Abstracts - Spring 2010 

Speaker: Denis Auroux 

Title: Mirror symmetry for blowups and hypersurfaces in toric varieties

Abstract: We will present joint work with Mohammed Abouzaid and Ludmil Katzarkov, investigating mirror symmetry for blowups from the perspective of the Strominger-Yau-Zaslow conjecture. We will start with a quick review of the SYZ approach to the construction of mirror pairs by dualization of Lagrangian torus fibrations. Our main goal will be to apply this program in the case of the blowup of a toric variety along a codimension 2 subvariety contained in a toric hypersurface. We will discuss the SYZ mirror and its instanton corrections, providing an explicit description of the mirror Landau-Ginzburg model (up to higher order corrections to the superpotential). This construction allows one to geometrically construct mirrors of essentially arbitrary hypersurfaces in toric varieties. We will focus on examples such as pairs of pants and curves of arbitrary genus.

Speaker: Vladimir Baranovksy

Title: The Chow space of effective cycles as a scheme

Abstract: Let X be a projective variety. By the classical construction of Chow forms, the set of effective cycles on X can be identified with the set of closed points of another algebraic variety Chow(X).  In general, this construction depends on the choice of a projective embedding of X, and different embeddings only lead to homeomorphic varieties Chow(X). We will discuss the problem of defining a scheme representing a certain functor (in the spirit of Grothendieck's approach to Hilbert schemes), which agrees with Chow(X) when the projective embedding of X is ample enough. If time permits, we will also discuss similar moduli spaces, such the Uhlenbeck moduli space of bundles.

Speaker: Renzo Cavalieri 

Title: Wall Crossings for double Hurwitz numbers

Abstract: This talk will explore the piecewise polynomial structure of double Hurwitz numbers. Initial results of Goulden-Jackson-Vakil (piecewise polynomiality) and Shadrin-Shapiro-Vahinstein (wall crossings in genus 0) are reproven and extended to arbitrary genus using a combinatorial interpretation of Hurwitz numbers as appropriate weighted sums over graphs. This is joint work with Paul Johnson and Hannah Markwig.

Speaker: Dan Rogalski

Title: Noncommutative projective geometry and birational maps of surfaces

Abstract: I will give a brief overview of some of the ideas of noncommutative projective geometry.  I then plan to discuss some recent results in the classification theory of noncommutative projective varieties which are related to the study of birational maps of (commutative!) surfaces, especially their dynamical properties.