Gel'fand and Cetlin [I. Gel'fand and M. Tsetlin, Finite-dimensional representations of the group of orthogonal matrices, Dokl. Akad. Nauk SSSR 17 (1950), 1017--1020; I. Gel'fand and M. Tsetlin, Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR 71 (1950), 825--828.] constructed in the 1950s a canonical basis for a finite-dimensional representation $V(\lambda)$ of $U(n,\C)$ by successive decompositions of the representation by a chain of subgroups. Guillemin and Sternberg constructed in the 1980s the Gel'fand--Cetlin integrable system on the coadjoint orbits of $U(n,\C)$, which is the symplectic-geometric version, via geometric quantization, of the Gel'fand-Cetlin construction. (Much the same construction works for representations of $SO(n,\R)$.) Molev [A. Molev, A basis for representations of symplectic Lie algebras, Comm. Math. Phys. 201(3) (1999), 591--618.] in 1999 found a Gel'fand--Cetlin-type basis for representations of the symplectic group, using essentially new ideas. An important new role is played by the Yangian $Y(2)$, an infinite-dimensional Hopf algebra, and a subalgebra of $Y(2)$ called the twisted Yangian $Y^{-}(2)$. In this paper, we use deformation theory to give the analogous symplectic-geometric results for the case of $U(n,\H)$, i.e., we construct a completely integrable system on the coadjoint orbits of $U(n,\H)$. We call this the Gel'fand--Cetlin--Molev integrable system.

We study generalized complex (GC) manifolds from the point of view of symplectic and Poisson geometry. We start by recalling that every GC manifold admits a canonical Poisson structure. We use this fact, together with Weinstein's classical result on the local normal form of Poisson, to prove a local structure theorem for GC, complex manifolds, which extends the result Gualtieri has obtained in the "regular'' case. Finally, we begin a study of the local structure of a GC manifold in a neighborhood of a point where the associated Poisson tensor vanishes. In particular, we show that in such a neighborhood, a "first-order approximation'' to the GC structure is encoded in the data of a constant B-field and a complex Lie algebra.

In this paper, we construct a family of symplectic 4-manifolds with positive signature for any given fundamental group G that approaches the BMY line. The family is used to show that one cannot hope to do better than the BMY inequality in finding a lower bound for the function $f=\chi+b\sigma$ on the class of all minimal symplectic 4-manifolds with a given fundamental group.

We introduce an analogue of the inflation technique of Lalonde--McDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic 4-manifolds. We consider the implications of this construction for the symplectic cones of Käahler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudo-holomorphic curves via a perturbation.

In this paper, we study contact structures on any open $3$-manifold $V$ that is the interior of a compact $3$-manifold. To do this, we introduce new proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for $T^2 \times [0, \infty)$, $T^2 \times \R$, and $S^1 \times \R^2$ using these concepts. The only other classification result on an open $3$-manifold is Eliashberg's classification on $\R^3.$ Our investigation uncovers a new phenomenon in contact geometry: There are infinitely many tight contact structures on $T^2 \times [0,1)$ that cannot be extended to a tight contact structure on $T^2 \times [0, \infty)$. Similar results hold for $T^2 \times \R$ and $S^1 \times \R^2$. Finally, we show that if every $S^2 \subset V$ bounds a ball or an $S^2$ end, then there are uncountably many tight contact structures on $V$ that are not contactomorphic, yet are isotopic. Similarly, there are uncountably many overtwisted contact structures on $V$ that are not contactomorphic, yet are isotopic. These uncountability results generalize work by Eliashberg when $V = S^1 \times \R^2$.