Schedule of talks:

Abstracts

Paul : In this talk we give new and elementary (without the use of derived categories) proofs of basic results of Gelfand, Kapranov, and Zelevinsky which express the dual of a smooth complex projective variety X as the determinant of a complex. For a space curve, this in turn is used to express the K-energy of X restricted to the Bergman metrics in terms of the dual of X. This relationship should extend to varieties of any dimension.

Viaclovsky : I will discuss various aspects of orthogonal complex structures on domains in Euclidean spaces, and the connection with twistor theory. A crucial tool in dimension six is a classification of threefolds of order one in the complex six-quadric. This is joint work with Lev Borisov and Simon Salamon.

Ache : We show that the Yamabe problem can be solved on compact Riemannian orbifolds under certain assumptions, adapting arguments of Aubin, Schoen, and others. We also present examples of compact orbifolds for which the Yamabe problem has no solution. This is joint work with Jeff Viaclovsky.

Lee : This is a joint work with Thomas H. Parker. We define a new type of symplectic “local Gromov-Witten invariant” of spin curves. When X is a Kahler surface with a smooth canonical divisor D, its (full) GW invariants are expressed in terms of such local invariants, which in turn are universal functions determined by the normal bundle of the canonical divisor D. We also show that how these local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. This yields an interesting theorem relat- ing two- and four-dimensional Gromov-Witten theory. We also explicitly compute these local invariants in some cases.

Tseng : Let G be a finite group. A G-gerbe over a space X may be intuitively thought of as a fiber bundle over X with fibers being the classifying space (stack) BG. In particular BG itself is the G-gerbe over a point. A more interesting class of examples consist of G-gerbes over BQ, which are equivalent to extensions of the finite group Q by G. Considerations from physics have led to conjectures asserting that the geometry of a G-gerbe Y over X is equivalent to certain "twisted" geometry of a "dual" space Y'. A lot of progresses have be made recently towards proving these conjectures in general. In this talk we'll try to explain theses conjectures in the elementary concrete examples of G-gerbes over a point or BQ.

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.

Ostrover : In this talk we discuss certain algebraic properties of the quantum homology algebra of toric Fano manifolds. In particular, motivated by applications in symplectic geometry, we describe an easily- verified sufficient condition for the semi-simplicity of the quantum homology. (This is a joint work with Ilya Tyomkin.)

McReynolds : The focus of this talk will be spectral geometry for Riemannian manifolds. I will give a summary of certain aspects of this subject and some recent work joint with Alan Reid and also Chris Leininger, Walter Neumann, and Alan Reid. The talk will focus on locally symmetric manifolds and more specifically hyperbolic n-manifolds.

Salur : Examples of n-dimensional Ricci flat manifolds are Riemannian manifolds whose holonomy groups Hol(g) are subgroups of SU(n), for n=2m, and subgroups of the exceptional Lie group G2, for n=7. We call them Calabi-Yau and G2 manifolds, respectively. They are also examples of manifolds with special holonomy. Calibrated submanifolds of Calabi-Yau and G2 manifolds are volume minimizing in their homology classes and their moduli spaces have many important applications in geometry, topology and physics. In particular, they are believed to play a crucial role in explaining the mysterious "mirror symmetry" between pairs of Calabi-Yau and G2 manifolds. In this talk we give an introduction to calibrated geometries and a report of recent research on the calibrations inside the manifolds with special holonomy.

Maulik : I will explain the recent proof of the Yau-Zaslow conjecture for genus 0 curve counts for all classes on K3 surfaces, primitive or not. The approach involves Noether-Lefschetz theory on the moduli space of K3 surfaces and (rigorous) mirror symmetry calculations. (Joint with Klemm, Pandharipande, and Scheidegger).

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