# MATH 595: Homological Mirror Symmetry (Spring 2018)

## Table of Contents

## Basic information

- Course meets
- MWF 12:00–12:50 p.m. in 243 Altgeld Hall
- Instructor: James Pascaleff
- Email: jpascale@illinois.edu

Office: 341B Illini Hall

Office hours: Tuesdays 11:00-12:00

## Texts

- D. Auroux,
*A beginner's introduction to Fukaya categories*. - P. Aspinwall,
*et al.*,*Dirichlet branes and mirror symmetry*, Clay Mathematics Monographs vol. 4. - P. Seidel,
*Fukaya categories and Picard-Lefschetz theory*, Zurich Lectures in Advanced Mathematics. - p. Seidel,
*Abstract analogues of flux as symplectic invariants*. - P. Seidel,
*Homological mirror symmetry for the quartic surface*, Memoirs of the American Mathematical Society. - N. Sheridan,
*Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space*, Inventiones Mathematicae 199 (2015), no. 1, 1-186.

For the example of mirror symmetry for \(\mathbb{P}^2\), see

- P. Seidel,
*More about vanishing cycles and mutation*, Proposition 3.2.

## Schedule and notes

The material in these lectures is mainly drawn from the following sources:

- Lectures 2–4: Weibel, Introduction to Homological Algebra.
- Lectures 5–27: Seidel, Fukaya Categories and Picard Lefschetz theory.
- Lectures 28–35: Seidel, Abstract analogues of flux as symplectic invariants, Chapter 2.
- Lectures 36–42: Seidel, Homological mirror symmetry for the quartic surface.

## Course Description

Homological Mirror Symmetry (HMS) is the study of the relations between three types of mathematical objects: \[\text{symplectic manifolds} \longleftrightarrow \text{triangulated categories} \longleftrightarrow \text{algebraic varieties}\] For a symplectic manifold \(X\), there is a triangulated category \(\mathcal{F}(X)\) called the Fukaya category, and for an algebraic variety \(Y\) there is a triangulated category \(\mathcal{D}(Y)\) called the derived category. We then pose the problem of finding pairs \(X\) and \(Y\) such that \[\mathcal{F}(X) \cong \mathcal{D}(Y)\] The origin of this relation is in theoretical particle physics, where the two categories are interpreted as collections of D-branes, and the relation expresses the duality between A-twisted topological string theory on \(X\) and B-twisted topological string theory on \(Y\).

The investigation of this relation raises many questions. How are the two sides actually defined? How do we compute the two sides, and what should the "answer" of such a computation look like? What general structure is present that constrains the problem? The goal of this course is to set up the machinery and understand the solution in a specific case: when \(X\) is a hypersurface in projective space, including the quintic threefold, following Seidel and Sheridan. Topics to include:

- Categories: triangulated, differential graded, \(A_\infty\).
- Algebraic varieties, categories of coherent sheaves.
- Symplectic manifolds, Lagrangian Floer cohomology, Fukaya categories.
- Case of surfaces, HMS for the two-torus, other relatively simple models.
- Hypersurfaces in projective space.

## Prerequisites

In order to have a good chance at learning something in this class, you should have a solid background in two things:

- 1. Abstract algebra
- Particularly commutative rings and modules over them.
- 2. Differential topology
- Smooth manifolds, vector bundles, tensors and differential forms.

If you are not familiar with these topics then that is where you should start. There are many books on these topics and you should find one that you like. The next things would be:

- 3. Homological algebra
- Some classic books are
*Methods of Homological Algebra*by Gelfand and Manin and*An Introduction to Homological Algebra*by Rotman and a book of the same title by Weibel. - 4. Symplectic geometry
- See
*Lectures on Symplectic Geometry*by Ana Cannas da Silva and*Introduction to Symplectic Topology*by McDuff and Salamon.

All of the books mentioned above except for McDuff-Salamon are available as e-books through the UIUC library.

The more background you have, the better, but 1 and 2 are the minimum. I will still give introductions to 3 and 4 in the course. The course on Symplectic Geometry taught concurrently by Prof. Tolman would be helpful, but is not required.