Alan Weinstein. (Berkeley): On the Poisson brackets of constraints for Einstein's equations.

Abstract: When the Einstein equations of general relativity are formulated as a hamiltonian system on the cotangent bundle T*M(S) of the space M(S) of riemannian metrics on a typical Cauchy hypersurface S, the initial conditions are subject to constraints. The space of initial conditions satisfying the constraints is involutive, suggesting that the constraints are components of the momentum map for a hamiltonian action of a symmetry group. In fact, work in progress with Christian Blohmann and Marco Cezar Fernandes leads to the conclusion that the relevant symmetries form a groupoid rather than a group. We construct a Lie algebroid over T*M(S) which is a trivial bundle and for which the constant sections map to the hamiltonian vector fields of the constraint functions. This Lie algebroid is closely related to a groupoid whose morphisms are diffeomorphisms between spacelike hypersurfaces of lorentzian manifolds.


Iain Aitchison. (Melbourne): A concept of genus for finitely presented groups.

Abstract: Every finitely presented group arises as the fundamental group of an orientable compact manifold of any chosen dimension 4 or greater. Most groups do not occur as fundamental groups of 3-dimensional manifolds. For any compact orientable 3-manifold with non-empty boundary, add a cone to a point from one of its boundary components. We prove that every finitely presentable group arises as the fundamental group of such a space, and define the genus of the group as the smallest possible genus of coned boundary component giving a space with this fundamental group. As a result of Thurston and Perelman’s results on geometrization of 3-manifolds, fundamental groups of orientable 3-manifolds are theoretically now understood, and are exactly the groups of genus 0. Moreover, we obtain a new non-trivial invariant of closed orientable 3-manifolds, which vanishes if the manifold embeds in the 4-dimensional sphere. Our work raises questions concerning the applicability of 3-manifold techniques to understanding finitely presented groups, to decidability questions for calculating the genus of a group, and the determination of whether or not two groups of the same genus are isomorphic.


Jeff Giansiracusa. (Oxford): Formality of the framed little discs operad and 3-dimensional handlebodies.

Abstract: Complexes of graphs appear in many settings. Kontsevich proved that the little 2-discs operad is formal by introducing an appropriate graph complex, and there are complexes of graphs computing the homology of automorphism groups of free groups and of moduli spaces of curves. The framed little 2-discs operad has the interesting feature of being a cyclic operad. P. Salvatore and I prove that it is formal in a way compatible with its cyclic operad structure. The proof introduces a new type of graph complex in which the differential is a combination of edge contractions and deletions. As an application, there is also a complex of graphs computing the cohomology of the handlebody subgroups of mapping class groups.


Paul Johnson. (Imperial & Princeton): Equivariant Gromov-Witten theory of orbifold curves and Integrable Hierarchies.

Abstract: In a series of three papers, Okounkov and Pandharipande completely determine the Gromov-Witten theory of curves. Their method relies heavily on the infinite wedge, an algebraic framework that helps shed light on connections to the Virasoro conjecture and integrable hierarchies. We present the first step in extending their work to orbifold curves: showing that the Equivariant Gromov-Witten theory of orbifold P^1s satisfy the 2-Toda hierarchy.


Julius Ross. (Cambridge): Constant scalar curvature orbifold metrics and stability of orbifolds through embeddings in weighted projective spaces.

Abstract: There is a conjectural relationship due to Yau-Tian-Donaldson between stability of projective manifolds and the existence of canonical Kahler metrics (e.g. Kahler-Einstein metrics). Embedding the projective manifold in a large projective space gives on one hand a Geometric Invariant Theory stability problem (by changing coordinates on the projective space) and on the other a notion of balanced metric which can be used to approximate the canonical Kahler metric in question. I shall discuss joint work with Richard Thomas that extends this framework to orbifolds with cyclic quotient singularities using embeddings in weighted projective space, and examples that show how several obstructions to constant scalar curvature orbifold metrics can be interpreted in terms of stability.


Andre Neves. (Imperial): Rigidity theorems for 3-manifolds with positive scalar curvature.

Abstract: A classical theorem in Geometry states that a 3-manifold with nonnegative scalar curvature having an area minimizing torus has universal cover isometric to R^3. I will talk about extensions of this result to the case where the scalar curvature is strictly positive.


Sven Meinhardt. (Oxford): How to interpret modern Donaldson-Thomas theory?

Abstract: I will give an interpretation of modern Donaldson-Thomas theory in the classical as well as in the motivic framework. The latter can be considered as a generalised deformation quantisation of the classical case which has a nice geometrical interpretation in the symplectic world. The main part of the talk will be an overview of the theory, but it will also contain some new results.


Cristina Manolache. (Humboldt): Virtual Intersections.

Abstract: I will try to answer the following question: Given a morphism of smooth projective varieties, when can we express (certain) Gromov-Witten invariants of the source variety in terms of Gromov-Witten invariants of the target variety?


Luis Alvarez-Consul (Madrid): Moduli of quiver sheaves.

Abstract: I will explain a construction of the moduli of semistable quiver sheaves over a projective scheme, extending previous joint work with Alastair King for coherent sheaves. By "quiver sheaf" here, I mean a representation of a quiver in coherent sheaves. The main differences with related previous work by Alexander Schmitt come from the choice of a different semistability condition. Embedding this moduli space in a moduli space for representations of a different quiver in vector spaces, I can use the invariant theory for quiver representations to obtain affine and homogeneous coordinates on the moduli of quiver sheaves, respectively similar to the Hitchin map for Higgs bundles and the generalized theta functions for vector bundles.


Simon Donaldson. (Imperial): Gauge theory and exceptional holonomy.
Abstract: This talk will review material, well-known to specialists, on calibrated geometry and Yang-Mills theory over manifolds with holonomy SU(3), G_{2} or Spin(7). We will also describe extensions of the standard set-up, modelled on Gromov's "taming forms" for almost-complex structures.


Simon Donaldson. (Imperial): Moduli of Calabi-Yau 3-folds and instantons on G_2 manifolds.
Abstract: This talk will be largely speculative. First we consider the formal properties that could be expected of a "topological field theory" in 6+1 dimensions defined by G_{2} instantons. We explain that this could lead to holomorphic bundles over moduli spaces of Calabi-Yau 3-folds whose ranks are the DT-invariants. We also discuss in more detail the compactness problem for G_{2} instantons and associative submanifolds.