Math 641: Topics in Topology, Fall 2020

Monday, Wednesday, 9:30-11:00am on Zoom

Teaching Staff

Course Description and Prerequisites

In Fall 2020, Math 641 will be a course on mirror symmetry. We will give an incomplete survey of some aspects of mirror symmetry, an unexpected link discovered heuristically by string theorists in the early 1990s between the symplectic geometry one space and the complex geometry of a `mirror' space. While the underpinnings of mirror symmetry are much better understood today (and known to work in much greater generality than initially proposed), large parts of this correspondence remain conjectural. Topics to be discussed may include (but are not limited to):

• a review of parts of symplectic and complex (and Kähler) and geometry, and mirror symmetry as an equivalence of moduli spaces of such structures
• combinatorial constructions of mirror pairs and mirror submanifolds (via toric and/or tropical geometry)
• the Strominger-Yau-Zaslow (SYZ) proposed geometric approach to constructing mirror pairs, and relationships to affine and tropical geometry
• `open string'/categorical, or homological mirror symmetry (HMS) (between the Fukaya category and categories of coherent sheaves)
• `closed string'/`enumerative' mirror symmetry as first discovered by string theorists (between cohomology groups with their Hodge structures and/or between curve counts and period integrals),
• expected relationships between various points of view.

The course cannot quite be self contained, as it will draw in parts from, and reference to, differential geometry, analysis, algebraic geometry, algebra/category theory, and combinatorics. We will particularly require familiarity with the geometry of manifolds and algebraic topology at the level of USC's Math 535a and Math 540 courses or equivalent. We will also largely assume Math 535b (Advanced Differential Geometry as covered here), though we will try to review some notions to be self-contained. Familiarity with homologial algebra at the level of the Math 510 series will also help at various points in the class. In general, we will try to overview important background when it is possible. Frequently, in place of precise definitions we will aim to focus on simple examples where phenomena can be understood.

In the last roughly one or two weeks of the course, the class will become student run: students will give a series of talks about advanced topics in one or more of these areas. There will be a wide array of topics possible, ranging from geometric to algebraic to combinatorial.

References

There is no official course textbook, but we will frequently refer to the following books and papers (and likely others):

• Ana Cannas da Silva, Lectures on symplectic geometry, available online.
• Michèle Audin, Torus actions in symplectic geometry 2nd ed (available online through SpringerLink).
• David Cox and Sheldon Katz, Mirror symmetry and algebraic geometry. Discusses enumerative mirror symmetry and various constructions of mirror pairs, largely from the viewpoint of algebraic geometry.
• Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry. Available online through the Clay Mathematics Institute here.
• Emily Clader and Yongin Ruan, Mirror symmetry constructions, available here.

We will also make key use of the following course and lecture notes:

• Denis Auroux's 2009 course notes on Mirror symmetry, 2018 course notes on Symplectic manifolds and Lagrangian submanifolds, and 2016 lecture series notes on Fukaya categories and mirror symmetry.
• James Pascaleff's 2014 lecture notes on Lagrangian Floer homology and 2018 lecture notes on Homological Mirror Symmetry.
• Nick Sheridan's 2014 lecture series on Homological Mirror Symmetry.

Grading System

Your grade in the course, a measure of your demonstrated understanding of course topics, will come from participation along with two (individual) final assignments:

• A lecture given in the final weeks of the course, on an advanced topic building on what is covered in the class. and
• A final expository paper, at least 5 pages long, about an advanced topic as above. This is allowed, but does not need to exactly coincide with your lecture topic. In fact, even if the topic is the same, the paper should necessarily have more details and/or differing emphasis from what is lectured about in class.

A list of possible topics for the above assignments will be provided later in the semester. You are also welcome to propose a different topic, subject to instructor approval.

Student lectures and final papers

Details about your final lecture and paper assignment will be posted later in the semester.

Lecture Notes

Lecture notes will be posted on blackboard.