Past Geometry and Analysis Seminar

7 November 2016
14:15
Felix Schulze
Abstract

We will discuss two recent short-time existence results for (1) mean curvature of surface clusters, where n-dimensional surfaces in R^{n+k}, are allowed to meet at equal angles along smooth edges, and (2) for planar networks, where curves are initially allowed to meet in multiple junctions that resolve immediately into triple junctions with equal angles. The first result, which is joint work with B. White, follows from an elliptic regularisation scheme, together with a local regularity result for flows with triple junctions, which are close to a static flow of the half-planes. The second result, which is joint work with T. Ilmanen and A.Neves, relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow. 
 

  • Geometry and Analysis Seminar
31 October 2016
14:15
Alex Ritter
Abstract

Abstract: (This is joint work with Mark McLean, Stony Brook University N.Y.).


The classical McKay correspondence is a 1-1 correspondence between finite subgroups G of SL(2,C) and simply laced Dynkin diagrams (the ADE classification). These diagrams determine the representation theory of G, and they also describe the intersection theory between the irreducible components of the exceptional divisor of the minimal resolution Y of the simple surface singularity C^2/G. In particular those components generate the homology of Y. In the early 1990s, Miles Reid conjectured a far-reaching generalisation to higher dimensions: given a crepant resolution Y of the singularity C^n/G, where G is a finite subgroup of SL(n,C), the claim is that the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This has led to much active research in algebraic geometry in recent years, in particular Batyrev proved the conjecture in 2000 using algebro-geometric techniques (Kontsevich's motivic integration machinery). The goal of my talk is to present work in progress, jointly with Mark McLean, which proves the conjecture using symplectic topology techniques. We construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the crepant setup.

 

 

  • Geometry and Analysis Seminar
24 October 2016
14:15
Dario Beraldo
Abstract

I will define the notion of "sheaf of categories with a local action of Hochschild cochains" over a stack. (This notion is analogous to D-modules, in the same way as the notion of "sheaf of categories" is analogous to quasi-coherent sheaves.) I will prove that both categories appearing in geometric Langlands carry this structure over the stack of de Rham {\check{G}}-local systems. Using this, I will explain how to glue D-mod(Bun_G) out of *tempered* D-modules associated to smaller Levi subgroups of G.

 

  • Geometry and Analysis Seminar
17 October 2016
14:15
Jason Lotay
Abstract

Since Donaldson-Thomas proposed a programme for studying gauge theory in higher dimensions, there has
been significant interest in understanding special Yang-Mills connections in Ricci-flat 7-manifolds with holonomy
G_2 called G_2-instantons.  However, still relatively little is known about these connections, so we begin the
systematic study of G_2-instantons in the SU(2)^2-invariant setting.  We provide existence, non-existence and
classification results, and exhibit explicit sequences of G_2-instantons where “bubbling" and "removable
singularity" phenomena occur in the limit.  This is joint work with Goncalo Oliveira (Duke).

 

  • Geometry and Analysis Seminar
10 October 2016
14:15
Andrew Dancer
Abstract

We review the concept of solitons in the Ricci flow, and describe various methods for generating examples, including some where the equations

may be solved in closed form

  • Geometry and Analysis Seminar
6 June 2016
14:15
Thomas Schick
Abstract

Question: Given a smooth compact manifold $M$ without boundary, does $M$
 admit a Riemannian metric of positive scalar curvature?

 We focus on the case of spin manifolds. The spin structure, together with a
 chosen Riemannian metric, allows to construct a specific geometric
 differential operator, called Dirac operator. If the metric has positive
 scalar curvature, then 0 is not in the spectrum of this operator; this in
 turn implies that a topological invariant, the index, vanishes.

  We use a refined version, acting on sections of a bundle of modules over a
 $C^*$-algebra; and then the index takes values in the K-theory of this
 algebra. This index is the image under the Baum-Connes assembly map of a
 topological object, the K-theoretic fundamental class.

 The talk will present results of the following type:

 If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has
 non-trivial index, what conditions imply that $M$ does not admit a metric of
 positive scalar curvature? How is this related to the Baum-Connes assembly
 map? 

 We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$),
 Engel and new generalizations. Moreover, we will show how these results fit
 in the context of the Baum-Connes assembly maps for the manifold and the
 submanifold. 
 

  • Geometry and Analysis Seminar
23 May 2016
14:15
Vasilisa Shramchenko
Abstract

In 1995 N. Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation starting with any Poncelet trajectory, that is a closed billiard trajectory inscribed in a conic and circumscribed about another conic. In this talk I will show that Hitchin's construction is the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation. Moreover, this Okamoto transformation can be written in terms of an Abelian differential of the third kind on the associated elliptic curve, which allows to write down solutions to the corresponding Schlesinger system in terms of this differential as well. This is a joint work with V. Dragovic.

  • Geometry and Analysis Seminar
16 May 2016
14:15
Luc Nguyen
Abstract

The classical Yamabe problem asks to find in a given conformal class a metric of constant scalar curvature. In fully nonlinear analogues, the scalar curvature is replaced by certain functions of the eigenvalue of the Schouten curvature tensor. I will report on quantitative Liouville theorems and fine blow-up analysis for these problems. Joint work with Yanyan Li.
 

  • Geometry and Analysis Seminar

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