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 Floertype theories. The seminar is conducted in English.
The seminar takes place this semester on Mondays, 15:0017:00 (c.t.) in room 1.023 (the BMS Seminar Room) at Rudower Chaussee 25. It will occasionally be preempted by the BerlinHamburg 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 superrigidity for multiply covered holomorphic curves, including some necessary background material on holomorphic curve theory and GromovWitten 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 15:0017:00 (c.t.) RUD 25, Room 1.023 
Informal meeting to discuss topics for the semester 

Monday October 21, 2019 15:0017:00 (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 Jholomorphic 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 finitedimensional 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 CauchyRiemann 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 15:0017:00 (c.t.) RUD 25, Room 2.006 (Achtung! room change) 
Speaker: Yoanna Kirilova Topic: The SardSmale theorem and transversality for somewhere injective Jholomorphic curves Abstract: The main result of this talk is the standard theorem that for generic choices of almost complex structure, all somewhere injective closed Jholomorphic curves are Fredholm regular. The main tools needed for this are (1) a unique continuation result for solutions of linear CauchyRiemann type equations, and (2) Smale's infinitedimensional version of Sard's theorem, both of which should be stated carefully (but not proved) since they will be needed again later. The Floer "Cepsilon" space of perturbed almost complex structures should also be introduced. Local elliptic regularity results for CauchyRiemann type operators (e.g. the fact that weak solutions of class L^{p} for p > 2 are smooth) should be taken as black boxes. Literature: Chris's holomorphic curve notes, Section 4.4 
Monday November 4, 2019  BerlinHamburg Symplectic Geometry Seminar (in Berlin) 
Monday November 11, 2019 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: Felix Noetzel Topic: The normal CauchyRiemann operator Abstract: The purpose of this talk is to explain the following facts: (1) Every nonconstant Jholomorphic curve has a "generalized" normal bundle which is well defined even at its (isolated) critical points. (2) Restricting the linearized CauchyRiemann operator of a Jholomorphic curve to its generalized normal bundle defines a linear CauchyRiemann type operator on that bundle. (3) A Jholomorphic curve is Fredholm regular if and only if its normal CauchyRiemann operator is surjective. (The proof of the nonimmersed case may be skipped if time is short.) (4) For any multiple cover of an immersed somewhere injective index zero Jholomorphic curve, if the normal CauchyRiemann operator is injective, then the only other Jholomorphic curves nearby are other multiple covers of the same index zero curve. Literature:

Monday November 18, 2019 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: Paramjit Singh Topic: Superrigidity, local GromovWitten invariants and obstruction bundles Abstract: An immersed somewhere injective index zero Jholomorphic curve is called superrigid if the normal CauchyRiemann operators of all its multiple covers are injective. This is equivalent to saying that the nonlinear CauchyRiemann operator along the moduli space of covers of a superrigid curve intersects the zero section "cleanly" (though not transversely), and e.g. in the setting of symplectic CalabiYau 3folds, it is the best case scenario. If all somewhere injective curves are superrigid, then one can associate to each one its socalled "local" GromovWitten invariants and calculate the global GromovWitten invariants as finite sums of such local contributions, which are essentially Euler numbers of certain welldefined "obstruction bundles" over the moduli space of covers. The aim of this talk should be to first define the notion of superrigidty for a closed Jholomorphic curve, then to introduce the general idea of obstruction bundles in computing GWinvariants, and explain why these bundles are well defined whenever the superrigidity hypothesis holds. (In other words, the goal is to explain Corollary 1.5 in the superrigidity paper.) Literature:

Monday November 25, 2019 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: Kangning Wei Topic: The stratification theorem and its applications Abstract: The equivariant transversality approach in the superrigidity 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 CauchyRiemann operators are constant. (3) Estimate the dimensions of all substrata in order to show that all curves for which the desired result (superrigidity 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 superrigidity 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 superrigid (Theorem A), and all unbranched multiple covers are Fredholm regular (Theorem B). Literature: The superrigidty paper, Sections 2.22.3, plus Appendix A 
Monday December 2, 2019  BerlinHamburg Symplectic Geometry Seminar (in Hamburg) 
Monday December 9, 2019 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: Milica Đukić Topic: CauchyRiemann type operators on punctured domains Abstract: While the results in the superrigidity paper deal exclusively with closed Jholomorphic 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 CauchyRiemann type operators on such spaces, and then the proof of the following result (Prop. 3.13 in the paper): given a CauchyRiemann 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 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: TBA Topic: Regular presentations and the twisted bundle decomposition of CauchyRiemann operators with symmetry Abstract: This talk does not require significant knowledge of holomorphic curve theory or CauchyRiemann 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 CauchyRiemann 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 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: TBA Topic: Fredholm indices of twisted CauchyRiemann operators Abstract: The goal here is to prove a precise formula for the indices of the twisted CauchyRiemann type operators that appeared in the direct sum decomposition of the previous talk. The main tool required is the punctured version of the RiemannRoch formula, expressed in terms of the relative first Chern number and ConleyZehnder 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  BerlinHamburg Symplectic Geometry Seminar (in Berlin) 
Monday January 27, 2020 15:0017:00 (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 infinitedimensional 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:

Monday February 3, 2020 15:0017:00 (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 CauchyRiemann 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 CauchyRiemann operators of Jholomorphic curves do satisfy Petri's condition. The proof of this requires only finitedimensional analysis, because instead of examining differential operators on function spaces, it considers the maps they induce on the space of kjets at a point for finite values of k. The main idea is then roughly to show that in this finitedimensional 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 "Cinfinity subvarieties," spaces that are not as nice as smooth manifolds but are nonetheless nice enough for Sard's theorem to apply. Literature: The superrigidty paper, Section 5 (excluding 5.5) and Appendix C 
Monday February 10, 2020 15:0017:00 (c.t.) RUD 25, Room 1.023 
Speaker: TBA Topic: loose ends 