Syzygies and mirror symmetry
September 5 to September 8, 2023
at the
American Institute of Mathematics,
Pasadena, California
organized by
Daniel Erman and Andrew Hanlon
Original Announcement
This workshop will be devoted to exploring the implications of recent work connecting multigraded commutative algebra, derived categories of toric varieties, and homological mirror symmetry. It is inspired by the recent breakthrough progress on conjectures related to Rouquier dimension and virtual resolutions for toric varieties, which involved techniques that were motivated by symplectic geometry and homological mirror symmetry. The goal of the workshop is to build on these results, develop connections among the communities studying these areasâ€”including symplectic geometry, algebraic geometry, and commutative algebraâ€”and foster further research that is bolstered by varying perspectives.
The main topics of exploration for the workshop will be the following:

Leverage the recent breakthroughs to explore and develop new conjectures on the structure of derived categories of toric varieties/stacks and implications for the general theory of derived categories of coherent sheaves and homological mirror symmetry.

Create an explicit implementation of the free resolutions that arose via symplectic methods and launch initial investigations into generalizations of the technique to other algebraic or geometric settings, such as homogeneous spaces, Mori dream spaces, or more.

Investigate algebraic structures, such as generation, Rouquier dimension, and explicit resolutions, captured by the geometry of partially wrapped Fukaya categories and their images under homological mirror symmetry.

Apply the HanlonHicksLazarev resolution of the diagonal to study algebraic invariants, such as multigraded CastelnuovoMumford regularity, Betti numbers, and more.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
Papers arising from the workshop: