Lecture summaries / reading suggestions / exercises (updated every week)
Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage)
Time and place:
I will assume that all students are comfortable with the essentials of differential geometry (smooth manifolds, vector fields and Lie bracket, differential forms and Stokes' theorem, de Rham cohomology, connections on vector bundles), as well as some algebraic topology (fundamental group, singular homology and cohomology) and functional analysis (continuous linear operators on Banach spaces, the standard Lp-spaces). Some previous knowledge of additional topics from topology (homological intersection theory, the first Chern class) and functional analysis or PDE theory (Fourier transforms, distributions, Sobolev spaces) will sometimes be helpful, though the relevant results can be taken as black boxes when necessary. For students who have not seen any symplectic geometry before, I will give a concise overview of the subject in the first one or two problem classes.
Gromov-Witten theory lies in the intersection of three subbranches of mathematics: symplectic geometry, algebraic geometry, and mathematical physics. This course will focus mainly on the symplectic perspective, but it may also be of interest to students and researchers from the other two subjects.
Symplectic manifolds were invented around the turn of the 20th century as the natural geometric setting in which to study Hamilton's equations of motion from classical mechanics. The subject of symplectic geometry has developed considerably since then, and it retains a close connection with theoretical physics despite being technically a branch of “pure” mathematics. In particular, the subfield known as symplectic topology, which deals with “global” rather than “local” properties of symplectic manifolds, has witnessed an explosion of activity since the introduction of techniques from elliptic PDE theory in the 1980s. The most spectacular advances came from Gromov's theory of pseudoholomorphic curves, which has led to a wide assortment of algebraic invariants of symplectic manifolds, some of them related to structures that physicists study in quantum field theory or string theory. One example of this is the Gromov-Witten invariants, which are interpreted as counts of holomorphic curves satisfying specified constraints in a symplectic manifold. Since many interesting examples of symplectic manifolds are also algebraic varieties, the Gromov-Witten invariants are also heavily studied by algebraic geometers and can be viewed as a modern approach to enumerative problems (i.e. generalizations of the question “how many lines are there through two points?”) that have been studied in algebraic geometry since the 19th century.
The first goal of this course will be to establish the basic analytical underpinnings of the Gromov-Witten invariants: we will study the local and global structure of moduli spaces of Riemann surfaces and holomorphic curves, elliptic regularity theory for the nonlinear Cauchy-Riemann equation, Fredholm theory, the Riemann-Roch formula, transversality results via the Sard-Smale theorem, and Gromov's compactness theorem for pseudoholomorphic curves. These ingredients are sufficient to give a mathematically rigorous definition of the Gromov-Witten invariants for symplectic manifolds that satisfy a technical condition known as “semi-positivity”, which is always satisfied for manifolds of dimension at most six. Once this is established, there are various additional topics we might discuss, depending on the time available and interests of the class:
I will not be writing detailed lecture notes for this course, but will write up a brief summary of what was covered at the end of each week, including reading suggestions and exercises. A considerable amount of the material we'll cover is contained in notes that I have written for other courses in the past, notably:
I will assign exercises sometimes. Sometimes I will discuss them in the problem class. They will not be graded.
Since this is an advanced course, I have a fairly relaxed attitude about grades. If you come to the course with adequate prerequisites and stay with it for the whole semester, you can come to my office at the end for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). The format is as follows: you pick one particular coherent topic from the course to focus on, typically the contents of four to six lectures, and we will talk about that. If you demonstrate that you learned something interesting from the course, you'll get a good grade.