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.
As in past semesters, we will switch the time of the seminar this semester to Mondays, 15:00-17:00 (c.t.) in room 1.315 at Rudower Chaussee 25. It will occasionally be preempted by the Berlin-Hamburg Symplectic Geometry Seminar.
| Monday October 15, 2018 | no seminar (dies academicus) |
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| Monday October 22, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Marc Kegel Topic: Parallelizability of 3-manifolds via surgery Abstract: Every 3-manifold is parallelizable, i.e. admits three linearly independent vector fields. In this talk, we will discuss new proofs of this classical result using surgery descriptions of 3-manifolds. To understand this talk, only basic knowledge on manifolds and their homology and fundamental groups are required. This is joint work with S. Durst, H. Geiges, and J. Gonzalo. |
| Monday October 29, 2018 | no seminar |
| Monday November 5, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Marc Kegel Topic: Parallelizability of 3-manifolds via surgery (continuation) |
| Monday November 12, 2018 | Berlin-Hamburg Symplectic Geometry Seminar (in Berlin) |
| Monday November 19, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Felix Nötzel Topic: Morse-Bott property of the action functional in symplectisations Abstract: In order to find periodic Reeb orbits in a contact manifold, one can study their symplectisations and punctured pseudoholomorphic curves and their asymptotic behaviour. I will explain that in the Morse-Bott case the action functional is a Morse-Bott function on a loop space and how this property can be used to show that at each puncture the pseudoholomorphic curves converge to a Reeb orbit. |
| Monday November 26, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Jonathan Bowden (Monash University) Topic: Symplectic caps of higher-dimensional contact manifolds |
| Monday December 3, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Felix Nötzel Topic: Morse-Bott property of the action functional in symplectizations (continued) |
| Monday December 10, 2018 | Berlin-Hamburg Symplectic Geometry Seminar (in Hamburg) |
| Monday December 17, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
no seminar (Weihnachtsfeier) |
| Monday January 7, 2018 | no seminar |
| Monday January 14, 2018 | Berlin-Hamburg Symplectic Geometry Seminar (in Berlin) |
| Monday January 21, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Dingyu Yang Topic: Introduction to Kuranishi structures Abstract: In symplectic geometry, the algebraic topology of moduli spaces of (perturbed) J-holomorphic curves in symplectic manifolds provides a rich source of information. Examples include the Gromov-Witten invariants and symplectic field theory, and they reveal deep topological and dynamical properties intrinsic to symplectic structures. J-curves can bubble off and degenerate into nodal or broken configurations, forming a compactified moduli space. In the exact and semi-positive settings, these phenomena either do not occur or are of harmless codimension respectively; in these cases it suffices to work with somewhere injective curves and perturb J generically, because the relevant moduli spaces are all nice manifolds simultaneously and everyone is happy. In the general situation, multiply covered curves cause problems due to the incompatibility between symmetry and transversality, and one cannot avoid working with compactifications directly. It is fair to say that "virtual techniques" have the maximal applicability to obtain a well-defined theory, where every J-curve in the compactification is treated on equal footing as a point in a generalized space. In two talks, I will gently introduce one such generalized space called Kuranishi structures (and its close cousin known as good coordinate systems). Its local model is elementary in differential geometry but not of a fixed dimension. I will cover some ingredients, and explain how transversality can be achieved in spite of non-uniform dimensions. |
| Monday January 28, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Dingyu Yang Topic: Introduction to Kuranishi structures (continued) |
| Monday February 4, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Mihai Munteanu (Berkeley) Topic: Nontrivial tori in spaces of symplectic embeddings Abstract: The problem of when one can embed one symplectic manifold into another has produced beautiful surprising results even when the spaces in question are relatively simple. Gromov's nonsqueezing theorem and McDuff's result about the connectivity of the space of symplectic embeddings between 4-dimensional symplectic ellipsoids are just two examples of this phenomenon. In this talk, we will tackle the study of the fundamental group of spaces of symplectic embeddings and show how certain contractible loops of unitary transformations become noncontractible when restricted as loops of symplectic embeddings between certain symplectic ellipsoids. Moreover, we will introduce recent work (joint with Julian Chaidez) showing that certain n-torus families of symplectic embeddings between 2n-dimensional ellipsoids become homologically nontrivial if certain inequalities involving symplectic invariants hold. |
| Monday February 11, 2018 15:00-17:00 (c.t.) RUD 25, Room 1.315 |
Speaker: Dingyu Yang Topic: Introduction to Kuranishi structures (continued) |