geodesic flows on Teichmüller spaces, Teichmüller curves in the moduli space M_g of smooth complex curves of genus g and billiards on polygonal billard tables. Although translation surfaces have been intensively studied since the 1980’s, there are still many natural questions wildly open. One of these questions is: Which translation surface of genus g has the largest shortest curve and how long is this curve? We describe an algorithmic approach to this question using a special class of translation surfaces called origamis or square-tiled surfaces.

• Karen Vogtmann (University of Warwick)

  The rational Euler characteristic of Out(F_n) (Wednesday 11h00)

  I will recall old joint work with J. Smillie and then discuss recent work of M. Borinsky that builds on that to determine the asymptotic behavior of the rational Euler characteristic of Out(F_n). Neat connections to various classical numbers and functions appear.

Impostor syndrome:

• Myriam Bechtoldt (EBS University)

  Winners who think they are losers – The impostor phenomenon (Wednesday 17h00)

  In the 1970s, two psychotherapists, Pauline Clance and Suzanne Imes, were puzzled by patients who expressed intense fears of failure. While these achievement-related fears per se were not exceptional, remarkably, though, they were experienced by successful women. Obviously these women's self-evaluations were incongruent with objective evidence regarding their abilities. Instead of gaining self-confidence from their professional or academic success, they felt uncertain about it and attributed it to some other factor than intelligence, such as charm, luck, or hard work. The significant discrepancy between these women's self-views and their achievements inspired Clance and Imes (1978) to coin the term “impostor phenomenon”.

  This talk will present recent empirical findings on this phenomenon. Among others, it will address the following questions: Who is affected? Do individuals scoring high on impostorism honestly think they are losers? How does the impostor phenomenon influence the behavior of people in leadership positions?

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