D. Auroux - Tuesdays 2-3:30, Room 939 Evans

This seminar will revisit the themes of past seminars (see here, here, here and here) on geometric aspects of mirror symmetry.

The details are to be determined based on the participants' interests. Possible topics for this semester are as follows:

- Lagrangian Floer homology and Fukaya categories
- Lagrangian fibrations, family Floer homology and HMS, after Abouzaid
- Liouville sectors and their Fukaya categories, after Ganatra-Pardon-Shende

We will also hear from some of the participants about their own work or related topics.

Prerequisite: some knowledge of symplectic geometry, ideally including prior exposure to J-holomorphic curves.

**On Sept 29, Oct 6, and Oct 13, we will meet FRIDAYS at
11am; then we return to Tuesdays 2-3:30pm.**

**Tuesday 8/29**: First meeting and overview (D. Auroux)**Tuesday 9/5**: Lagrangian Floer homology (D. Tonkonog)**Tuesday 9/12**: Lagrangian Floer homology continued (D. Tonkonog)**Tuesday 9/19**: Families of Lagrangian tori with local systems and analytic dependence (D. Auroux) (Notes)**Tuesday 9/26**: Family Floer theory: the elliptic curve (D. Auroux) (Notes)**Friday 9/29**: Family Floer theory after Abouzaid: twisting cocycles and gerbes (J. Hicks) (Notes)**Friday 10/6**: Rigid analytic geometry: affinoid domains (J. Hicks)**Friday 10/13**: Umut Varolgunes (MIT):*Mayer-Vietoris sequence for relative symplectic cohomology*

Abstract: I will first recall the definition of an invariant that assigns to any compact subset K of a closed symplectic manifold M a module SH_{M}(K) over the Novikov ring. I will go over the case of M=S^2 to illustrate various points about the invariant. Finally I will state the Mayer-Vietoris property and explain under what conditions it holds.**Tuesday 10/17**: Continuation maps; open-closed maps (J. Hicks / D. Tonkonog)**Tuesday 10/24**: Faithfulness of family Floer (D. Tonkonog)**Tuesday 10/31**: HMS for toric varieties (Abouzaid's thesis) (M. Jeffs)**Tuesday 11/7**: Monomial admissibility and monodromy (A. Hanlon)

Abstract: We will discuss a new interpretation of the Fukaya-Seidel category mirror to a compact toric variety and several applications of the construction. In particular, we will see how a natural monodromy of these categories is mirror to tensoring by a line bundle.**Tuesday 11/14 in 748 Evans**: Homological mirror symmetry for the genus 2 curve and its SYZ mirror (C. Cannizzo)

Abstract: Homological mirror symmetry (HMS) for the genus 2 surface on the symplectic side is known by work of Seidel. Here we consider it on the complex side, and a mirror is constructed following Abouzaid-Auroux-Katzarkov's paper on SYZ for hypersurfaces of toric varieties. We describe the manifolds involved, and discuss progress made towards proving HMS.**Tuesday 11/21:**no meeting**Tuesday 11/28:**Localization and Fukaya categories (A. Hanlon)**Tuesday 12/5:**Liouville sectors and their Fukaya categories (A. Ward)

Notes from past seminars and courses on the topic:

- Fall 2009 topics course on mirror symmetry (Oct 15-Dec 1 lectures)
- Fall 2011 seminar (see also the Spring 2011 seminar)
- Fall 2016 Eilenberg lectures on Fukaya categories and HMS

- P. Seidel,
*Fukaya categories and Picard-Lefschetz theory*, Zurich Lectures in Advanced Mathematics, European Math. Soc., Zurich, 2008. - D. Auroux,
*A beginner's introduction to Fukaya categories*, Contact and Symplectic Topology, Bolyai Soc. Math. Stud.**26**, Springer, 2014, pp. 85-136. (arXiv:1301.7056)

- M. Abouzaid,
*Family Floer cohomology and mirror symmetry*, Proceedings of 2014 ICM (arXiv:1404.2659) - M. Abouzaid,
*The family Floer functor is faithful*, (arXiv:1408.6794) - M. Abouzaid,
*Homological mirror symmetry without corrections*, (arXiv:1703.07898)

- M. Abouzaid,
*A geometric criterion for generating the Fukaya category*, Publ. Math. IHES**112**(2010), 191-240 (arXiv:1001.4593) - S. Ganatra, J. Pardon, V. Shende,
*Covariantly functorial Floer theory on Liouville sectors*, (arXiv:1706.03152)