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.


Albrecht Klemm. (Bonn): Direct integration in Matrix Models and Topological String Theory.

Abstract: Direct integration is a technique to solve the B-model holomorphic equation using modular invariance and physical boundary conditions. Together with a choice of flat coordinates it allows to extract the perturbative part of topological string partition function everywhere in the moduli space. As explained in the first part of the talk, the formalism solves the perturbative sector of matrix models, topological string theory on local Calabi-Yau manifolds and supersymmetric gauge theories. In the second part of the talk we focus on the possible non-perturbative completion of the perturbative result mainly in the example of the matrix model.


Alessandro Ghigi. (Milan): Satake compactifications, moment map and first eigenvalue of the Laplacian.

Abstract: n 1994 Bourguignon, Li and Yau studied upper bounds for the first eigenvalue of the Laplacian of Kaehler metrics in a fixed Kaehler class. Their main tool is a map from SL(n)/SU(n) to the set of Hermitian matrices with trace 1. We will discuss joint work with Leonardo Biliotti showing that this map is a particular case (corresponding to projective space) of a more general construction that works for arbitrary flag manifolds. Given a complex semisimple Lie group G, a maximal compact subgroup K and an irreducible unitary representation of K, one constructs a flag manifol M and a Satake compactification of G/K. Let O be the image of M by the moment map. We will show that the Bourguignon-Li-Yau map is a homeomorphism of the compactification onto the convex envelope of O. From this we will deduce that on Hermitian symmetric spaces the first eigenvalue is maximal for the symmetric metric.


Julien Grivaux. (Jussieu): Topological properties of symplectic and almost-complex punctual Hilbert schemes.

Abstract: If X is a smooth projective complex surface, the punctual Hilbert schemes of X have been the objects of intensive study in the past twenty years, starting with the works of Göttsche, Grojnowski, Nakajima, Lehn and others. More recently, Voisin constructed punctual Hilbert schemes for almost-complex four-manifolds which are stable almost-complex differentiable manifolds. They are symplectic if the initial four-manifold is symplectic. In this talk, I will explain the methods developed to study the cohomology ring of symplectic Hilbert schemes, and the complex cobordism class of almost-complex ones.


Nick Addington (Imperial): Derived Categories of Intersections of Quadrics.

Abstract: If X is a complete intersection of hypersurfaces in P^n with canonical bundle O(-k) for some k >= 0, its derived category has a semi-othogonal decomposition
D^b(X) = < O(-k+1), ..., O(-1), O, A >,
where A is some triangulated category that should be regarded as the "interesting piece" of D^b(X). Orlov describes it as the "derived category of singularities", and when X is a hypersurface, as the "derived category of matrix factorizations". When X is a complete intersection of quadrics, Kuznetsov describes A as the derived category of a non-commutative variety.
But we would like to see some geometry. For complete intersections of even-dimensional quadrics, we can understand A in terms of a moduli problem with a very classical flavor. I will discuss its history, which includes Reid's thesis and ultimately goes back to Weil, and then my own result, which is that for the intersection of four even-dimensional quadrics, A is the derived category of twisted sheaves on a certain non-Kaehler complex threefold. If time permits I may speculate about rationality of cubic fourfolds.


Vincent Minerbe. (Jussieu): On ALF gravitational instantons.

Abstract: Basically, ALF gravitational instantons are complete non-compact hyperkahler manifolds whose geometry at infinity is asymptotic to a circle fibration over the Euclidean three-space, with fibres of asymptotically constant length. In this talk, I will describe examples and explain a classification result, fitting into a conjecture inspired from string theory and P. B. Kronheimer's earlier works.


Brendan Guilfoyle. (Tralee): Neutral Kaehler geometry, mean curvature flow and holomorphic discs.

Abstract: In this talk we discuss Kaehler 4-manifolds in which the symplectic structure does not tame the complex structure, as is usually assumed. Rather, the metric formed by the complex structure and symplectic form is indefinite, of signature (2,2). Submanifold theory in such manifolds turns out to be very rich. We show how the classical Caratheodory conjecture on the number of umbilic points on a closed convex surface in Euclidean 3-space can be reformulated as a question of the index of isolated complex points on Lagrangian surfaces in the space of oriented affine lines of 3-space. We then explain how mean curvature flow can be used to prove the existence of holomorphic discs satisfying a Lagrangian boundary condition. This restricts the Keller-Maslov index of the boundary and completes the proof of the Conjecture.


Kentaro Nagao. (Oxford and Kyoto): Vertex operators in Donaldson-Thomas theory.

Abstract: I will introduce Okounkov-Reshetikhin-Vafa type vertex operators to compute the generating function of Donaldson-Thomas invariants of a small crepant resolution of a 3-dimensional toric Calabi-Yau variety. The commutator relation of the vertex operators gives the wall-crossing formula of Donaldson-Thomas type invariants.


Huy Nguyen. (Warwick): Generalized Sphere Theorems and Curvature Flows.

Abstract: In this talk we will discuss how recent advances in curvature flows such as Brendle-Schoen proof of the differentiable sphere theorem by Ricci flow and Huisken-Sinestrari's classification of two-convex hypersurfaces in Euclidean space by mean curvature have led us to formulate new versions of classical sphere theorems, that is for certain curvature conditions invariant under geometric flows, we can classify the manifolds as connected sums of a finite collection of well understood manifolds. In particular, we will consider mean curvature flow in the sphere with a quadratic curvature condition, a condition different from two-convexity. Under this condition, we will investigate singularity formation of type I and II, a priori gradient estimates for the second fundamental form and surgeries and connected sums.


Rosa Sena-Dias. (Lisbon): Scalar-flat Kahler metrics on non-compact toric surfaces.

Abstract: In this talk we will discuss a new construction of scalar-flat Kahler toric metrics on non-compact 4-manifolds. The construction actually gives a new perspective on Gibbons-Hawking's so-called gravitational instantons. Perhaps more interestingly, it allows us to write down some new examples of complete scalar-flat Kahler metrics on some important non-compact toric varieties. We will give some background on such metrics and show symplectic toric geometry is well suited to tackle them. This is joint work with Miguel Abreu.


Andras Juhasz. (Cambridge): Cobordisms of sutured manifolds.

Abstract: Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology induced by decorated knot cobordisms.