## Northern California Symplectic Geometry Seminar, 2020-2021 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 takes place in October, during the first meeting of the seminar.

### Upcoming NCSGS meetings

Monday, April 5, 2021, TBA.

### Previous NCSGS meetings

Monday, Oct 5, 2020, @Stanford (virtually), poster.

1pm-2pm (Pacific Time) on ZOOM
Andreas Floer Memorial Lecture
Grisha Mikhalkin (Geneva University), Exotic embeddings of symplectic cubes, slides

Abstract: By a symplectic cube C(A) we mean the product of n disks of area $$A>0$$. For $$1/2 < A < 2/3$$ we exhibit symplectic embeddings of C(A) into C(1) that are not equivalent with respect to Hamiltonian isotopy and coordinate permutations. The number of such embeddings grows arbitrary large when A approaches 1/2. The construction continues the series of symplectic applications of (toric) geometry of Markov's equation $$a^2+b^2+c^2=3abc$$ (and its clones like $$a^2+2b^2+c^2=4abc$$) uncovered in the 2010 preprint of Galkin and Usnich. Joint work with Joé Brendel and Felix Schlenk.

2:30pm-3:30pm (Pacific Time) on ZOOM
Mohammed Abouzaid (Columbia University), Arnol'd Conjecture and Morava K-theory
, slides

Abstract: I will give an overview of the proof of the following joint result with Blumberg: for every closed symplectic manifold, the number of time-1 periodic orbits of a non-degenerate Hamiltonian is bounded below by the rank of the cohomology with coefficients in any field. The case of characteristic 0 was proved by Fukaya and Ono as well as Li and Tian. The new ingredient in our proof is the construction of generalised Floer cohomology groups with coefficients in Morava K-theory. This means that we have to use higher dimensional moduli spaces of pseudo-holomorphic curves, and extract from their Kuranishi structures "fundamental chains" in generalized cohomology.

Monday, Nov 2, 2020, @Davis (virtually), poster.

2:30pm-3:30pm (Pacific Time)
John Etnyre (GaTech), Legendrian cables; notes and video.

Abstract: Legendrian and transverse knots have played a large role in the development of 3 dimensional contact geometry, and studying Legendrian and transverse representatives of torus knots and cable knots has played a large role in our understanding of the general behavior of Legendrian and transverse knots. In this talk I will discuss some recent progress in the area. In particular, I will discuss joint work with Jennifer Dalton and Lisa Traynor about torus links and cable links. For example, for nice knot types (uniform thick) we show that when considering the symmetries of Legendrian cables all topological symmetries, only cyclic symmetries, or no symmetries are allowed if the cabling slope is larger than, less than, or equal to the maximal Thurston-Bennequin invariant, respectively (and all the components have maximal Thurston-Bennequin invariant). This is one of the first classifications of infinite families of Legendrian links where restrictions on symmetries are present. We will also discuss joint work with Apratim Chakraborty and Hyun Ki Min that completely describes Legendrian representatives of large positive cable knots in terms of the underlying knot as well as explain the phenomena of Legendrian large cables. The latter, which were discovered in a beautiful but mysterious construction of Yasui, are Legendrian representatives of a cabled knot type with Thurston-Benniquin invariant larger that was classically expected.

4pm-5pm (Pacific Time)
Eric Zaslow (Northwestern), Branes, Wavefunctions, Sheaves, Clusters; notes and video.

Abstract: I will try to weave together ideas of many physicists and mathematicians into a semi-coherent mathematical yarn. We identify the wavefunction of a Lagrangian brane with a potential function for its moduli space, then interpret it as the generating function for open Gromov-Witten invariants. For a particular class of branes, we relate these to quiver invariants as well — the quiver depending, curiously, on a “framing.” The mathematical setting is a Legendrian surface defined by a planar triangulation and the moduli space of constructible sheaves microsupported on it. This moduli space is a Lagrangian subspace inside a symplectic leaf of a cluster variety that has a quantization, one chart of which is described by the triangulation. The defining function for the corresponding ideal is conjecturally the all-genus open Gromov-Witten generating function for a non-exact Lagrangian filling depending on the framing. Placing the problem in the context of clusters allows one to exploit mutations to transfer knowledge from a patch where the wavefunction is known to patches where it is not, providing a scheme for calculation. This is work in progress with Linhui Shen.

Monday, Dec 7, 2020, @Berkeley (virtually).

1pm-2pm (Pacific Time)
Alex Oancea (Sorbonne University), Aspects of duality for Rabinowitz-Floer homology; notes and video.

