WAGS Spring 2015

WAGS / Spring 2015

Abstracts

Qile Chen
The decomposition formula for log Gromov-Witten invariants
One major goal of log Gromov-Witten theory is to reconstruct usual Gromov-Witten invariants of a smooth projective variety X using appropriate invariants of a degeneration of X. The decomposition formula is the first step toward this. It breaks the invariants of a degeneration of X into terms which can be classified by the dual complexes/tropical curves of stable log maps. This leads to a decomposition on the level of virtual cycles.

The decomposition formula for log Gromov-Witten invariants

One major goal of log Gromov-Witten theory is to reconstruct usual Gromov-Witten invariants of a smooth projective variety X using appropriate invariants of a degeneration of X. The decomposition formula is the first step toward this. It breaks the invariants of a degeneration of X into terms which can be classified by the dual complexes/tropical curves of stable log maps. This leads to a decomposition on the level of virtual cycles.

David Eisenbud
BGG in a new context
The Bernstein-Gel'fand-Gel'fand correspondence relates sheaves on projective space and free resolutions over an exterior algebra. I'll describe the basic idea of the correspondence—which is a lovely and elementary bit of linear algebra—and a new application of it related to high syzygy modules over complete intersections. This is part of ongoing joint work with Irena Peeva and Frank-Olaf Schreyer.

David Eisenbud

BGG in a new context

The Bernstein-Gel'fand-Gel'fand correspondence relates sheaves on projective space and free resolutions over an exterior algebra. I'll describe the basic idea of the correspondence—which is a lovely and elementary bit of linear algebra—and a new application of it related to high syzygy modules over complete intersections. This is part of ongoing joint work with Irena Peeva and Frank-Olaf Schreyer.

Christopher Hacon
On the boundedness of the functor of KSBA stable varieties
Let X be a canonically polarized smooth n-dimensional projective variety over C (so that ω_X is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of X in projective space. It then follows easily that if we fix certain invariants of X then X belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized n-dimensional projective varieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.

Christopher Hacon

On the boundedness of the functor of KSBA stable varieties

Let X be a canonically polarized smooth n-dimensional projective variety over C (so that ω_X is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of X in projective space. It then follows easily that if we fix certain invariants of X then X belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized n-dimensional projective varieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.

Motohico Mulase
Topological recursion and quantum curves
The "topological recursion" was first formulated by physicists Eynard and Orantin in 2007 in their study of random matrix theory. Its significance in algebraic geometry was then noticed by string theorists. For example, its application to Gromov-Witten theory was proposed by Bouchard, Klemm, Marino, and Pasquetti, and to algebro-geometric study of knot invariants by Gukov and Sulkowski. The talk is aimed at presenting the simplest mathematical examples of this physics originated theory. We will give a simultaneous three-line proof of the Witten-Kontsevich theorem and the lambda-g formula of Faber and Pandharipande, using the topological recursion. The notion of quantum curves is also introduced, with its still mysterious relation to quantum invariants. The talk is based on my joint papers with Bouchard, Dumitrescu, Eynard, Norbury, Safnuk, Shadrin, Sulkowski, Zhang, and others.

Motohico Mulase

Topological recursion and quantum curves

The "topological recursion" was first formulated by physicists Eynard and Orantin in 2007 in their study of random matrix theory. Its significance in algebraic geometry was then noticed by string theorists. For example, its application to Gromov-Witten theory was proposed by Bouchard, Klemm, Marino, and Pasquetti, and to algebro-geometric study of knot invariants by Gukov and Sulkowski. The talk is aimed at presenting the simplest mathematical examples of this physics originated theory. We will give a simultaneous three-line proof of the Witten-Kontsevich theorem and the lambda-g formula of Faber and Pandharipande, using the topological recursion. The notion of quantum curves is also introduced, with its still mysterious relation to quantum invariants. The talk is based on my joint papers with Bouchard, Dumitrescu, Eynard, Norbury, Safnuk, Shadrin, Sulkowski, Zhang, and others.

Jason Starr
Mendebaldeko lecture: Picard groups of spaces of rational curves in Fano manifolds
Fano manifolds, e.g., low degree hypersurfaces in projective spaces, are precisely the projective manifolds whose spaces of rational curves with specified (effective) curve class "grow" as we pass to multiples of the curve class. What is the geometry of this (large) space of curves? The key is a description of the Picard group, the canonical divisor class, and the effective and ample cones in this Picard group. I will survey older work, joint with Coskun, de Jong, Harris, and Roth, and then I will explain new joint work with Zhiyu Tian that describes the Picard group, canonical divisor and ample cones for spaces of rational curves on "most" Fano hypersurfaces.

Jason Starr

Mendebaldeko lecture: Picard groups of spaces of rational curves in Fano manifolds

Fano manifolds, e.g., low degree hypersurfaces in projective spaces, are precisely the projective manifolds whose spaces of rational curves with specified (effective) curve class "grow" as we pass to multiples of the curve class. What is the geometry of this (large) space of curves? The key is a description of the Picard group, the canonical divisor class, and the effective and ample cones in this Picard group. I will survey older work, joint with Coskun, de Jong, Harris, and Roth, and then I will explain new joint work with Zhiyu Tian that describes the Picard group, canonical divisor and ample cones for spaces of rational curves on "most" Fano hypersurfaces.

Bianca Viray
Arithmetic intersection numbers and cryptography
Cryptosystems that are based on the discrete logarithm problem require an instantiation of a group of large prime order. For instance, for certain primes p, one could use a subgroup of the group of units in F_p, the F_p-points on an elliptic curve, or the F_p-points on the Jacobian of a genus 2 curve. (The group of integers modulo p results in an insecure cryptosystem.) This raises the question of how to construct a genus 2 curve C over F_p whose Jacobian has N F_p-points, where N is a large prime. Perhaps surprisingly, this problem is related to arithmetic intersection numbers. Indeed, a formula (or a sharp upper bound) for the (finite) arithmetic intersection of the locus of products of elliptic curves with the locus of abelian surfaces with complex multiplication by a fixed field K enables one to construct genus 2 curves whose Jacobians have a specified number of points. In this talk, I will explain this connection between cryptography and arithmetic intersection numbers. In addition, I will give a formula, joint with Kristin Lauter, for this arithmetic intersection number.

Bianca Viray

Arithmetic intersection numbers and cryptography

Cryptosystems that are based on the discrete logarithm problem require an instantiation of a group of large prime order. For instance, for certain primes p, one could use a subgroup of the group of units in F_p, the F_p-points on an elliptic curve, or the F_p-points on the Jacobian of a genus 2 curve. (The group of integers modulo p results in an insecure cryptosystem.) This raises the question of how to construct a genus 2 curve C over F_p whose Jacobian has N F_p-points, where N is a large prime. Perhaps surprisingly, this problem is related to arithmetic intersection numbers. Indeed, a formula (or a sharp upper bound) for the (finite) arithmetic intersection of the locus of products of elliptic curves with the locus of abelian surfaces with complex multiplication by a fixed field K enables one to construct genus 2 curves whose Jacobians have a specified number of points. In this talk, I will explain this connection between cryptography and arithmetic intersection numbers. In addition, I will give a formula, joint with Kristin Lauter, for this arithmetic intersection number.

Department of Mathematics
University of California, Davis

Webpage maintained by Brian Osserman