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The homotopy theory of differential graded (dg) categories plays a significant role in various fields, including algebraic geometry, representation theory, higher categories, and symplectic geometry. In particular, understanding dg-categories is crucial for formulating and interpreting homological mirror symmetry.

In this talk, I will present our approach to the homotopy theory of dg-categories by establishing a cofibration structure, which can be viewed as a half-model structure. This structure enables a combinatorial description of derived constructions and offers computational advantages. This is joint work with Sangjin Lee (arXiv:2109.03411 and arXiv:2405.03258). Some key applications of our approach, particularly in symplectic and contact geometry, include:

• Combinatorial description of homotopy colimits of dg categories, which gives a local-to-global formula computing wrapped Fukaya categories of symplectic manifolds,
• Local-to-global construction of functors between wrapped Fukaya categories that are induced by symplectomorphisms,
• A simple description of internal Hom and Hochschild cohomology of dg-categories. This ongoing work aims to provide useful tools for addressing the Weinstein conjecture, which concerns the existence of periodic orbits of Reeb vector fields.

I plan to cover as much of this content as time permits, and according to the audience's interest.

Zoom uygulaması Bilim Akademisi tarafından sağlanmaktadır./Zoom link is provided by The Science Academy.