Seminar: Symplektische Topologie / Symplectic Topology — Wintersemester 2017-18, HU Berlin

This is a working group seminar run by Klaus Mohnke and Chris Wendl on recent developments in symplectic geometry. 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 will meet this semester on Mondays, 15:00-17:00 (c.t.) in room 1.114 at Rudower Chaussee 25 (subject to change). It will occasionally be preempted by the Berlin-Hamburg Symplectic Geometry Seminar.

Organizational note: The first Monday this semester is a dies academicus, but we will nonetheless have a special meeting of the seminar on that day since there is a visiting speaker. The first official meeting of the seminar for this semester will thus be on October 23, when we will discuss plans for the rest of the semester. Students looking for further information about the seminar in the mean time should feel free to contact Profs. Mohnke or Wendl.

Monday October 16, 2017
15:00-17:00 (c.t.)
RUD 25, Room 1.114
Speaker: Oliver Fabert (VU Amsterdam)
Topic: J-holomorphic curves in infinite dimensions
Abstract: Many important PDEs can be viewed as infinite-dimensional Hamiltonian systems. Although modern Hamiltonian dynamics has started with P. Rabinowitz' work on nonlinear wave equations, the infinite-dimensional case has received much less attention --- partly as the J-holomorphic curve techniques are not readily available. In my talk I will present an efficient model-theoretic method to transfer results from finite to infinite dimensions. I will illustrate its use by proving a nonsqueezing result for smooth infinite-dimensional Hamiltonian flows and a generalized Arnold conjecture for nonlinear Schrödinger equations.
Monday October 23, 2017
15:00-17:00 (c.t.)
RUD 25, Room 1.315
initial meeting to discuss further plans for the semester
Monday October 30, 2017 NO SEMINAR THIS WEEK
Monday November 6, 2017
15:00-17:00 (c.t.)
RUD 25, Room 1.315
Speaker: Chris Wendl
Topic: Introduction to Gromov-Witten invariants of symplectic manifolds
Abstract: This will be an introductory survey aimed at people unfamiliar with either Gromov-Witten theory or symplectic topology. In the simplest cases, Gromov-Witten invariants associate to any symplectic manifold M with a compatible almost complex structure J, and choices of a 2-dimensional homology class A and a nonnegative integer g, the signed count of J-holomorphic curves in M of genus g homologous to A; the count turns out to be independent of the chosen almost complex structure but depends on the deformation class of the symplectic structure. I will give an overview of the technical results that are needed in order to make this idea precise in all closed symplectic manifolds of dimension up to 6, following the standard approaches due to Ruan-Tian and McDuff-Salamon. I will then introduce the idea of obstruction bundle calculations and "multiple cover contributions", with particular focus on the setting of symplectic Calabi-Yau 3-folds, which have the special property that all moduli spaces of holomorphic curves have virtual dimension zero. This is meant as preparation for the talks on November 20 and 27, which will also involve obstruction bundle methods.
References:
Monday November 13, 2017
14:30-17:00 (s.t.)
HU Berlin
RUD 25, Room 1.115
Berlin-Hamburg Symplectic Geometry Seminar with speakers Jean Gutt (Cologne) and Patrick Massot (Orsay)
Tuesday November 14, 2017
15:30-17:00 (s.t.)
RUD 25, Room 1.315
Speaker: Patrick Massot (Orsay)
Topic: TBA
Monday November 20, 2017
15:00-17:00 (c.t.)
RUD 25, Room 1.315
Speaker: Chris Wendl
Topic: Transversality and super-rigidity for pseudoholomorphic curves, part 1
Abstract: Following up on my introductory talk from November 6, I will discuss some of the technical aspects of the transversality results behind standard symplectic constructions of Gromov-Witten theory, and then explain a much more recent result which establishes that for generic tame almost complex structures on a symplectic Calabi-Yau 3-fold, the Gromov-Witten invariants are fully "localized", meaning each count can be computed in terms of multiple cover contributions for a finite collection of disjoint embedded holomorphic curves. The technical reason for this is that for generic J, all simple holomorphic curves of index zero are "super-rigid", which implies among other things that none of their multiple covers are in the closure of the space of simple curves. The proof is inspired in part by Taubes's work on the Gromov invariant: it uses a representation-theoretic splitting of Cauchy-Riemann type operators with symmetries to stratify the space of all multiply covered curves into smooth submanifolds on which the kernels and cokernels of certain twisted Cauchy-Riemann operators have fixed dimension.
References:
Monday November 27, 2017
15:00-17:00 (c.t.)
RUD 25, Room 1.315
Speaker: Chris Wendl
Topic: Transversality and super-rigidity for pseudoholomorphic curves, part 2 (continued from November 20)
Monday December 4, 2017
15:00-17:00 (c.t.)
RUD 25, Room 1.315
Speaker: Janko Latschev (Hamburg)
Topic: Multifold symplectic sums after Tehrani-McLean-Zinger
Monday December 11, 2017
14:30-17:00 (s.t.)
Uni Hamburg
Berlin-Hamburg Symplectic Geometry Seminar with speakers Ailsa Keating (Cambridge) and Lei Zhao (Augsburg)
Tuesday December 12, 2017
15:30-17:00 (s.t.)
RUD 25, Room 1.315
Speaker: Alexandru Cioba (UCL)
Topic: Nicely embedded curves in symplectic cobordisms
Monday December 18, 2017 NO SEMINAR THIS WEEK
Monday January 8, 2018
15:00-17:00 (c.t.)
RUD 25, Room 1.315
Speaker: Alex Fauck (TBC)
Topic: TBA
Monday January 15, 2018
14:30-17:00 (s.t.)
HU Berlin
Berlin-Hamburg Symplectic Geometry Seminar with speakers Ivan Smith (Cambridge) and Oleg Lazarev (Columbia)
Monday January 22, 2018
15:00-17:00 (c.t.)
RUD 25, Room 1.315
TBA
Monday January 29, 2018
15:00-17:00 (c.t.)
RUD 25, Room 1.315
TBA
Monday February 5, 2018
15:00-17:00 (c.t.)
RUD 25, Room 1.315
TBA
Monday February 12, 2018
15:00-17:00 (c.t.)
RUD 25, Room 1.315
TBA

last semester's symplectic seminar