GTM Seminar

Speaker: Dogancan Karabas (Kavli IPMU)
Title: Categorical and computational methods in symplectic geometry
Date (JST): Fri, Apr 05, 2024, 15:30 - 17:00
Place: Seminar Room B
Abstract: In homological mirror symmetry, the A-side corresponds to the Fukaya category of a symplectic manifold M. When M is a (Wein)stein manifold, the A-side can be represented by the dg category of "microlocal sheaves" defined on the "skeleton" of M. These sheaves are constructed through local computations and assembled using gluing techniques. Moreover, the singularities of such skeleta are classified by trees and are known as "arboreal singularities". Microlocal sheaves on arboreal singularities are well-understood, so the main challenge lies in the gluing process, which involves taking the "homotopy colimit" of dg categories.

In my talk, I will explain the concepts mentioned above and introduce our approach to the homotopy theory of dg categories by establishing a cofibration structure, which enables a combinatorial expression of homotopy colimit and offers other advantages. This approach allows us to compute Fukaya categories efficiently, as I will demonstrate with examples. If time permits, I will also present an approach to attack the Weinstein conjecture, which concerns the existence of periodic orbits of Reeb vector fields, using local-to-global methods. This work is currently in progress in collaboration with Sangjin Lee.