Symplectic Geometry

In this note, we give item-by-item responses to the criticisms raised in [TZ] by Tehrani ad Zinger on our paper [LR]. We illuminate the main ideas and contributions in [LR] in section 2, itemize the responses to issues raised in [TZ] and…

We prove that if the unit codisc bundle of a closed Riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest nontrivial closed geodesic is at most half the area of the unit disc.

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy…

We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we…

We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated…

In this paper we prove that the Torelli part of the symplectomorphism groups of the $n$-point ($n\leq 4$) blow-ups of the projective plane is trivial. Consequently, we determine the symplectic mapping class group. It is generated by…

Let $\gamma$ be a non-degenerate Ustilovsky geodesic in $Ham (M, \omega)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $ \gamma$, as a critical point…

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K\"ahler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very…

This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see Lemma 3.9). In the second part, we use…

In this paper we survey some recent works that take the first steps toward establishing bilateral connections between symplectic geometry and several other fields, namely, asymptotic geometric analysis, classical convex geometry, and the…

This paper presents the theory of Bohr-Sommerfeld-Heisenberg quantization of a completely integrable Hamiltonian system in the context of geometric quantization. The theory is illustrated with several examples.

Many mechanical systems are large and complex, despite being composed of simple subsystems. In order to understand such large systems it is natural to tear the system into these subsystems. Conversely we must understand how to invert this…

Let $X$ be the unit circle bundle of a positive line bundle on a Hodge manifold. We study the local scaling asymptotics of the smoothed spectral projectors associated to a first order elliptic T\"{o}plitz operator $T$ on $X$, possibly in…

In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $\rho$ such that the open ball of radius $\rho$ in the Hofer metric centered at the identity on…

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with Gromov-Witten/Floer…

We use the hyperK\"aler geometry define an disc-counting invariants with deformable boundary condition on hyperK\"ahler manifolds. Unlike the reduced Gromov-Witten invariants, these invariants can have non-trivial wall-crossing phenomenon…

We show that all closed symplectic 4-manifolds have the packing stability property: there are no obstructions beyond volume to embedding a collection of sufficiently small balls. This generalizes a theorem of Biran which gives the same…

In this paper is established a topological constraint for monotone Lagrangian embeddings in certain complex hypersurfaces of integral K\"ahler manifolds. As an application, we prove that it is impossible to embed a connected sum of…

In the context of the two dimensional sigma model, we show that classical field theory naturally defines a functor from Segal's category of Riemann surfaces to the Guillemin-Sternberg/Weinstein category of canonical relations in symplectic…