Northern California Symplectic Geometry
Seminar, 2022-2023
Berkeley - Davis - Santa Cruz - Stanford
The Northern California Symplectic Geometry Seminar usually meets on the first Monday of each month. Established in 1992,
the Andreas Floer Memorial Lecture usually takes place in the Fall, during the first meeting of the seminar.
For slides or videos of the talks (when available), please click on link after the title of the talk.
Monday Nov 7, 2022
, @ Stanford, room 383N
2:30pm-3:30pm, room 383N Andreas Floer Memorial Lecture Ivan Smith (Cambridge University),
Quantum (Morava) K-theory Abstract:
Quantum cohomology is a well-known enumerative invariant of symplectic manifolds. For smooth projective varieties, quantum K-theory was defined around twenty years ago using the `virtual structure sheaf'. Recently we have constructed quantum K-theory, and quantum Morava K-theories, for general compact symplectic manifolds. We will describe something of this theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.
4pm-5pm, room 383N Egor Shelukhin (University of Montreal),
Exact Lagrangians, fixed points, and flux Abstract: We describe the interrelation between the exactness of Lagrangian graphs, the strong Arnol'd conjecture, and the flux conjectures. In particular we answer a question of Lalonde-McDuff-Polterovich and make new progress on the \(C^0\) flux conjecture. This is joint work in progress with Marcelo Atallah.
Monday March 6, 2023, @ Stanford, room 383N
2:30pm-3:30pm, room 383N Mike Sullivan (University of Massachusetts),
\(C^0\)-limits of Legendrian submanifolds
Abstract: Consider a sequence of contactomorphisms whose \(C^0\)-limit is a homeomorphism, and a Legendrian submanifold whose limiting image under this sequence is a smooth submanifold. I will show that the limit is in fact Legendrian, and moreover, when the contact manifold is 3-dimensional, the limit is contactomorphic to the initial Legendrian knot. This is joint work with Georgios Dimitroglou Rizell.
4pm-5pm, room 383N Oliver Edtmair (Berkeley),
Symplectic Weyl laws
Abstract:
Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics and C^0 symplectic geometry. For example, it plays a key role in the proof of the smooth closing lemma for three-dimensional Reeb flows and area preserving surface diffeomorphisms, and in the proof of the simplicity conjecture. ECH and PFH are highly sophisticated theories whose construction in particular relies on Seiberg-Witten theory. I will explain how one can use much more elementary methods (no Floer or gauge theory) to define spectral invariants satisfying an analogous Weyl law with similar applications.
Monday May 1st, 2023, @ Stanford, room 383N
2:30pm-3:30pm, room 383N Mohan Swaminathan (Stanford),
Counting embedded curves in Calabi-Yau 3-folds using bifurcation analysis
Abstract:
I will report on joint work with Shaoyun Bai where we study bifurcations of embedded holomorphic curves in Calabi-Yau 3-folds in order to define an integer-valued invariant analogous to Taubes' Gromov invariant. We are able to understand the bifurcations completely in some cases and this results in the definition of an invariant which is conjecturally equivalent to Gopakumar-Vafa's BPS invariants. An interesting feature of the problem is that global virtual techniques don't seem to help so much and we instead need to build on a far-reaching generalization of usual generic transversality results due to Wendl.
4pm-5pm, room 383N Tim Perutz (UT Austin),
Counting pseudo-holomorphic sections of Lefschetz fibrations
Abstract:
A sequence of simple closed curves on a closed surface determines a symplectic 4-manifold with boundary, the total space of the Lefschetz fibration over the disc with the specified fiber and the curves as vanishing cycles. One can ask how to compute the invariants of this symplectic 4-manifold – notably the relative Seiberg-Witten invariants or their symplectic avatars - in terms of the surface data. One such symplectic avatar is the Donaldson-Smith count of pseudo-holomorphic multi-sections of the Lefschetz fibration representing a prescribed second homology class. In this talk I will describe a project, joint with my UT Austin Ph.D. student Riccardo Pedrotti, that addresses the computation of the count of pseudo-holomorphic sections (not yet multi-sections) in a prescribed homology class.