Bonn Symplectic Geometry Seminar

Abstract: We show that the existence of pseudo-holomorphic lines passing through two generic points on a symplectic manifold X, phrased using Gromov-Witten invariants, implies that X is homotopy equivalent to CP^n with identical first Chern class and small quantum cohomology. We then deduce some rigidity results regarding symplectic hyperplanes in CP^{n+1}. The proof is based on Rabinowitz Floer homology. 

Abstract: Work of Diogo, Diogo-Lisi, and Ganatra-Pomerleano have explored the idea of computing symplectic cohomology of an ample divisor complement. In particular, we may compute the associated graded of the standard action filtration on symplectic cohomology in terms of the topology of the divisor complement and the topology of the circle bundle associated to the normal bundle of the divisor, and the obstruction to splitting into the associated graded is encoded by (zero-dimensional) genus 0 relative Gromov-Witten type moduli spaces. In this talk, we will explore how to lift this to Floer homotopy theory; here, (twisted) framed bordism classes of higher-dimensional genus 0 relative Gromov-Witten type moduli spaces obstruct the Floer homotopy type from splitting into its associated graded, where the latter is computed topologically. Time permitting, we will discuss example computations.

Abstract: The concept of orbi-topological space is that of a space which is locally the quotient of a topological space by the action of a group (usually assumed to be finite or compact Lie). This concept can be conveniently formalized using the language of stacks. In 2007, Gepner-Henriques proved a theorem to the effect that the homotopy theory of orbi-topological spaces is equivalent to a certain category of presheaves (of anima)---a "global" generalization of a classic theorem of Elmendorf in equivariant homotopy theory. I will explain a modern reformulation of their theorem using ∞-categories. I will also try to say something about what any of this has to do with symplectic geometry!

Abstract: The systolic ratio in metric geometry compares the length of the shortest

nontrivial closed geodesic with the volume of a Riemannian manifold. On some spaces,

highly symmetric metrics are known or expected to be locally optimal for the

systolic ratio. For instance, Zoll metrics on $S^2$, whose geodesics are all closed

with the same period, are known to be locally optimal for the systolic ratio. The

dynamical nature of this statement motivates a broader (dynamical) question: do

similar fully periodic Hamiltonian systems locally maximize an analogous systolic

ratio? In the contact case, this question has a complete answer: the local

maximizers of the contact systolic ratio are precisely those contact forms whose

Reeb flow induces a free circle action on the contact manifold. Such contact forms

are called Zoll contact forms.

In this talk, I will introduce a local analogue of the contact systolic ratio for

autonomous Hamiltonian systems and show that its local maximizers, in the

$C^2$-topology, are precisely the Zoll systems: those whose Hamiltonian flow induces

a free circle action. I will also present examples of Zoll Hamiltonian systems that

are not of contact type. Time permitting, I will discuss applications of this local

systolic ratio to twisted geodesic flows on Kähler manifolds of constant holomorphic

sectional curvature.

Abstract: It is known that partially wrapped Fukaya categories of marked surfaces are closely related to derived categories of gentle algebras. We discuss how this relation changes when the surface has nodal singularities. The main algebraic objects are pinched gentle algebras, which arise from localizing graded gentle algebras at certain spherical band objects. We explain how one can associate a minimal A_\infty-category to a nodal surface, giving a minimal model for the formal localization of the topological Fukaya category. We will also discuss some examples coming from simple pinching operations on marked surfaces. This is based on a joint work with Severin Barmeier, Pierre Bodin, and Sibylle Schroll.

Abstract: Fixed point Floer cohomology is a Floer theoretic invariant associated to symplectomorphisms of a symplectic manifold. It has a "product" structure which we can assemble into a ring. In previous joint work with Maxim Jeffs and Ziwen Zhao we fully computed this ring for Dehn twists on surfaces, and the result verified predictions from mirror symmetry. Furthermore, we constructed a "quantum cohomology" for singular symplectic surfaces using this ring that also matched expectations from mirror symmetry. I will survey this body of work and discuss some work in preparation towards computing this ring structure in higher dimensional Dehn twists - for these computations we shall see some interplay between string topology and Morse homology.

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