Past Geometry and Analysis Seminar

16 October 2017
14:15
Lorenzo Foscolo
Abstract

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds.
In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.

 

  • Geometry and Analysis Seminar
12 June 2017
14:15
Bill Goldman
Abstract

We describe a general program for the classification of flat connections on topological manifolds. This is motivated by the classification of locally homogeneous geometric structures on manifolds, in the spirit of Ehresmann and Thurston.  This leads to interesting dynamical systems arising from mapping class group actions on character varieties. The mapping class group action is a discrete version of a continuous object, namely the extension of the Teichmueller flow to a  unversal character variety over over the tangent bundle of Teichmuller space. We give several examples of this construction
and discuss joint work with Giovanni Forni on a mixing property of this suspended flow.

  • Geometry and Analysis Seminar
29 May 2017
14:15
Philip Boalch
Abstract

The theory of connections on curves and Hitchin systems is something like a “global theory of Lie groups”, where one works over a Riemann surface rather than just at a point. We’ll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of “Dynkin diagrams” as a step towards classifying some examples of such objects.

  • Geometry and Analysis Seminar
22 May 2017
14:15
Simon Salamon
Abstract

I shall discuss Zauner's conjecture about the existence of n^2 mutually equidistant points in complex projective space CP^{n-1} with its standard Fubini-Study metric. This is the so-called SIC-POVM problem, and is related to properties of the moment mapping that embeds CP^{n-1} into the Lie algebra su(n). In the case n=3, there is an obvious 1-parameter family of such sets of 9 points under the action of SU(3) and we shall sketch a proof that there are no others. This is joint work with Lane Hughston.

 

  • Geometry and Analysis Seminar
15 May 2017
14:15
Lucas Branco
Abstract

The moduli space M(G) of Higgs bundles for a complex reductive group G on a compact Riemann surface carries a natural hyperkahler structure and it comes equipped with an algebraically completely integrable system through a flat projective morphism called the Hitchin map. Motivated by mirror symmetry, I will discuss certain complex Lagrangians (BAA-branes) in M(G) coming from real forms of G and give a proposal for the mirror (BBB-brane) in the moduli space of Higgs bundles for the Langlands dual group of G.  In this talk, I will focus on the real groups SU^*(2m), SO^*(4m) and Sp(m,m). The image under the Hitchin map of Higgs bundles for these groups is completely contained in the discriminant locus of the base and our analysis is carried out by describing the whole
(singular) fibres they intersect. These turn out to be certain subvarieties of the moduli space of rank 1 torsion-free sheaves on a non-reduced curve. If time permits we will also discuss another class of complex Lagrangians in M(G) which can be constructed from symplectic representations of G.

 

  • Geometry and Analysis Seminar
8 May 2017
14:15
Abstract

The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real
Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane. In this talk, I will explain how to compute the mod 2 Betti numbers of M_R using Morse theory. A key role in this calculation is played by the Abel-Jacobi map from symmetric products of X to the Jacobian of X.

  • Geometry and Analysis Seminar
1 May 2017
14:15
Marina Logares
Abstract

Character varieties have been studied largely by means of their correspondence to the moduli space of Higgs bundles. In this talk we will report on a method to study their Hodge structure, in particular to compute their E- polynomials. Moreover, we will explain some applications of the given method such as, the study of the topology of the moduli space of doubly periodic instantons. This is joint work with A. González, V.Muñoz and P. Newstead.

 

  • Geometry and Analysis Seminar
24 April 2017
14:15
Roland Grinis
Abstract

We will give an overview of the Soliton Resolution Conjecture, focusing mainly on the Wave Maps Equation. This is a program about understanding the formation of singularities for a variety of critical hyperbolic/dispersive equations, and stands as a remarkable topic of research in modern PDE theory and Mathematical Physics. We will be presenting our contributions to this field, elaborating on the required background, as well as discussing some of the latest results by various authors.

  • Geometry and Analysis Seminar
6 March 2017
14:15
Abstract

Instanton bundles were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970s as the holomorphic counterparts, via twistor
theory, to anti-self-dual connections (a.k.a. instantons) on the sphere S^4. We will revise some recent results regarding some of the basic
geometrical features of their moduli spaces, and on its possible degenerations. We will describe the singular loci of instanton sheaves,
and how these lead to new irreducible components of the moduli space of stable sheaves on the projective space.

  • Geometry and Analysis Seminar

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