Past 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
16 November 2015
14:15
Marta Mazzocco
Abstract

The famous Greek astronomer Ptolemy created his well-known table of chords in order to aid his astronomical observations. This table was based on the renowned relation between the four sides and the two diagonals of a quadrilateral whose vertices lie on a common circle.

In 2002, the mathematicians Fomin and Zelevinsky generalised this relation to introduce a new structure called cluster algebra. This is a set of clusters, each cluster made of n numbers called cluster variables. All clusters are obtained from some initial cluster by a sequence of transformations called mutations. Cluster algebras appear in a variety of topics, including total positivity, number theory, Teichm\”uller theory and computer graphics. A quantisation procedure for cluster algebras was proposed by Berenstein and Zelevinsky in 2005.

After introducing the basics about cluster algebras, in this talk we will link cluster algebras to the theory of Painlevé equations. This link will provide the foundations to introduce a new class of cluster algebras of geometric type. We will show that the quantisation of these new cluster algebras provide a geometric setting for the Berenstein–Zelevinsky construction.  

  • Geometry and Analysis Seminar

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