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.


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.


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.


Andreas Ott. (ETH Zurich): Gauged Gromov-Witten invariants via perturbation of the symplectic vortex equations.

Abstract: Gauged Gromov-Witten invariants are the gauge-theoretic generalization of Gromov-Witten invariants for symplectic manifolds equipped with a Hamiltonian Lie group action. These invariants are defined by counting solutions of the symplectic vortex equations. They were introduced by Cieliebak, Gaio, and Salamon for actions of arbitrary compact Lie groups on aspherical manifolds (i.e. where the symplectic form vanishes on all spherical homology classes) and by Mundet for semi-free circle actions on compact monotone manifolds. The main reason for the additional assumptions are complications in obtaining transversality for the boundary strata of the compactified moduli space of solutions of the vortex equations occurring in the presence of a group action. In this talk, I will present a perturbation scheme for the vortex equations that solves these transversality problems in a natural way, and explain how to define the gauged Gromov-Witten invariants for actions of arbitrary compact Lie groups on monotone symplectic manifolds.


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.


Joel Fine. (Brussels): A gauge theoretic approach to the anti-self-dual Einstein equations.

Abstract: I will describe work in progress which is joint with Dmitri Panov. Given an SO(3)-bundle E over a 4-manifold M, I will talk about SO(3) connections in E satisfying a certain curvature inequality, called "definite connections". It turns out that a definite connection defines a Riemannian metric on M. There is an "energy functional" on the space of definite connections and the absolute minimum, when it exists, defines an anti-self-dual Einstein metric on M, with non-zero scalar curvature. In this way we turn the anti-self-dual Einstein equations into a first order PDE for a connection. I will discuss this energy functional and it's gradient flow. I will try to persuade you that it is analogous to the Yang-Mills functional. I will also discuss the analogy between this set-up and a symplectic setting considered by Donaldson related to hyperkahler 4-manifolds (these are precisely the anti-self-dual Einstein metrics with zero scalar curvature). Finally, if there is any time left, I will explain how to fit the problem of minimising the energy functional into the framework of moment map geometry. In this way we can see the anti-self-dual Einstein equations as the zeros of a certain moment map.

Kirill Krasnov (Nottingham): Renormalized volume of hyperbolic 3-manifolds.

Abstract: The notion of renormalized volume of, in general, asymptotically hyperbolic Einstein manifolds has origins in the AdS/CFT correspondence of string theory. We review its applications for the case of 3 dimensions, which is special as the relevant manifolds are then completely characterized by the conformal structures of their boundary components and the renormalized volume becomes a function on several copies of Teichmuller space. This function is very interesting, as it can be shown to be a Kahler potential for the Weil-Petersson metric on Teichmuller space. The arising relation between 2-dimensional (boundary) structures and 3-dimensional geometrical ones turns out to be quite useful, for it allows to prove some non-trivial 2-dimensional statements by rather elementary computations done in the 3-dimensional setting. Examples include McMullen's quasifuchsian reciprocity and the fact that Thurston's grafting map is symplectic. The talk is based on a recent review arXiv:0907.2590.

Hans-Joachim Hein (Princeton): Complete hyperkaehler metrics from rational elliptic surfaces.

Abstract: Yau's solution to the Calabi conjecture shows that every Kaehler class on a compact complex manifold with c_1 = 0 contains a unique Ricci-flat Kaehler metric. In complex dimension 2, these are hyperkaehler metrics on the K3 surface. I will talk about a non-compact analog of this story: existence and geometric properties of complete hyperkaehler metrics on the complement of a fiber in a rational elliptic surface. The asymptotic geometry of these metrics depends in an interesting fashion on the structure of the fiber D that was removed: cylindrical ("ALH") if D is smooth, cone times torus ("ALG") if D is singular of finite monodromy, and more irregular in the remaining cases. ALG and ALH spaces can be expected to play a role as instanton bubbles for Ricci-flat metrics on K3 in certain collapsing limits. The results are based on analytic work due to Tian-Yau and a geometric construction due to Gross-Wilson.

Grigory Mikhalkin (Geneva): Tropical rational curves on K3 and related enumerative problems.

Abstract: Tropical geometry can be used not only in the context of generic enumerative problems (such as those directly captured by the Gromov-Witten theory), but also for some non-generic configurations. As an example we consider degree d rational curves in P^2 passing through 3d-1 points contained in the union of three coordinate lines and see tropically 3200 rational plane section in a quartic surface in P^3.

Elmas Irmak (Bowling Green): Mapping Class Groups and Complexes of Curves on Orientable/Nonorientable Surfaces.

Abstract: will talk about the relation between the mapping class groups of surfaces and the automorphism groups and the superinjective simplicial maps of the complexes of curves on surfaces for both orientable and nonorientable surfaces. I will also talk about the proof that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface if g + n is at least 5 or at most 3, where g is the genus and n is the number of boundary components of the surface.

