|sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me)|
|Office hours (virtual)||TBD|
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.
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.
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:
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.
Details about your final lecture and paper assignment will be posted later in the semester.
Lecture notes will be posted on blackboard.