Mathematics

We prove that ( RP 2n−1 , ξ std ) is not exactly fillable for any n≠ 2 k and there exist strongly fillable but not exactly fillable contact manifolds for all dimension ≥5 .

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we only assume to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is " 2+2+2 " consisting of three independent ordered pairs, with the involution exchanging the members of each pair. A key feature of the classification is that any indecomposable (co)isotropic triple is either "split" or "non-split". The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the "split" case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a "symplectification". In the course of the paper we develop the framework of "symplectic poset representations", which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples. The framework lends itself equally well to studying poset representations on spaces carrying a non-degenerate symmetric bilinear form; we mainly keep our focus, however, on the symplectic side.

We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). Here the group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of the more general classification (arXiv:1612.03843) of multiplicity free manifolds in the special case of rank one. As a result we obtain numerous new concrete examples of multiplicity free quasi-Hamiltonian manifolds or, equivalently, Hamiltonian loop group actions.

Let X??R 4 be a convex domain with smooth boundary Y . We use a relation between the extrinsic curvature of Y and the Ruelle invariant Ru(Y) of the natural Reeb flow on Y to prove that there exist constants C>c>0 independent of Y such that c< Ru(Y ) 2 vol(X) ?�sys(Y)<C Here sys(Y) is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of Y divided by twice the volume of X . We then construct dynamically convex contact forms on S 3 that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomão. These are the first examples of dynamically convex contact 3 -spheres that are not strictly contactomorphic to a convex boundary Y .

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal S^1-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry. We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result without referring to additional structures. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner--Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

Motivated by the Atiyah-Floer conjecture, we consider SO(3) Santi-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov-Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a type of singular objects which may have both instanton and holomorphic curve components. This result is the first step towards constructing a natural bounding cochain proposed by Fukaya for the SO(3) Atiyah-Floer conjecture.

Hofer's metric is a bi-invariant metric on Hamiltonian diffeomorphism groups. Our main result shows that the topology induced from Hofer's metric is weaker than C^1-topology if the symplectic manifold is closed.

We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional f:X→ A 1 k , the derived critical locus has a natural Lagrangian fibration Crit(f)→X . In the case where f is non-degenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.

The 2016 papers of J. Solomon and S. Tukachinsky use bounding chains in Fukaya's A ∞ -algebras to define numerical disk counts relative to a Lagrangian under certain regularity assumptions on the moduli spaces of disks. We present a (self-contained) direct geometric analogue of their construction under weaker topological assumptions, extend it over arbitrary rings in the process, and sketch an extension without any assumptions over rings containing the rationals. This implements the intuitive suggestion represented by their drawing and P. Georgieva's perspective. We also note a curious relation for the standard Gromov-Witten invariants readily deducible from their work. In a sequel, we use the geometric perspective of this paper to relate Solomon-Tukachinsky's invariants to Welschinger's open invariants of symplectic sixfolds, confirming their belief and G. Tian's related expectation concerning K. Fukaya's earlier construction.

We use sheaves of spectra to quantize a Hamiltonian ∐ n BO(n) -action on lim − → N T ∗ R N that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment of stratified Morse theory by the J -homomorphism. This provides a key step in the following work [Jin] on the proof of a claim in [JiTr]: the classifying map of the local system of brane structures on an (immersed) exact Lagrangian submanifold L⊂ T ∗ R N is given by the composition of the stable Gauss map L→U/O and the delooping of the J -homomorphism U/O→BPic(S) . We put special emphasis on the functoriality and (symmetric) monoidal structures of the categories involved, and as a byproduct, we produce several concrete constructions of (commutative) algebra/module objects and (right-lax) morphisms between them in the (symmetric) monoidal (∞,2) -category of correspondences, generalizing the construction out of Segal objects in [GaRo], which might be of interest by its own.