Schedule of talks:

Abstracts

Oh : In this talk, I will present a proof of Weinstein's conjecture on strongly symplectically-fillable contact manifolds in all dimensions: Any compact strongly symplectically fillable contact manifold carries a closed characteristic.

Getzler : The moduli space of Poisson brackets on a manifold is actually a 2-groupoid. I explain how a sense in which the automorphisms of this 2-groupoid form a 3-group, and show how the de Rham complex gives an abelian 3-subgroup.

Weinberger : This talk will be an overview of some ideas related to the role of scale in thinking about various sorts of objects. I hope to discuss the problems of clustering in datasets, quasi-isometry types of discrete groups, closed geodesics on Riemannian manifolds, and methods of obtaining critical points for functionals that go beyond Morse theory.

Orr : L^2 signatures play a central role in the study of knot concordance, a classical relation on knots first defined and studied by Fox and Milnor, and closely allied with deep considerations in singularity theory and the classification of 4-manifolds. Using a new approach which subsumes past results, we extend the above techniques to a broader class of problems, and significantly extend key results concerning invariance of L^2 signatures and betti numbers.

Cappell : The talk explores classifications of topological group actions using their singularities. The methods, developed in joint work with Min Yan and Shmuel Weinberger, yield functorialities for such classifications of actions.

Schuermann : Given a compact singular Whitney stratified subset X of an oriented manifold M, one can associate to a constructible function on X a corresponding conic Lagrangian cycle in the cotangent bundle T*M. This can be done for example with the help of stratified Morse theory in the sense of Goresky-MacPherson. Then one defines for a connected component of the stratified critical locus of a one form on M an index as a micro-local intersection number in T*M. The sum of these indices calculates the Euler characteristic of X weighted by the constructible function. Suitable choices of the constructible function give different notions of indices used in the literature.

Leung : We incorporate Fourier transform to the current understanding of SYZ mirror symmetry in the case of a toric Calabi-Yau manifold equipped with the non-toric Lagrangian fibration. In particular we give a geometric construction of the Landau-Ginzburg mirror via Fourier transform of generating functions of open Gromov-Witten invariants.

Lerman : This is joint work with Yael Karshon. The paradigmatic result in symplectic toric geometry is the paper of Delzant that classifies compact connected symplectic manifolds with effective completely integrable torus actions, the so called (compact) symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a simple unimodular ("Delzant") polytope. This gives a bijection between simple unimodular polytopes and isomorphism classes of compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map of the quotient need not be an embedding. Moreover, even when the map of the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class makes an appearance. None the less there is a classification non-compact symplectic toric manifolds and I will explain what it is.

Suvaina : The existence or non-existence of Einstein metrics on a topological 4-manifold is strongly related to the differential structure considered. We analyze this dependency for manifolds with small topology. We also consider 4-manifolds with the canonical smooth structure, and for a large class of manifolds we prove non-existence theorems of group invariant metrics. The main techniques used come from Seiberg-Witten theory, the geometry of complex surfaces and symplectic topology.

Tamarkin : Using microlocal analysis of sheaves on \Re^n we prove a generalization of a theorem by Eliashberg-Kim-Polterovich on contact non-squezability of B\times S^1\subset \Re^{2n}\times S^1 where B is a ball of radius more than 1 in \Re^{2n}.

Hind : I will describe joint work with Ely Kerman giving restrictions on symplectic embeddings which in some cases improve on Gromov's nonsqueezing theorem. Constructions show that in these cases the new restrictions turn out to be asymptotically sharp. If time permits we will also discuss some consequences for displacement energies.

Unal : Harvey and Lawson showed that for any calibration phi there is an integer bound for the homotopy dimension of a strictly phi-convex domain and constructed a method to get these domains by using phi-free submanifolds. I will explain how to get phi-free submanifolds with different homotopy types in certain calibrated geometries and talk about the recent results about their existence.

Karakurt : We will discuss a practical method to calculate c^+, a Floer theoretic invariant of contact structures defined by Ozsváth and Szabó, on certain plumbed 3-manifolds. The method uses only formal TQFT properties of Heegaard-Floer theory and seems to be carried out in other kinds of Floer homologies, particularly the monopole Floer homology introduced by Kronheimer and Mrowka. Several applications of this formalism will be discussed if time permits.

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