for this workshop
Syzygies and mirror symmetry
American Institute of Mathematics, Pasadena, California
Daniel Erman and Andrew Hanlon
This workshop, sponsored by AIM and the NSF, 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 areas—including symplectic geometry, algebraic geometry, and commutative algebra—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 Hanlon-Hicks-Lazarev resolution of the diagonal to study algebraic invariants, such as multigraded Castelnuovo-Mumford regularity, Betti numbers, and more.
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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