Past Geometry and Analysis Seminar

25 April 2016
14:15
Abstract

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, 
with K-contact manifolds corresponding to symplectic manifolds. It is an interesting problem to find
obstructions for a closed manifold to admit such types of structures and in particular, to construct
K-contact manifolds which do not admit Sasakian structures. In the simply-connected case, the
hardest dimension is 5, where Kollar has found subtle obstructions to the existence of Sasakian 
structures, associated to the theory of algebraic surfaces.
In this talk, we develop methods to distinguish K-contact manifolds from Sasakian ones in 
dimension 5. In particular, we find the first example of a closed 5-manifold with first Betti number 0 which is K-contact but which carries no semi-regular Sasakian structure.

 (Joint work with J.A. Rojo and A. Tralle).

  • Geometry and Analysis Seminar
29 February 2016
14:15
Otis Chodosh
Abstract

Work of Schoen--Yau in the 70's/80's shows that area-minimizing (actually stable) two-sided surfaces in three-manifolds of non-negative scalar curvature are of a special topological type: a sphere, torus, plane or cylinder. The torus and cylinder cases are "borderline" for this estimate. It was shown by Cai--Galloway in the late 80's that the torus can only occur in a very special ambient three manifold. We complete the story by showing that a similar result holds for the cylinder. The talk should be accessible to those with a basic knowledge of curvature in Riemannian geometry.

  • Geometry and Analysis Seminar
22 February 2016
14:15
OAC-manifolds meeting: Diarmuid Crowley
Abstract
An exotic (n+1)-sphere has disc of origin D^k if k is the smallest integer such that some clutching diffeomorphism of the n-disc which builds the exotic sphere can be written as an (n-k)-parameter family of diffeomorphisms of the k-disc.
 
In this talk I will present a new method for constructing exotic spheres with small disc of origin via Toda brackets.  
 
This method gives exotic spheres in all dimensions 8j+1 and 8j+2 with disc of origin 6 and with Dirac operators of non-zero index (such spheres are often called "Hitchin spheres").
 
I will also briefly discuss implications of our results for the space of positive scalar curvature metrics on spin manifolds of dimension 6 and higher, and in particular the relationship of this project to the work of Botvinnik, Ebert and Randal-Williams.
 
This is part of joint work with Thomas Schick and Wolfgang Steimle.
  • Geometry and Analysis Seminar
1 February 2016
02:15
Marina Logares
Abstract

In the same way that the classical Torelli theorem determines a curve from its polarized Jacobian we show that moduli spaces of parabolic bundles and parabolic Higgs bundles over a compact Riemann surface X  also determine X. We make use of a theorem of Hurtubise on the geometry of algebraic completely integrable systems in the course of the proof. This is a joint work with I. Biswas and T. Gómez 

  • Geometry and Analysis Seminar
30 November 2015
14:15
Daniel Halpern-Leistner
Abstract

I will discuss theta-stability, a framework for analyzing moduli problems in algebraic geometry by finding a special kind of stratification called a theta-stratification, a notion which generalizes the Kempf-Ness stratification in geometric invariant theory and the Harder-Narasimhan-Shatz stratification of the moduli of vector bundles on a Riemann surface.

  • Geometry and Analysis Seminar

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