Mathematics

In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold T ϕ associated to a Weinstein domain M , and an exact, compactly supported symplectomorphism ϕ . T ϕ is another Weinstein domain and its contact boundary is independent of ϕ . In this paper, we distinguish T ϕ from T 1 M , under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and symplectic cohomology, but that are different as Liouville domains. Previously, we have suggested a categorical model M ϕ for the wrapped Fukaya category W( T ϕ ) , and we have distinguished M ϕ from the mapping torus category of the identity. In this paper, we prove W( T ϕ ) and M ϕ are derived equivalent (Theorem 1.9); hence, deducing the promised Theorem 1.1. Theorem 1.9 is of independent interest as it preludes an algebraic description of wrapped Fukaya categories of locally trivial symplectic fibrations as twisted tensor products.

We look at methods to select triples (M,ω,J) consisting of a symplectic manifold (M,ω) endowed with a compatible positive almost complex structure J , in terms of the Nijenhuis tensor N J associated to J . We study in particular the image distribution $\Image N^J$.

We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable (−1,−1) -curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.

I describe some of McDuff's contributions to symplectic geometry, with a focus on symplectic embedding problems.

The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension 2n+1 any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least n+1 simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examples of antipodally symmetric dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has non-empty C 1 -interior.

ECH capacities were developed by Hutchings to study embedding problems for symplectic 4 -manifolds with boundary. They have found especial success in the case of certain toric symplectic manifolds where many of the computations resemble calculations found in cohomology of Q -line bundles on toric varieties, or in lattice point counts for rational polytopes. We formalise this observation in the case of convex toric lattice domains X Ω by constructing a natural polarised toric variety ( Y Σ(Ω) , D Ω ) containing the all the information of the ECH capacities of X Ω in purely algebro-geometric terms. Applying the Ehrhart theory of the polytopes involved in this construction gives some new results in the combinatorialisation and asymptotics of ECH capacities for convex toric domains.

Let (Y,A) be a smooth rational surface or a possibly singular toric surface with ample divisor A . We show that a family of ECH-based, algebro-geometric invariants c alg k (Y,A) proposed by Wormleighton obstruct symplectic embeddings into Y . Precisely, if (X, ω X ) is a 4 -dimensional star-shaped domain and ω Y is a symplectic form Poincaré dual to [A] then (X, ω X ) embeds into (Y, ω Y ) symplectically ⟹ c ECH k (X, ω X )≤ c alg k (Y,A) We give three applications to toric embedding problems: (1) these obstructions are sharp for embeddings of concave toric domains into toric surfaces; (2) the Gromov width and several generalizations are monotonic with respect to inclusion of moment polygons of smooth (and many singular) toric surfaces; and (3) the Gromov width of such a toric surface is bounded by the lattice width of its moment polygon, addressing a conjecture of Averkov--Hofscheier--Nill.

The group Ham(M,ω) of all Hamiltonian diffeomorphisms of a symplectic manifold (M,ω) plays a central role in symplectic geometry. This group is endowed with the Hofer metric. In this paper we study two aspects of the geometry of Ham(M,ω) , in the case where M is a closed surface of genus 2 or 3. First, we prove that there exist diffeomorphisms in Ham(M,ω) arbitrarily far from being a k -th power, with respect to the metric, for any k≥2 . This part generalizes previous work by Polterovich and Shelukhin. Second, we show that the free group on two generators embeds into the asymptotic cone of Ham(M,ω) . This part extends previous work by Alvarez-Gavela et al. Both extensions are based on two results from geometric group theory regarding incompressibility of surface embeddings.

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism. We fill in some technical details, including the Morse-Bott direct limit argument and some writhe bounds. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings--Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang. In a subsequent paper, we plan to extend these methods for Seifert fiber spaces and relate the ECH U-map to orbifold Gromov-Witten invariants of the base.

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.