A representation of the Jacobi algebra by first order differential operators with polynomial coeficients on a Kähler manifold which as set is the product of the complex multidimensional plane times the Siegel ball is presented.

We study some geometrical and topological properties of the electric fields created by point charges on Riemannian manifolds. Particularly, we characterize the spaces on which the electric lines emanating from a point charge are geodesics, and describe the topological properties of the basin boundary for $N$ point charges. Several open problems will be posed.

In a preprint from 1982, John Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In this paper, we describe some of the answers (and partial answers) obtained in the preceding years.

We discuss a special case of the gauge theory to string theory correspondence where exact results are available. We show how the Witten-Reshetikhin-Turaev invariant in $SU(n)$ Chern-Simons theory on $S^3$ is related via conifold transition to the all genus generating function of the topological string amplitudes on a Calabi-Yau manifold.

In this note we present an abstract approach, based on Lie algebra cohomology, to the Lie algebra extensions associated to symplectic manifolds. We associate to any Lie algebra cocycle of degree at least two an abelian extension by some space a and central extensions of subalgebras analogous to the Lie algebras of symplectic, respectively, hamiltonian vector fields. We even obtain a Poisson bracket on a compatible with the hamiltonian Lie subalgebra. We then describe how this general approach provides a unified treatment of cocycles defined by closed differential forms on Lie algebras of vector fields on possibly infinite-dimensional manifolds.

The first Pontryagin form of a compact orientable surface $M$ determines a canonical pre-symplectic structure on the space of Riemannian metrics on $M$. The first equivariant Pontryagin form determines a canonical moment map for it. We study the corresponding symplectic reduction and we state (Theorem 6) that the symplectic quotient is the Teichmüller space of the surface with the Weil-Petersson symplectic form.

The study of Riemann surfaces with parametrized boundary components was initiated in conformal field theory (CFT). Motivated by general principles from Teichmüller theory, and applications to the construction of CFT from vertex operator algebras, we generalize the parametrizations to quasisymmetric maps. For a precise mathematical definition of CFT (in the sense of G. Segal), it is necessary that the moduli space of these Riemann surfaces be a complex manifold, and the sewing operation is holomorphic. We report on the recent proofs of these results by the authors.

It is the general impression that deformation problems are always governed by cohomology spaces. In this contribution we consider the deformation of Lie algebras. There this close connection is true for finite-dimensional algebras, but fails for infinite dimensional ones. We construct geometric families of infinite dimensional Lie algebras over the moduli space of complex one-dimensional tori with marked points. These algebras are algebras of Krichever-Novikov type which consist of meromorphic vector fields of certain type over the tori. The families are non-trivial deformations of the (infinite dimensional) Witt algebra, and the Virasoro algebra respectively, despite the fact that the cohomology space associated to the deformation problem of the Witt algebra vanishes, and hence the algebra is formally rigid. A similar construction works for current algebras. The presented results are jointly obtained with Alice Fialowski.

We consider the Taubes correspondence between solutions of Seiberg-Witten equations on a compact four-dimensional symplectic manifold and pseudo-holomorphic curves. We start from Kähler surfaces, in which case there is a direct correspondence between solutions of Seiberg-Witten equations and holomorphic curves. The general Taubes correspondence for symplectic four-manifolds involves, in contrast with the Kähler case, a limiting procedure, called the scaling limit. Under this scaling limit solutions of Seiberg-Witten equations reduce to families of solutions of certain vortex equations in the normal bundle of the limiting pseudoholomorphic curve.

In this paper we present a new notion of a smooth manifold with corners and relate it to the commonly used concept in the literature. We also introduce complex manifolds with corners show that if $M$ is a compact (respectively, complex) manifold with corners and $K$ is a smooth (respecti vely, complex) Lie group, then $C^{\infty}(M, K)$ (respectively, $\mathcal{O}(M, K)$) is a smooth (respectively, complex) Lie group.