Abstract: I will explain a duality theorem for Rabinowitz Floer homology which takes into account product structures. This result has numerous applications, and I will place particular emphasis on string topology. The talk will be based on recent joint work with Kai Cieliebak and Nancy Hingston.

3pm-4pm (Pacific Time)
Richard Hind (University of Notre Dame), Embedded Lagrangian tori in 4 dimensional domains; notes and video.

Abstract: We determine the possible area classes of Lagrangian tori in 4 dimensional ellipsoids and polydisks. Optimal embeddings, those whose area class cannot appear in smaller domains, are either inclusions of product tori or images of product tori under a kind of winding and lifting. The proofs involve studying punctured holomorphic curves asymptotic to thin ellipsoids sitting on the Lagrangian, resembling closed curves with a high degree tangency. Finally we discuss some analogous questions for disjoint unions of tori. This is joint work with Ely Kerman, Emmanuel Opshtein and Jun Zhang.

Monday, Feb 8, 2021, @UC Santa Cruz (virtually), poster.

1pm-2pm (Pacific Time)
Rich Schwartz (Brown University), Inscribing rectangles in Jordan curves; slides and video.

Abstract: The notorious square peg problem asks if every Jordan curve has an inscribed square -- i.e., 4 points which make the vertices of a square. I don't know how to solve this problem but I will explain my result that all but at most 4 points of any Jordan curve are vertices of inscribed rectangles, and show lots of computer pictures. I'll also discuss the recent results of Josh Greene and Andrew Lobb about these kinds of problems -- results which make a connection between inscribed rectangles and Lagrangian Klein bottles.

3pm-4pm (Pacific Time)
Julian Chaidez (University of California, Berkeley), Convex contact forms and the Ruelle invariant; slides and video.

Abstract: The boundary of a strictly convex domain in 4-space is equipped with a natural contact form acquired by restricting the standard radial Liouville form. A contact form on the 3-sphere is called dynamically convex if every Reeb orbit has Conley-Zehnder index greater than or equal to 3. Every contact form arising by restriction of the standard Liouville form to the boundary of a strictly convex domain is dynamically convex. It has been a longstanding open problem to determine if, conversely, every dynamically convex contact form arises in this way.
In this talk, I will explain my recent joint work with Oliver Edtmair constructing dynamically convex contact forms on the 3-sphere that are not convex. Our main tool is a bound on the Ruelle invariant, which can be viewed as a sort of spacetime averaged rotation number or Conley- Zehnder index. We construct dynamically convex contact forms that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomao.

Monday, March 1, 2021, @Stanford (virtually), poster.

1pm-2pm (Pacific Time)
Tobias Ekholm (Uppsala University), Skein module curve counts and recursion; notes

Abstract: Counting holomorphic curves in a Calabi-Yau 3-fold X with Maslov zero Lagrangian boundary condition L by their boundaries in the framed skein module of L gives a deformation invariant quantity. We review this construction briefly and compare the resulting invariants to real Gromov-Witten invariants when there is an involution. We then study the toric brane in complex 3-space and knot conormals. Here we show that holomorphic curves on the Legendrian which is the ideal boundary of the Lagrangian stores the information of the curve count in terms of a skein valued recursion relation, which is comparatively easy to compute. The talk reports on joint works with Penka Georgieva, Lenhard Ng, and Vivek Shende.

2:30pm-3:30pm (Pacific Time)
Daniel Cristofaro-Gardiner (UC Santa Cruz and IAS), PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric; notes and video.

Abstract: The group of Hamiltonian diffeomorphisms of a symplectic manifold admits a remarkable bi-invariant metric, called Hofer’s metric. My talk will be about a recent joint work with Vincent Humilière and Sobhan Seyfaddini resolving the following two open-questions related to the geometry of this metric. The first, due to Kapovich and Polterovich, asks whether the two-sphere, equipped with Hofer’s metric, is quasi-isometric to the real line; we show that it is not. The second, due to Fathi, asks whether the group of area and orientation preserving homeomorphisms of the two-sphere is a simple group; we show that it is not. Key to our proofs is a new sequence of spectral invariants defined via Hutchings’ Periodic Floer Homology.

Organizers: Roger Casals (Davis), Yasha Eliashberg (Stanford), Dmitry Fuchs (Davis), Dan Gardiner (Santa Cruz), Viktor Ginzburg (Santa Cruz), Michael Hutchings (Berkeley), Eleny Ionel (Stanford), Richard Montgomery (Santa Cruz), Vivek Shende (Berkeley), Laura Starkston (Davis), Katrin Wehrheim (Berkeley), Alan Weinstein (Berkeley/Stanford).