Commutative Algebra Seminar

Log Gromov-Witten invariants, introduced by Abramovich-Chen-Gross-Siebert, are counts of curves in pairs (X,D) consisting of a smooth projective variety X together with a normal crossing divisor D, with prescribed tangency conditions along D. These invariants play a key role in mirror symmetry for log Calabi-Yau pairs (X,D), in which case D is an anticanonical divisor. After briefly reviewing log Gromov-Witten theory, I will explain a combinatorial recipe based on tropical geometry and wall-crossing algorithms to calculate such curve counts when (X,D) is obtained as a blow-up of a toric variety along hypersurfaces in the toric boundary divisor. This is based on joint work with Mark Gross.

A fundamental problem at the confluence of algebraic geometry, commutative algebra, and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. I will describe an answer in the case of the incidence correspondence (the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it), and highlight surprising connections to other questions of interest: the splitting of jet bundles on the projective line, the Han-Monsky representation ring, or Lefschetz properties for Artinian monomial complete intersections. This is based on joint work with Annet Kyomuhangi, Emanuela Marangone, and Ethan Reed.

In this talk, I will discuss some examples of the relative Langlands duality (introduced by Ben-Zvi-Sakellaridis-Venkatesh) for strongly tempered spherical varieties. In some cases, I will introduce a relative trace formula comparison and prove the fundamental lemma/smooth transfer. This is a joint work with Zhengyu Mao and Lei Zhang.