Northern California Symplectic Geometry
Seminar, 2024-2025
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.
2:30pm-3:30pm, room 384H Basak Gurel (University of Central Florida),
Invariant Sets and Hyperbolic Periodic Orbits
Abstract:
The presence of hyperbolic periodic orbits or invariant sets often has an effect on the global behavior of a symplectic dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards a higher-dimensional Franks’ theorem or forced existence of periodic orbits for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Ginzburg and asserting that no periodic orbit of a Hamiltonian pseudo-rotation of a complex projective space is locally maximal. The talk is based on a joint work with Erman Cineli, Viktor Ginzburg and Marco Mazzucchelli.
4pm-5pm, room 383N Jae Hee Lee (Stanford),
3D mirror symmetry in positive characteristic
Abstract:
Abstract: In positive characteristic, on one hand the quantum D-module (from Gromov--Witten theory) carries extra central endomorphisms known as quantum Steenrod operations. On the other hand, in representation theory, the algebra of differential operators also carry a "large center" in positive characteristic given by a version of the Frobenius map. I will explain how these centers are related to each other by a duality of algebraic symplectic varieties. Joint with Shaoyun Bai.
Abstract: Symplectic cohomology is a fundamental invariant of a symplectic manifold M with
contact type boundary that is defined in terms of dynamical information and counts of pseudoholomorphic
genus zero curves, and carries algebraic structures that parallel the algebraic
structures on the Hochschild (co)homology of the Fukaya category of M. We show, under natural
topological assumptions, that the symplectic cohomology is the homology of a genuine
p-cyclotomic spectrum in the sense of Nikolaus-Scholze. The cyclotomic structure arises geometrically
from the map which sends loops in M to their p-fold iterates. The data of this
refinement is expected to produce many new algebraic structures of an arithmetic nature on
symplectic cohomology, analogously to the way that prismatic cohomology refines the de Rham
cohomology of a variety. The talk will explain the result and, if time permits, discuss concrete
connections to equivariant string topology, equivariant Gromov-Witten theory, and to arithmetic
geometry.
4pm-5pm, room 383N Andreas Floer Memorial Lecture Ko Honda (UCLA),
A Morse \( A_\infty\)-model for the higher-dimensional Heegaard Floer homology of cotangent fibers
Abstract:
Given a smooth closed n-manifold M and a \(\kappa\)-tuple of basepoints \({\bf q}\subset M\), we define a Morse-type \(A_\infty\)-algebra called the based multiloop \(A_\infty\)-algebra and show the equivalence with the higher-dimensional Heegaard Floer \(A_\infty\)-algebra of \(\kappa\) disjoint cotangent fibers of \(T^*M\).
Abstract: A fundamental theorem of Giroux states that every closed surface in a contact 3-
manifold can be smoothly approximated by a convex surface. Recently, Honda-Huang partially
generalized Giroux’s theorem to higher dimensions, by proving that any hypersurface in a contact
manifolds can be continuously approximated by a convex one.
In this talk, I will explain a proof that Giroux’s theorem is false in higher dimensions.
Precisely, there are hypersurfaces in any contact manifold of dimension five or greater that
cannot be smoothly approximated by convex hypersurfaces. The main technical step is the
construction of a Bonatti-Diaz type blender in the contact setting.
4pm-5pm, room 732 Evans Eric Kilgore (Stanford),
Legendrian non-squeezing via microsheaves
Abstract:
In this talk I will explain some quantitative embedding results for Legendrian submanifolds
of pre-quantization spaces. To start, I will recall some contact non-squeezing results
for domains, and present an elementary proof of Legendrian non-squeezing for lifts of integral
Lagrangian loops in \(T^∗\mathbb R\), using a notion of normal ruling. Then I will explain a high dimensional
generalization of this technique in the language of microsheaves. If time permits, as an
application, I will show that the Legendrian lifts of certain Clifford and Chekanov tori are not
squeezable.
Abstract:
The ellipsoid embedding function generalizes symplectic ball packing problems. For a symplectic manifold, this function determines the minimum scaling factor required for a standard ellipsoid with a given eccentricity to embed symplectically into the manifold. If the function has infinitely many nonsmooth points, it is said to have an infinite staircase. An infinite staircase implies that an infinite number of distinct obstructions are needed to characterize the function. In this talk, we will present partial results addressing the question: when does the ellipsoid embedding function for a convex toric domain have an infinite staircase? This will include joint work with McDuff-Weiler, Pires-Weiler, and upcoming work with Cristofaro-Gardiner and McDuff.
4pm-5pm, room 383N Viktor Ginzburg (UC Santa Cruz),
Towards the HZ- and multiplicity conjectures for dynamically convex Reeb flows
Abstract:
In this talk, based on a joint work with Erman Cineli and Basak Gurel, we discuss the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain. The first of our two main results asserts that such a flow has at least n prime closed Reeb orbits, settling a conjecture which is usually attributed to Ekeland. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly n or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks’ celebrated 2-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture.
2:30pm-3:30pm, room 732 Evans Leonid Polterovich (Tel Aviv University and University of Chicago)
Coarse geometry of Hamiltonian diffeomorphisms
Abstract:
I will discuss the asymptotic growth of autonomous Hamiltonian flows with respect
to the Hofer metric. This includes a dichotomy on the two-sphere: the Hofer norm either
grows linearly or remains bounded in time by a universal constant. I will also touch on some
connections to dynamics. Based on joint works with Lev Buhovsky, Ben Feuerstein, and Egor
Shelukhin.
4pm-5pm, room 732 Evans Adriano Tomassini (University of Parma)
Solvmanifolds with special structures
Abstract:
We will present some recent results on the existence of special structures on compact
non Kahler manifolds obtained as quotients of \( \mathbb C^n \ltimes \mathbb C^m\). More in particular, we will focus on
p-Kahler structures and on symplectic structures satisfying the hard Lefschetz condition. Bott-Chern and Aeppli cohomologies are explicitly computed.
2:30pm-3:30pm, room 384H Jean Gutt (IAS and Toulouse Univ)
Ekeland-Hofer capacities from positive S^1 equivariant symplectic homology
Abstract:
Associated to a star-shaped domain in R^{2n} are two increasing sequences of capacities: the Ekeland-Hofer capacities and the so-called Gutt-Hutchings capacities. I shall recall both constructions and then present the main theorem that they are the same. This is joint work with Vinicius Ramos.
4pm-5pm, room 383N Constantin Teleman (UC Berkeley)
Integral Kirwan formality and surjectivity
Abstract:
Relying on Morse theory and an Euler class argument of Atiyah and Bott, Frances Kirwan proved two important results about the rational cohomology of compact symplectic manifold $X$ with the Hamiltonian action of a connected, compact group $G$: equivariant formality, or the triviality of the $G$-action on the \(H^*(X;\mathbb{Q})\), and the subjectivity of the natural restriction map $H^*_G(X;\mathbb{Q}) \to H^*(X//G;\mathbb{Q})$ (when the latter symplectic quotient is an orbifold). In this slightly `retro’ talk, I will explain an integral (and even stable homotopy) improvement of these results, subject to natural restrictions. This is joint work with Dan Pomerleano (and is based one older ideas of Harada-Landweber and related results of Bai-Pomerleano).
Organizers: Mohammed Abouzaid (Stanford), Roger Casals (Davis), Yasha Eliashberg (Stanford), Dmitry Fuchs (Davis), 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).
For further information, please contact
Eleny Ionel