We prove that all maximal-tb positive Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m), (3,3), (3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group PSL(2,Z) and the mapping class group M0,4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

An almost K\"ahler structure on a symplectic manifold (N,ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form ω satisfies ω(⋅,⋅)=g(J(⋅),⋅) . Any symplectic manifold admits an almost K\"ahler structure and we refer to (N,ω,g,J) as an almost K\"ahler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost K\"ahler manifolds. A metric and complex connection $\hn$ on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N . We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection $\hn$ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in K\"ahler-Einstein manifolds where the connection $\hn$ is the Levi-Civita connection of g . Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost K\"ahler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt \cite{b} in K\"ahler manifolds that are almost Einstein.

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.

Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.

In this paper we start the classification of strongly bracket-generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.