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Abstracts for BICMR-HCM: Beijing meets Bonn

Alternatively have a look at the program.

Opening/BICMR-HCM: Beijing meets Bonn

Posted in
Date: 
Mon, 2015-11-02 09:00 - 09:15
Location: 
MPIM Lecture Hall

BICMR postdoc program presentation

Posted in
Date: 
Mon, 2015-11-02 09:15 - 09:45
Location: 
MPIM Lecture Hall

Dual complex of pairs

Posted in
Speaker: 
Chenyang Xu
Affiliation: 
BICMR
Date: 
Mon, 2015-11-02 10:15 - 11:15
Location: 
MPIM Lecture Hall

Dual complex characterizes how components of a divisor intersect each other. We will sketch the recent progress on using minimal model program to study it, with applications in various natural settings.

Small eigenvalues of the Laplacian on surfaces

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Speaker: 
Werner Ballmann
Affiliation: 
MPIM
Date: 
Mon, 2015-11-02 11:15 - 12:15
Location: 
MPIM Lecture Hall

Eigenvalues of the Laplacian on hyperbolic surfaces are called small, if they lie below $1/4$, the bottom of the spectrum of the Laplacian on the hyperbolic plane. Buser showed that, for any $n \in \mathbb{N}$ and $\epsilon > 0$, the closed surface $S$ of genus $g\ge 2$ carries a hyperbolic metric with $2g - 2$ eigenvalues below $\epsilon$ and $n$ eigenvalues below $1/4 + \epsilon$. Buser's results were refined by Schmutz, and they conjectured that a hyperbolic metric on $S$ has at most $2g - 2$ small eigenvalues.

Genus-2 generating functions for semisimple cohomological field theory

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Speaker: 
Xiaobo Liu
Affiliation: 
BICMR
Date: 
Mon, 2015-11-02 14:00 - 15:00
Location: 
MPIM Lecture Hall

An axiomatic definition of cohomological field theories (CohFT) was introduced by Kontsevich and Manin. This theory includes Gomov-Witten thoery and quantum singularity theory as special cases. The genus-0 part of a CohFT introduces a Frobenius manifold structure. When the Frobenius manifold is semisimple, the genus-2 potential function can be solved from universal equations or Virasoro constraints. The solution depends on the so called canonical coordinates on the Frobenius manifolds. Recently B. Dubrovin, S. Liu, and Y.

BKMP remodeling conjecture and its applications

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Speaker: 
Bohan Fang
Affiliation: 
BICMR
Date: 
Mon, 2015-11-02 15:00 - 16:00
Location: 
MPIM Lecture Hall

Mirror symmetry predicts Gromov-Witten theory for a Calabi-Yau manifold from the B-model of its mirror. The BKMP (Bouchard-Klemm-Marino-Pasquetti) remodeling conjecture is a mirror symmetry statement predicting all genus open-closed Gromov-Witten theory for a toric CY 3-orbifold from the topological recursion on its mirror curve. Nice features of the topological recursion as B-model give many desired properties of GW invariants, which are usually difficult to prove by other means. I will sketch a proof of BKMP conjecture and a construction of the global mirror curve over the Kahler moduli.

Survey on $L^2$-invariants

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Speaker: 
Wolfgang Lück
Affiliation: 
U Bonn
Date: 
Mon, 2015-11-02 16:30 - 17:30
Location: 
MPIM Lecture Hall

We give a survey on $L^2$-invariants such as $L^2$-Betti numbers and $L^2$-torsion. We describe some of their applications to questions in topology, geometry, group theory and algebra and discuss some open problems.

HCM presentation

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Speaker: 
Michael Meier
Affiliation: 
Chief Administrator HCM
Date: 
Tue, 2015-11-03 09:15 - 09:45
Location: 
MPIM Lecture Hall

Presentation of the Hausdorff Center for Mathematics, in particular its opportunities for international postdocs.

Super-Ricci flows of metric measure spaces

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Speaker: 
Karl-Theodor Sturm
Affiliation: 
U Bonn
Date: 
Tue, 2015-11-03 10:15 - 11:15
Location: 
MPIM Lecture Hall

A time-dependent family of Riemannian manifolds is a super-Ricci flow if $2 Ric + \partial_t g \ge 0.$ This includes all static manifolds of nonnegative Ricci curvature as well as all solutions to the Ricci flow equation. We extend this concept of super-Ricci flows to time-dependent metric measure spaces. In particular, we present characterizations in terms of dynamical convexity of the Boltzmann entropy on the Wasserstein space as well in terms of Wasserstein contraction bounds and gradient estimates. And we prove stability and compactness of super-Ricci flows under mGH-limits.

K-stability implies CM-stabilit

Posted in
Speaker: 
Gang Tian
Affiliation: 
BICMR
Date: 
Tue, 2015-11-03 11:15 - 12:15
Location: 
MPIM Lecture Hall
Both K-stability and CM-stability were first introduced on Fano manifolds in 90s and generalized to any polarized projective manifolds. In this talk, I will show how the K-stable implies CM-stable. I will also discuss their relation to Geometric Invariant Theory and the existence problem of Kahler metrics with constant scalar curvature.
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