Geometry, Groups and Topology

Oct 10, 2016, 8:00 AM → Oct 14, 2016, 6:00 PM Europe/Berlin

GGT Karlsruhe 2016

The conference will highlight recent advances in geometry, topology, geometric analysis and geometric group theory and aims at fostering communication between experts and young researchers as well as identifying new directions of further research in these fields.

It is part of the conference series "Geometry, Groups and Topology" held at KIT since 2011. This time it also acts as a follow-up to the first Chinese-German Workshop on Metric Riemannian Geometry, which took place at Shanghai Jiao Tong University in October 2015.

Speakers

• Christian Bär (University of Potsdam)
• Mauricio Bustamante (Yau Mathematical Sciences Center Tsinghua University)
• Esther Cabezas-Rivas (Goethe University Frankfurt)
• Anand Dessai (University of Fribourg)
• Fuquan Fang (Capital Normal University)
• Thomas Farrell (Yau Mathematical Sciences Center Tsinghua University)
• Bobo Hua (Fudan University)
• Huihong Jiang (Shanghai Jiao Tong University)
• Yi Jiang (Yau Mathematical Sciences Center Tsinghua University)
• Jürgen Jost (MPI MIS)
• Mijia Lai (Shanghai Jiao Tong University)
• Tobias Lamm (Karlsruhe Institute of Technology)
• Yi Li (Shanghai Jiao Tong University)
• Yi Liu (Peking University)
• Xiaochun Rong (Rutgers and Capital Normal University)
• Guofang Wang (University of Freiburg)
• Anna Wienhard (Heidelberg University)
• Yihu Yang (Shanghai Jiao Tong University)
• Miaomiao Zhu (Shanghai Jiao Tong University)

Scientific Committee

• Fuquan Fang (Capital Normal University, China)
• Enrico Leuzinger (Karlsruhe Institute of Technology, Germany)
• Xiaochun Rong (Rutgers University, USA and Capital Normal University, China)
• Roman Sauer (Karlsruhe Institute of Technology, Germany)
• Wilderich Tuschmann (Karlsruhe Institute of Technology, Germany)
• Yihu Yang (Shanghai Jiao Tong University, China)

Local Organizing Team

• Diego Corro
• Fernando Galaz-García
• Sebastian Grensing
• Jan-Bernhard Kordaß
• Ute Peters
• Wilderich Tuschmann

Background image courtesy of H.-J. Sommerfeld

Poster GGT 2016 Conference Poster

summary Programme

Monday, October 10

Registration, Certificates & Reimbursements (Frau Peters)

1 A Splitting Theorem for Decomposable Non-negatively Curved Polar Manifolds

Polar actions constitute a special yet rich and geometrically significant class of isometric actions on Riemannian manifolds, including actions with orbits of codimension one and isotropy actions of symmetric spaces. Fang-Grove-Thorbergsson proved that any polar action on a closed simply connected Riemanian manifold M with positive (sectional) curvature is equivariantly diffeomorphic to a polar action on a rank one symmetric space, as long as its orbit space has dimension at least two. In this talk I will address to polar actions on Riemanian manifold with non-negative (sectional) curvature where the orbit space splits into a product of Alexandrov spaces. A splitting theorem for the polar manifold will be explained. This is a joint work with K.Grove.

Speaker: Fuquan Fang (Capital Normal University)

2 Collapsed manifolds with Ricci curvature and local rewinding volume bounded below

A Riemannian manifold is collapsed, if any unit ball has small volume. The local rewinding volume of a metric ball $B_r(x)$ is the volume of $B_r(x^*)$, where $(U^*(x,r),x^*)\to B_r(x),x)$ denotes the Riemannian universal covering space. We will report recent work on collapsed Riemannian manifolds with Ricci curvature and local rewinding volume bounded below.

Speaker: Xiaochun Rong (Capital Normal University and Rutgers University)

12:00 PM Lunch Break

Welcome Address by KIT Vice President Thomas Hirth

3 Harmonic maps and singularities of period mappings

We use simple methods from harmonic maps to investigate singularities of period mappings at infinity. More precisely, we derive a harmonic map version of Schmid’s nilpotent orbit theorem. This is a joint work with J. Jost and K. Zuo.

