Related papers: A construction of symplectic connections through r…
We construct Grassmann spaces associated with the incidence geometry of regular and tangential subspaces of a symplectic copolar space, show that the underlying metric projective space can be recovered in terms of the corresponding…
We present a new approach for constructing covariant symplectic structures for geometrical theories, based on the concept of adjoint operators. Such geometric structures emerge by direct exterior derivation of underlying symplectic…
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower…
Bielavsky introduced and investigated the class of symmetric symplectic spaces, that is, symmetric spaces endowed with a symplectic form invariant with respect to symmetries. Since the theory of symmetric spaces has generalizations, we ask…
These are (heavily revised) notes from lectures given at the AMS Algebraic Geometry meeting in Seattle, 2005. The main topic is symplectic homology seen from the point of view of Lefschetz fibrations. Most of the content is speculative, but…
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between…
In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure.…
We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces…
These lecture notes explain the construction and basic properties of the wonderful compactification of a complex semisimple group of adjoint type. An appendix discusses the more general case of a semisimple symmetric space.
We obtain the symplectic group as an amalgam of low rank subgroups akin to Levi components. We do this by having the group act flag-transitively on a new type of geometry and applying Tits' lemma. This provides a new way of recognizing the…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be…
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong's canonical coordinate form of such connections is simplified.
We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the…
In this short note we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about next steps.
There are studied in details 5-dimensional pseudo-Riemannian manifolds equipped with the structure analogous to the almost cosymplectic (almost coKaehler) structure. The curvature by assumption commutes with the structure affinor and all…
We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary…
Take a compact Sasakian threefold $M$ and consider the associated irreducible $\text{SL}(r,{\mathbb C})$-character variety ${\mathcal R} := \text{Hom}(\pi_1(M, x_0), \text{SL}(r, {\mathbb C}))^{ir}/ \text{SL}(r, {\mathbb C})$ of $M$, where…
In this paper we calculate the curvature of the Hitchin connection. We further show that a slight (possibly trivial) modification of the Hitchin connection has curvature equal to an explict given multiple of the Weil-Petersen symplectic…