# MATH 595: Homological Mirror Symmetry (Spring 2018)

## 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

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

## 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.
 W Jan 17 1. Introduction to HMS, statement for the quartic surface. (got through top of page 4.) F Jan 19 2. Triangulated categories. (got through most of page 4.) M Jan 22 3. Cochain complexes of modules. W Jan 24 4. Derived categories of modules. F Jan 26 5. Differential graded categories. M Jan 29 6. $$A_\infty$$-algebras. W Jan 31 7. $$A_\infty$$-category theory. F Feb 2 8. Associativity and Riemann surfaces. M Feb 5 9. The moduli space of stable pointed disks. W Feb 7 10. Moduli spaces and operads. F Feb 9 11. Closed and open TFT. M Feb 12 12. TFT from symplectic manifolds. W Feb 14 13. Almost complex structures and pseudo-holomorphic curves. F Feb 16 14. The Lagrangian Floer TFT. M Feb 19 15. The Lagrangian Floer TFT, II. W Feb 21 16. Gromov compactification. F Feb 23 17. Proving relations using compactified moduli spaces. M Feb 26 18. Proving relations, II. W Feb 28 19. Examples of Floer cohomology. F Mar 2 20. Example: the mirror of $$\mathbb{P}^2$$. M Mar 5 21. Fukaya's $$A_\infty$$-category. W Mar 7 22. Lagrangian Grassmannian. F Mar 9 23. Graded Lagrangian submanifolds. M Mar 12 24. Indices of graded Lagrangian intersections. W Mar 14 25. Index theory and dimensions of moduli spaces. F Mar 16 Finish previous lecture. M Mar 26 26. Spin, Pin, and orientations of moduli spaces. W Mar 28 27. Fukaya categories away from characteristic 2. F Mar 30 28. Beginning the case of the two-torus. M Apr 2 29. A quiver algebra from the two-torus. W Apr 4 30. Deformation and classification of $$A_\infty$$ structures. F Apr 6 31. $$A_\infty$$ structures on $$Q$$. M Apr 9 32. Twisted complexes and triangulated $$A_\infty$$-categories. W Apr 11 33. Some twisted complexes over $$Q_p$$. F Apr 13 34. Twisted complexes on the two-torus. M Apr 16 35. Conclusion of HMS for the two-torus. W Apr 18 36. Beginning the quartic surface. F Apr 20 37. $$A_\infty$$ structures on $$Q_4$$. M Apr 23 38. One-parameter deformation theory. W Apr 25 39. Results for the mirror of the quartic surface. F Apr 27 40. Affine, relative, and projective Fukaya categories. M Apr 30 41. Lagrangian spheres in the quartic surface. W May 2 42. Results 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.