Who?

This semester's symplectic seminar is a cooperative effort between the HU's symplectic geometry (Chris Wendl, Klaus Mohnke, Thomas Walpuski) and algebraic geometry (Bruno Klingler, Gavril Farkas, Thomas Krämer) groups.

The target audience for this seminar includes all mathematicians from student to professor level who are interested in learning about the commonalities between symplectic, complex and algebraic geometry. If you are interested in participating actively but aren't sure whether you have the right background, feel free to speak with Chris Wendl (wendl at math dot hu dash berlin dot de) or Gerard Bargallo in advance, if you haven't already.

If you plan to participate in the seminar, please join the moodle; the enrolment key is "moduli". The moodle will be used for occasional time-sensitive announcements. If you have no HU computer account, you can still register for the HU Moodle with an external e-mail address after clicking on the button marked "Externen Zugang anlegen". (Sorry, that page does not appear to exist in English.)

When and where?

The seminar meets on Mondays, 15:15-16:45 in room 1.114 at the HU math department (Rudower Chaussee 25). See below for a precise schedule.

What?

This will be a learning seminar, meaning it does not aim to cover any recent research results; the goal is instead to foster more interaction and understanding between symplectic/differential geometers and algebraic/complex geometers. As a topic to focus on that is of interest in both fields, we will aim to present at least two parallel perspectives on the definition of the Gromov-Witten invariants and Kontsevich's recursion formula for the enumeration of rational curves in CP².

We will start with about four talks covering a selection of fundamental topics in complex and symplectic geometry that probably everyone needs to know (but not everyone does). These talks could be given by Masters-level students without so much specialized knowledge:

• Sheaves, cohomology and deRham/Dolbeault theorems
• Basics on complex manifolds (sheaves, line bundles and divisors, blowups)
• Basics of symplectic topology (Darboux and Moser theorems, almost complex structures, sketch of Gromov non-squeezing)
• Elliptic operators on closed manifolds (symbol, Sobolev spaces, regularity, Fredholm property)

After this initial phase, the remainder of the seminar will be aimed at understanding Gromov-Witten invariants from both symplectic and algebraic perspectives. Topics covered from the symplectic perspective should include:

• Gromov's compactness theorem (and some mumbling about gluing)
• The Sard-Smale theorem and transversality arguments
• Inhomogeneous perturbations and the Gromov-Witten pseudocycle

• Moduli spaces in algebraic geometry
• Rational curves and their deformations
• Virtual fundamental cycles (probably more than one talk)
• Localization

• WDVV equations and the Kontsevich recursion formula
• Topological sigma models (i.e. "what is GW theory according to physicists?")