Speaker: Yihu Yang (Shanghai Jiao Tong University)

4 Curvature bounds for discrete metric spaces

Usually, sectional curvature bounds in the sense of Alexandrov or Busemann, as inspired by the properties of Riemannian manifolds with global bounds on their sectional curvature, are formulated for geodesic length spaces. In this talk, it is shown how to formulate such bounds for arbitrary metric spaces, and the relation with the classical formulations is explored.

Speaker: Jürgen Jost (MPI MIS)

4:30 PM Welcome Reception

Tuesday, October 11

Registration, Certificates & Reimbursements (Frau Peters)

5 Blow-up analysis at the boundary for approximate (Dirac-)harmonic maps from surfaces

In this talk, we shall present some recent progress on the blow-up analysis at the free boundary for approximate harmonic maps from surfaces. Also, we shall briefly discuss the Dirichlet boundary case for (Dirac-)harmonic maps and some applications to (Dirac-)harmonic map flows from surfaces with smooth boundary. These are joint works with Jürgen Jost and Lei Liu.

Speaker: Miaomiao Zhu (Shanghai Jiao Tong University)

6 Conformal Willmore tori in $\mathbb{R}^4$

In this talk I am going to present recent existence and non-existence results for conformal Willmore Tori in $\mathbb{R}^4$ which were obtained in a collaboration with Reiner M. Schätzle (Tübingen).

Speaker: Tobias Lamm (Karlsruhe Institute of Technology)

12:00 PM Lunch Break

7 Harmonic functions on discrete groups

For a finitely generated group, one defines a graph structure, so-called Cayley graph, via a specified generating set. For this graph, it associates with a discrete Laplace operator. The kernel of the operator consists of (discrete) harmonic functions. Global behaviors of harmonic functions on infinite groups are of interest. In this talk, we will discuss several known results for harmonic functions on discrete groups.

Speaker: Bobo Hua (Fudan University)

2:50 PM Coffee Break

8 The Ricci Flow on manifolds with almost non-negative curvature operator

We show that n-manifolds with a lower volume bound v and upper diameter bound D whose curvature operator is bounded below by $-\varepsilon(n,v,D)$ also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound.

Speaker: Esther Cabezas-Rivas (Goethe Universität Frankfurt)

Wednesday, October 12

Registration, Certificates & Reimbursements (Frau Peters)

9 Topology of local symplectic conifold transitions on $CP^{1}$-bundles

Symplectic conifold transitions are certain symplecitc surgeries on symplectic 6-manifolds introduced by I.Smith, R.P.Thomas and S.-T.Yau. It is an open question that if symplectic conifold transitios preserve Kahler structures. In this talk we will recall some related basic notions and motivations, and then discuss some results on local symplectic conifold transitions on all $CP^{1}$-bundles over symplectic 4-manifolds.

Speaker: Yi Jiang (Yau Mathematical Sciences Center Tsinghua University)

10 Diameter growth and bounded topology of complete manifolds with nonnegative Ricci curvature

A manifold is said to be of finite topological type if it is homeomorphic to the interior of a compact manifold with boundary. In this talk, I will show that a complete $n$-dim Riemannian manifold with nonnegative Ricci curvature is of finite topological type provided that the diameter growth of $M$ is of order $o(r^{((n-1)\alpha+1)/n})$ and the sectional curvature is no less than $-{\frac{c}{r^{2\alpha}}}$ (here $0 \le \alpha \le 1$ and $c$ is some positive constant) outside a geodesic ball large enough. This can be considered as a generalization of Abresch-Gromoll Theorem. This is based on a joint work with Yihu Yang.

Speaker: Huihong Jiang (Shanghai Jiao Tong University)

11 Smooth bundles with nonpositively curved fibers

I will discuss some features of the topology of smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. Specifically, I will show (topological) rigidity results for the associated vertical tangent bundle and a vanishing theorem for the generalized Miller-Morita-Mumford classes. This is joint work with Tom Farrell and Yi Jiang.

