Algebraic Geometry Seminar
Department of Mathematics
Fall 2016 Lectures and events
Place: Fine Hall 322
|October 4||Junliang Shen
|Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories|
By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. I will explain a mathematical approach to prove (part of) the HKK Conjecture. Our method is to construct an involution on the derived category and use wall-crossing techniques. The talk is based on joint work with Georg Oberdieck.
|October 11||Dawei Chen
|Compactification of strata of abelian differentials|
Many questions about Riemann surfaces are related to study their flat structures induced from abelian differentials. Loci of abelian differentials with prescribed type of zeros form a natural stratification. The geometry of these strata has interesting properties and applications to moduli of complex curves. In this talk we focus on the question of compactifying the strata of abelian differentials from the viewpoints of algebraic geometry, complex analytic geometry, and flat geometry. In particular, we provide a complete description of the strata compactification over the Deligne-Mumford moduli space of pointed stable curves. The upshot is a global residue condition compatible with a full order on the dual graph of a stable curve. This is joint work with Bainbridge, Gendron, Grushevsky and Moeller, based on arXiv:1604.08834.
|October 18||Brian Lehmann
|Convexity in divisor theory|
For toric varieties there is a dictionary relating the geometry of divisors to the theory of polytopes. I will discuss how certain aspects of this dictionary can be extended to divisors on arbitrary smooth projective varieties. These results build upon ideas of Khovanskii and Teissier; as in their work, geometric inequalities and convexity theory play an important role. This is joint work with Jian Xiao.
|October 25||David Anderson
Ohio State University
|Old and new formulas for degeneracy loci
A very old problem asks for the degree of a variety defined by rank conditions on matrices. The story of the modern approach begins in the 1970's, when Kempf and Laksov proved that the degeneracy locus for a map of vector bundles is given by a certain determinant in their Chern classes. Since then, many variations have been studied -- for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. I will describe recent extensions of these results -- beyond determinants and Pfaffians, and beyond ordinary cohomology -- including my joint work with W. Fulton, as well as work of several others.
|November 8||Dhruv Ranganathan
|Tropical curve counting in superabundant geometries|
I will discuss a general framework using Artin fans -- certain logarithmic algebraic stacks -- in which to understand the relationship between logarithmic stable maps and tropical curve counting. These objects provide a flexible tool to study correspondences between algebraic and tropical curves. In particular, we obtain new realization theorems for tropical curves in superabundant settings. After explaining some remarkably cheap consequences of this setup, I will discuss an application, joint with Yoav Len, to the enumerative geometry of elliptic curves on toric surfaces.
|November 15||Han-Bom Moon
|December 6||Wenfei Liu
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