Seminar: Symplektische Geometrie / Symplectic Geometry — Wintersemester 2019-20, HU Berlin

This is a working group seminar run by Klaus Mohnke and Chris Wendl on recent developments in symplectic geometry. It generally runs every semester on Mondays. Participants are expected to be familiar with the basics of symplectic geometry, including some knowledge of holomorphic curves and/or Floer-type theories. The seminar is conducted in English.

The seminar takes place this semester on Mondays, 15:00-17:00 (c.t.) in room 1.023 (the BMS Seminar Room) at Rudower Chaussee 25. It will occasionally be preempted by the Berlin-Hamburg Symplectic Geometry Seminar.

Note for new students: If you think you might be interested in this seminar but have neither attended before nor spoken with Profs. Mohnke or Wendl about it, it is a good idea to get in touch with one of us ahead of time!


Topic for this semester: Unlike most semesters, the seminar this semester will be focused on a particular topic. The plan is to work through Wendl's recent paper Transversality and super-rigidity for multiply covered holomorphic curves, including some necessary background material on holomorphic curve theory and Gromov-Witten invariants. Chris Wendl will give a brief overview of what it's about during the first meeting on October 14. For a more comprehensive but informal overview of the ideas in the paper, we recommend the following series of blog posts:

Monday October 14, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Informal meeting to discuss topics for the semester
Monday October 21, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: Michael Rothgang
Topic: Teichmüller slices, Fredholm regularity and the implicit function theorem
Abstract: The main goals are to define the moduli space of closed unparametrized J-holomorphic curves in an almost complex manifold, the notion of Fredholm regularity for such curves, and prove the standard theorem endowing the (open) set of regular curves in the moduli space with the structure of a smooth finite-dimensional orbifold. Some facts about the moduli space of Riemann surfaces and Teichmüller space will also need to be mentioned, most likely without proof; the fact that the linearized Cauchy-Riemann operator is Fredholm should likewise be taken as a black box.
Literature: Chris's holomorphic curve notes, Section 4.3 (with some background material in Sections 4.1 and 4.2)
Monday October 28, 2019
15:00-17:00 (c.t.)
RUD 25, Room 2.006
(Achtung! room change)
Speaker: Yoanna Kirilova
Topic: The Sard-Smale theorem and transversality for somewhere injective J-holomorphic curves
Abstract: The main result of this talk is the standard theorem that for generic choices of almost complex structure, all somewhere injective closed J-holomorphic curves are Fredholm regular. The main tools needed for this are (1) a unique continuation result for solutions of linear Cauchy-Riemann type equations, and (2) Smale's infinite-dimensional version of Sard's theorem, both of which should be stated carefully (but not proved) since they will be needed again later. The Floer "C-epsilon" space of perturbed almost complex structures should also be introduced. Local elliptic regularity results for Cauchy-Riemann type operators (e.g. the fact that weak solutions of class Lp for p > 2 are smooth) should be taken as black boxes.
Literature: Chris's holomorphic curve notes, Section 4.4
Monday November 4, 2019 Berlin-Hamburg Symplectic Geometry Seminar (in Berlin)
Monday November 11, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: Felix Noetzel
Topic: The normal Cauchy-Riemann operator
Abstract: The purpose of this talk is to explain the following facts: (1) Every nonconstant J-holomorphic curve has a "generalized" normal bundle which is well defined even at its (isolated) critical points. (2) Restricting the linearized Cauchy-Riemann operator of a J-holomorphic curve to its generalized normal bundle defines a linear Cauchy-Riemann type operator on that bundle. (3) A J-holomorphic curve is Fredholm regular if and only if its normal Cauchy-Riemann operator is surjective. (The proof of the non-immersed case may be skipped if time is short.) (4) For any multiple cover of an immersed somewhere injective index zero J-holomorphic curve, if the normal Cauchy-Riemann operator is injective, then the only other J-holomorphic curves nearby are other multiple covers of the same index zero curve.
Literature:
Monday November 18, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: Paramjit Singh
Topic: Super-rigidity, local Gromov-Witten invariants and obstruction bundles
Abstract: An immersed somewhere injective index zero J-holomorphic curve is called super-rigid if the normal Cauchy-Riemann operators of all its multiple covers are injective. This is equivalent to saying that the nonlinear Cauchy-Riemann operator along the moduli space of covers of a super-rigid curve intersects the zero section "cleanly" (though not transversely), and e.g. in the setting of symplectic Calabi-Yau 3-folds, it is the best case scenario. If all somewhere injective curves are super-rigid, then one can associate to each one its so-called "local" Gromov-Witten invariants and calculate the global Gromov-Witten invariants as finite sums of such local contributions, which are essentially Euler numbers of certain well-defined "obstruction bundles" over the moduli space of covers. The aim of this talk should be to first define the notion of super-rigidty for a closed J-holomorphic curve, then to introduce the general idea of obstruction bundles in computing GW-invariants, and explain why these bundles are well defined whenever the super-rigidity hypothesis holds. (In other words, the goal is to explain Corollary 1.5 in the super-rigidity paper.)
Literature:
Monday November 25, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: Kangning Wei
Topic: The stratification theorem and its applications