Speaker: Mauricio Bustamante (Yau Mathematical Sciences Center Tsinghua University)

12:00 PM Conference Photo

12:15 PM Lunch Break

Group Discussions and Excursion

Thursday, October 13

Registration, Certificates & Reimbursements (Frau Peters)

12 An index theorem for Lorentzian manifolds with boundary

We show that the Dirac operator on a globally hyperbolic Lorentzian spacetime with compact spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. If time permits, an application to quantum field theory will be sketched. This is the first index theorem for Lorentzian manifolds and, from an analytic perspective, the methods to obtain it are quite different from the classical Riemannian case. This is joint work with Alexander Strohmaier.

Speaker: Christian Bär (Universität Potsdam)

13 Degree of L2 Alexander torsion for 3-manifolds

For an irreducible orientable compact 3-manifold N with empty or incompressible toral boundary, the L2 Alexander torsion has been introduced by Dubois, Friedl, and Lueck. In this talk, I'll explain the idea to prove that the full L2 Alexander torsion is a continuous everywhere positive and asymptotically monomial function in both ends. It can be further shown that the degree of the full L2 Alexander torsion is equal to the Thurston norm of the defining first cohomology class.

Speaker: Yi Liu (Beijing International Center for Mathematical Research)

12:00 PM Lunch Break

14 A new geometric flow about cscK metrics in Kahler geometry

In this talk I will give a new flow of Kahler metrics whose stable point is a constant scalar curvature Kahler metric, and discuss the long time existence. This is a joint work with Yuan Yuan and Yuguang Zhang.

Speaker: Yi Li (Department of mathematics, Shanghai Jiao Tong University)

2:50 PM Coffee Break

15 Transversal Yamabe problem

In this talk I will introduce a Yamabe type problem on a Riemannian foliation and give a sufficient condition under which it has a solution. Moreover, I will talk about its difficulty.

Speaker: Guofang Wang (University of Freiburg)

7:00 PM Conference Dinner

Friday, October 14

Registration, Certificates & Reimbursements (Frau Peters)

16 Special metrics on conic spheres

I will present the problem of finding special metrics on conic spheres. The first one is the "least-pinched" metric when the conic sphere does not admit constant curvature metrics, the second is a metric realizing minimal volume for conic sphere in the sense of Gromov. This is a joint work with Hao Fang.

Speaker: Mijia Lai (Shanghai Jiao Tong University)

17 Moduli spaces of metrics of nonnegative sectional curvature

In my talk I will give a survey on results concerning the topology of moduli spaces of metrics of nonnegative sectional curvature for closed manifolds.

Speaker: Anand Dessai (University of Fribourg)

12:00 PM Lunch Break

18 Positivity and higher Teichmüller theory

Classical Teichmüller space describes the space of conformal structures on a given topological surface S. It plays an important role in several areas of mathematics as well as in theoretical physics. Higher Teichmüller theory generalizes several aspects of classical Teichmüller theory to the context of Lie groups of higher rank, such as the symplectic group PSp(2n; R) or the special linear group PSL(n; R). So far, two families of higher Teichmüller spaces are known. The Hitchin component, which is defined when the Lie group is a split real forms, and the space of maximal representations, which is defined for Lie groups of Hermitian type. Interestingly, both families are linked with various notions of positivity in Lie groups. In this talk I will give an introduction to higher Teichmüller theory, introduce new positive structures on Lie groups and discuss the (partly conjectural) relation between the two.

Speaker: Anna Wienhard (Heidelberg University)

2:50 PM Coffee Break

19 Approximately fibering a manifold over an aspherical one.

This talk is a report on joint work with W. Lueck and W. Steimle. Let $p:M \to B$ be a continuous map between closed connected manifolds such the induced map $P$ on fundamental groups is an epimorphism and $B$ is aspherical. Let $F(p)$ denote the homotopy fiber of $p$. An explicit model for $F(p)$ is the covering space of $M$ corresponding to the kernel of $P$.) The question addressed in this talk is to give useful sufficient conditions which guarantee that $p$ is homotopic to an approximate manifold fibration $q:M \to B$; i.e. a continuous map such that $q^{-1}(U)$ is homotopy equivalent to $F(p)$ for each open subset $U$ of $B$ which is homeomorphic to $R^n$ where $n= \dim B$. We do this for a large class of aspherical manifolds $B$ including all negatively curved manifolds of dimension different from 4.

Speaker: F. Thomas Farrell (YMSC and Dept. of Mathematics Tsinghua University)

4:30 PM Farewell Coffee