Felix Schulze (Berlin): Stability of Hyperbolic Space under Ricci flow.

Abstract: We consider metrics, which are C^0-perturbations of the hyperbolic metric on H^n. If the perturbation is bounded in L^2 and small enough in C^0, we show that for dimensions 4 and higher the scaled Ricci harmonic map heat flow of such a metric converges exponentially fast in C^\infty to the hyperbolic metric as time approaches infinity. We also prove a related result for Ricci flow and for the 2-dimensional conformal Ricci flow. This is joint work with O. Schnuerer and M. Simon.

Robert Berman (Chalmers): Twisted analytic torsion on Fano manifolds and vortices.

Abstract: Inspired by their arithmetic Riemann-Roch theorem Gillet-Soule conjectured in the nineties that the regularized determinant of the Kodaira Laplacian is bounded from above when seen as a functional on the space of all Hermitian metrics on a line bundle over a Kahler manifold. In the case of the two-sphere this was confirmed by Fang who in turn conjectured that the functional is maximized precisely on the Fubini-Study metric (up to automorphisms). In this talk I will explain who to deduce Fang's conjecture by maximizing another functional recently introduced by Berndtsson. This latter functional can be seen as an adjoint version of Donaldson's L-functional and generalizes the Ding-Tian functional, whose extremals are Kahler-Einstein metrics on Fano manifolds. I will also point out applications to the study of the twisted analytic torsion on Fano manifolds - i.e. an alternating sums of determinants as above - and to the uniqueness problem for vortices in the mean field and Chern-Simons-Higgs equations on a Riemann surface.

Jonny Evans (Cambridge and ETH): Isotopy of Lagrangian submanifolds.

Abstract: Lagrangian submanifolds are an important class of objects in symplectic geometry. They arise in diverse settings: as vanishing cycles in complex algebraic geometry, as invariant sets in integrable systems, as Heegaard tori in Heegaard-Floer theory and of course as "branes" in the A-model of mirror symmetry. We ask the difficult question: when are two Lagrangian submanifolds isotopic? Restricting to the simplest case of Lagrangian spheres in rational surfaces we will give examples where this question has a complete answer. We will also give some very pictorial examples (due to Seidel) illustrating how two Lagrangians can fail to be isotopic.

Andriy Haydys (Imperial): Invariants of low-dimensional manifolds arising from higher dimensional gauge theory.

Abstract: The anti-self-duality equation has been generalized to higher dimensions by Donaldson and Thomas in the mid 90s. In this talk I will discuss some gauge-theoretical problems on low-dimensional manifolds arising from the higher dimensional asd equation. I will show that one can rediscover certain well-known invariants of low-dimensional manifolds in this way as well as obtain some new problems.

Ken Baker (Miami): Cabling open books and contact structures

Abstract: A knot with fibered exterior induces a contact structure on the 3-manifold. Aside from certain exceptions, each cable of such a knot also has fibered exterior. We'll examine how the supported contact structure changes under the operation of cabling. This extends results of Hedden to manifolds other than S^3 and to rationally null homologous knots.

Simon Brendle (Stanford): Curvature, sphere theorems, and the Ricci flow.

Abstract: In 1926, Hopf proved that any compact, simply connected Riemannian manifold with constant curvature 1 is isometric to the standard sphere. Motivated by this result, Hopf posed the question of whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere. This question has been studied by many authors over the past six decades, a milestone being the Topological Sphere Theorem proved by Berger and Klingenberg in 1960. In this lecture, I will discuss the history of this problem, and describe the proof (joint with R. Schoen) of the Differentiable Sphere Theorem. This theorem classifies all manifolds with 1/4-pinched curvature up to diffeomorphism. The distinction between homeomorphism and diffeomorphism is significant in light of the exotic spheres constructed by Milnor; the proof uses the Ricci flow technique introduced by Hamilton.

Fernando Coda Marques (IMPA): Deformations of the hemisphere that increase scalar curvature.

Abstract: The Positive Mass Theorem of General Relativity (Schoen-Yau'79, Witten'81) implies the following well-known rigidity result: if g is a metric of nonnegative scalar curvature on Euclidean space, and g coincides with the Euclidean metric outside a compact set, then g is flat. Inspired by that, Min-Oo (1988) proved the analogous rigidity statement for the hyperbolic space, and made the following conjecture for the spherical setting: if g is a metric of scalar curvature greater than or equal to n(n-1) on the standard hemisphere, and g coincides with the standard metric in a neighborhood of the equator, then g has constant sectional curvature 1. In this talk we will describe the construction of counterexamples to Min-Oo's conjecture in any dimension greater than or equal to three. This is joint work with Simon Brendle and Andre Neves.