Math 641: Topics in Topology, Fall 2020

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

Teaching Staff

Instructor Sheel Ganatra
Office KAP 266D
E-mail sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me)
Office hours (virtual) TBD

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):

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.

A more detailed lecture plan (updated on an ongoing basis, after each lecture) will be posted below.

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):

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

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 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 here later in the semester.

Lecture Plan

Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.

Day Lecture topics References and remarks Notes
Aug. 17 Lecture 1: Welcome + an overview of the class.