Schedule of talks:

Abstracts

Treumann : This is a talk on joint work with Bohan Fang, Chiu-Chu Melissa Liu, and Eric Zaslow. An ample, equivariant line bundle on a toric variety X determines a polytope in R^n. We extend this correspondence to an equivalence between the derived category of equivariant coherent sheaves on X and a certain derived category of constructible sheaves on R^n. We connect this result to mirror symmetry through a recent theorem of Nadler and Zaslow, which identifies the category of constructible sheaves on R^n with a Fukaya category of Lagrangian submanifolds of the cotangent bundle.

Viaclovsky : We determine the group of conformal automorphisms of the self-dual metrics on #nCP^2 due to LeBrun for n>=3, and Poon for n=2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H^3 minus a finite number of points, called monopole points. We show that for n>=3 connected sums, any conformal automorphism is a lift of an isometry of H^3 which preserves the set of monopole points. Furthermore, we prove that for n=2, such lifts form a subgroup of index 2 in the full automorphism group, which we show is a semi-direct product (U(1) x U(1)) |x D_4, the dihedral group of order 8. This is joint work with Nobuhiro Honda.

Kim : On affine algebraic variety, we can study both the symplectic homology and the Gromov-Witten invariants. In this talk, I will give a brief introduction followed by a blow-up approach to understand relationship between the symplectic homology and the Gromov-Witten invariants. If time permits, I will discuss how the approach can be applied on del-Pezzo surfaces and the cotangent bundle of 2-sphere.

Paul : Let (X,L) be a polarized algebraic manifold. I have recently proved that the Mabuchi energy of (X,L) is bounded from below along any degeneration if and only if the Hyperdiscriminant polytope contains the Chow polytope (with respect to the various Kodaira embeddings). This completes the analysis initiated by Ding and Tian in their 1992 Inventiones paper "Kahler Einstein metrics and the Generalized Futaki Invariant", and therefore gives the final form to Tian's concept of K-semistability.

Kalafat : We construct handlebody diagrams of families of non-simply connected Locally Conformally Flat(LCF) 4-manifolds realizing rich topological types, which are obtained from conformal compactifications of the 3-manifolds, that are built from the Panelled Web Groups. These manifolds have strictly negative scalar curvature and the underlying topological 4-manifolds do not admit any Einstein metrics. This is a joint work with S. Akbulut.

Szekelyhidi : On Fano manifolds we study the supremum of the possible t such that there exists a metric in the first Chern class with Ricci curvature bounded below by t. This is the same as the existence time for Aubin's continuity method for finding Kahler-Einstein metrics. For the projective plane blown up in one point we show that this supremum is 6/7.

Savelyev : I will describe how to use (cohomological) field theoretic ideas to study "Morse theory" for the Hofer length functional on the path spaces of the group of Hamiltonian symplectomorphisms of a symplectic manifold. The virtual aspect is due to the fact that there is nothing resembling gradient flow, yet we show that this functional has properties of a perfect Morse-Bott function (with a flow). As an application we show how this can be used to study topology-geometry of \Omega Ham(G/T).

Auroux : Following the philosophy of the Strominger-Yau-Zaslow conjecture, we will focus on the construction of the mirror manifold of a blowup. Namely, we first describe how to construct a Lagrangian torus fibration on the blowup of a toric variety along a codimension 2 subvariety contained in a toric hypersurface. Then we discuss the SYZ mirror and its instanton corrections, to provide an explicit description of the mirror Landau-Ginzburg model (possibly up to higher order corrections to the superpotential). This construction allows one to recover geometrically the predicted mirrors in various interesting settings: pairs of pants, curves of arbitrary genus, etc. This is joint work with Mohammed Abouzaid and Ludmil Katzarkov.

Ivansic : Many hyperbolic 3-manifolds are known to be link complements in the 3-sphere. Extending this phenomenon to dimension 4, we discuss the situation and provide examples that are complements of tori and/or Klein bottles in the 4-sphere and other simply-connected 4-manifolds. One of our tools is a procedure that converts a side-pairing of a polyhedron to a handle decomposition. In dimension 3, this procedure helps produce the link diagram.

Garcia Leon : We prove isoperimetric inequality on a Riemannian manifold, assuming that the Cheeger constant for balls satisfies a certain estimation.

Lanneau : In this talk, I will recall the classification of SL(2,R) orbits'closures in the moduli space of genus 2 translation surfaces (after McMullen). Then, I will give some examples in genus 3 explaining the difference between genus 2 and 3. (This is a joint work with P. Hubert and M. Moeller).

Koca : In 1997 Tonnesen-Friedman constructed a family of extremal Kahler metrics on special kind of ruled surfaces called "generalized Hirzebruch surface". Using a variational theorem due to Chen-LeBrun-Weber we will show that for every genus >1 there is a generalized Hirzebruch surface such that one of the extremal Kahler metrics in that family is actually conformally Einstein outside a certain hypersurface.

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