Abstract: The equivariant transversality approach in the super-rigidity paper consists essentially of three steps: (1) Decompose the moduli space of all holomorphic curves into strata, which are smooth submanifolds (with easily computable dimensions) in which all elements have the same symmetries. (2) Decompose each stratum further into substrata on which the dimensions of kernels/cokernels of normal Cauchy-Riemann operators are constant. (3) Estimate the dimensions of all substrata in order to show that all curves for which the desired result (super-rigidity or regularity) fails belong to strata that have negative dimension, and are therefore empty. Step (2) is the hardest part, and the main result of that step is called the "stratification theorem" (Theorem D in the super-rigidity paper). The goal of this talk should be to state that theorem, along with a couple of related lemmas on index computations, and then explain why these imply the other main results in the paper, namely that for generic J, all somewhere injective index zero curves in dimensions greater than four are super-rigid (Theorem A), and all unbranched multiple covers are Fredholm regular (Theorem B).
Literature: The super-rigidty paper, Sections 2.2-2.3, plus Appendix A
Monday December 2, 2019 Berlin-Hamburg Symplectic Geometry Seminar (in Hamburg)
Monday December 9, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: Milica Đukić
Topic: Cauchy-Riemann type operators on punctured domains
Abstract: While the results in the super-rigidity paper deal exclusively with closed J-holomorphic curves, the setup in the proofs requires some analytical results that originate with the theory of punctured holomorphic curves. This requires a brief introduction to Sobolev spaces on punctured domains with exponential weights, the statements of the standard results about the Fredholm property, index and asymptotic behavior of Cauchy-Riemann type operators on such spaces, and then the proof of the following result (Prop. 3.13 in the paper): given a Cauchy-Riemann type operator on a closed domain and any finite set of points, one can remove those points and introduce suitable exponential weights at the punctures so that the index and kernel of the operator remain unchanged.
Literature:
Monday December 16, 2019
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: TBA
Topic: Regular presentations and the twisted bundle decomposition of Cauchy-Riemann operators with symmetry
Abstract: This talk does not require significant knowledge of holomorphic curve theory or Cauchy-Riemann operators, but instead requires some standard facts about covering spaces and vector bundles, and bit of the representation theory of finite groups (e.g. orthogonormality of characters). The goals are to explain (1) the canonical regular cover that is determined by any branched cover of Riemann surfaces, and how to identify its automorphism group; (2) why the normal Cauchy-Riemann operator of any multiply covered holomorphic curve has a natural direct sum decomposition with summands corresponding to the irreducible representations of the (generalized) automorphism group. This discussion culminates in a dimension formula (Corollary 3.21) that is crucial for computing the dimensions of the strata in the stratification theorem.
Literature:
Monday January 6, 2020
no seminar
Monday January 13, 2020
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: TBA
Topic: Fredholm indices of twisted Cauchy-Riemann operators
Abstract: The goal here is to prove a precise formula for the indices of the twisted Cauchy-Riemann type operators that appeared in the direct sum decomposition of the previous talk. The main tool required is the punctured version of the Riemann-Roch formula, expressed in terms of the relative first Chern number and Conley-Zehnder indices with respect to arbitrarily chosen asymptotic trivializations. The main challenge is to find a choice of trivializations in which both of the latter are computable.
Literature:
Monday January 20, 2020 Berlin-Hamburg Symplectic Geometry Seminar (in Berlin)
Monday January 27, 2020
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: TBA
Topic: Proof of the stratification theorem modulo Petri's condition
Abstract: The purpose of this talk is to set up the version of the infinite-dimensional implicit function theorem needed for the proof of stratification (Theorem D), and then discuss what is required for proving that the linearized operator in that setup is surjective. This leads naturally to the definition of Petri's condition, a desirable local property of linear differential operators that resembles (but does not follow from) unique continuation.
Literature:
  • The super-rigidty paper, Sections 3.5, 5.1 and 5.5, and about the first four pages of Section 6
  • The blog post Super-rigidity is fixed gives an overview of what Petri's condition is and how it naturally arises in equivariant transversality problems for differential operators
Monday February 3, 2020
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: TBA
Topic: Petri's condition is generic
Abstract: The bad news about Petri's condition is that not all Cauchy-Riemann type operators satisfy it, despite what one might intuitively expect from unique continuation. The last necessary ingredient for the proof of stratification is therefore to prove that for generic almost complex structures, all operators that arise as normal Cauchy-Riemann operators of J-holomorphic curves do satisfy Petri's condition. The proof of this requires only finite-dimensional analysis, because instead of examining differential operators on function spaces, it considers the maps they induce on the space of k-jets at a point for finite values of k. The main idea is then roughly to show that in this finite-dimensional setting, the space of operators that fail to satisfy Petri's condition will have larger codimension than the dimension of the ambient space of operators as soon as k is large enough. This dimension counting argument is made precise in the language of "C-infinity subvarieties," spaces that are not as nice as smooth manifolds but are nonetheless nice enough for Sard's theorem to apply.
Literature: The super-rigidty paper, Section 5 (excluding 5.5) and Appendix C
Monday February 10, 2020
15:00-17:00 (c.t.)
RUD 25, Room 1.023
Speaker: TBA
Topic: loose ends

Other symplectic events going on this semester

...and next semester

In Sommersemester 2020, Prof. Chris Wendl will be teaching a lecture course (4 hours/week + Übung) on symplectic field theory. It will be based roughly on this book (to appear in the EMS Lectures in Mathematics series).

last semester's symplectic seminar