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.)
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.
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:
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:
| Monday October 16, 2023 15:15-16:45 RUD 25, Room 1.114 |
No seminar (dies academicus) |
|---|---|
| Monday October 23, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Gerard Bargallo Topic: Introduction and planning meeting Abstract: We will start with a general heuristic introduction to the Gromov-Witten invariants and sketch (with all technical details brushed under the rug) how they lead to an enumeration of the rational curves of all degrees in CP2. We will then distribute topics for the rest of the semester. Gerard's typed notes (with useful references!) |
| Monday October 30, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Bálint Zsigri Topic: Sheaves, cohomology and de Rham / Dolbeault theorems |
| Monday November 6, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Sophia Bugarija Topic: Basics on complex manifolds Abstract: examples, basic sheaves, line bundles and divisors, blowups |
| Monday November 13, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Apratim Choudhury Topic: Elliptic operators on closed manifolds Abstract: principal symbols, Sobolev spaces, regularity, Fredholm property |
| Monday November 20, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Kanishka Katipearachchi Topic: Basics of symplectic topology Abstract: Darboux and Moser theorems, almost complex structures, sketch of Gromov non-squeezing |
| Monday November 27, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: David Suchodoll Topic: Gromov's compactness theorem Abstract: The moduli space of stable nodal J-holomorphic curves, bubbling analysis, Deligne-Mumford degenerations of stable Riemann surfaces |
| Monday December 4, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Michael Rothgang Topic: Generic transversality Abstract: The nonlinear Cauchy-Riemann operator and its linearization, what it means for a moduli space to be "cut out transversely", the Sard-Smale theorem, sketch of the proof that for generic J, the moduli space of somewhere-injective J-holomorphic curves is cut out transversely |
| Monday December 11, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Chris Wendl Topic: Inhomogeneous perturbations and the Gromov-Witten pseudocycle Abstract: Achieving transversality via inhomogeneous perturbations, the universal curve, perturbed moduli spaces in the genus zero case, pseudocycles, definition of GW for genus zero on semipositive symplectic manifolds, higher-genus case via Prym covers |
| Monday December 18, 2023 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Vasily Rogov Topic: Moduli problems and ways to solve them Abstract: I will discuss the difficulties that typically arise on the way to constructing a moduli space of some geometric objects, with an emphasis on the moduli space of smooth complex projective curves. I will explain how these difficulties naturally lead to the concept of a stack and discuss its different versions (algebraic stacks, Deligne-Mumford stacks), as well as their relation to the GIT point of view. Time permitting, I will also cover a bit of deformation theory (infinitesimal deformations and their relation to cohomology, obstructions etc.) |
| Monday January 8, 2024 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Vasily Rogov Topic: (continuation) |
| Monday January 15, 2024 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Andrea Di Lorenzo Topic: Deformation theory, a.k.a. the local theory of moduli Abstract: In this talk I will give an overview of deformation theory from an algebraic geometer's perspective. Time permitting, I will explain why this theory is relevant also for studying the global geometry of moduli problems. |
| Monday January 22, 2024 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Gerard Bargallo Topic: Fantasy impromptu: a coda to what has been done so far Abstract: In this almost-impromptu talk we do a recap of what we have done so far trying not to get too technical, as if we were in mild fantasyland. I will tell you my main takeaways of the talks, trying to ground them in specific examples, as well as try to emphasize algebraic-analytic connections and hopefully tie a couple of loose ends. |
| Monday January 29, 2024 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Raju Krishnamoorthy Topic: Localization in equivariant cohomology Abstract: We sketch the localization formula in equivariant cohomology, due to Atiyah-Bott, along with basic applications. Time permitting, we indicate the applications to GW theory. |
| Monday February 5, 2024 15:15-16:45 RUD 25, Room 1.114 |
No seminar |
| Monday February 12, 2024 15:15-16:45 RUD 25, Room 1.114 |
Speaker: Josh Lam Topic: Virtual fundamental classes Abstract: I'll attempt to give an idea of what a virtual fundamental class is, which a priori has little to do with Gromov-Witten theory. The key ingredient is the intrinsic normal cone, due to Behrend and Fantechi, which is a canonical object attached to any complex algebraic variety (or even Deligne-Mumford stack); roughly, this is an invariant which tells you how singular a variety is. I'll then illustrate how the theory applies to the moduli of stable maps. I'll mostly try to compute some explicit examples. |
| Friday February 16, 2024 11:15-12:45 RUD 25, Room 1.114 |
Speaker: Davide Scazzuso Topic: On topological sigma models Abstract: String theory is a theoretical framework that provides deep understanding of many aspects of fundamental physics, and in doing so leads to a host of surprising mathematical conjectures and results. In this talk, I will review some constructions and tools that allowed fruitful interaction between physics and mathematics in the past 35 years, most notably the topological sigma model. On the way we will discuss 2d Topological QFTs, 2d Topological CFTs, 2d N=(2,2) superconformal field theory and the topological twist. |
| Monday February 19, 2024 13:15-14:45 RUD 25, BMS Seminar Room (1.023) |
Speaker: Davide Scazzuso Topic: (continuation) notes from Davide's talk a physicist's bibliography on Gromov-Witten theory Davide has provided his notes and bibliography with the following caveats: (1) No one should assume that the notes are complete or free of mistakes, and (2) The bibliography concerns over 30 years of research and is thus obviously incomplete. |
| Monday April 8, 2024 11:15-12:45 RUD 25, Room 1.114 |
Speaker: Apratim Choudhury Topic: Kontsevich's recursion formula Abstract: In this talk, I will discuss the main idea behind Kontsevich’s recursion, which leads to the computation of the genus zero Gromov-Witten invariants of CP2, answering a long-standing question in enumerative geometry. If time permits, I will try to talk about how this idea can be modified and applied to certain questions about counting curves with singularities. |