We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real analytic curves to real analytic curves. Under mild completeness conditions the second requirement can be replaced by: real analytic along affine lines. Enclosed and necessary is a careful study of locally convex topologies on spaces of real analytic mappings. As an application we also present the theory of manifolds of real analytic mappings: the group of real analytic diffeomorphisms of a compact real analytic manifold is a real analytic Lie group.

We give a review of the extended conformal algebras, known as $W$ algebras, which contain currents of spins higher than 2 in addition to the energy-momentum tensor. These include the non-linear $W_N$ algebras; the linear $W_\infty$ and $W_{1+\infty}$ algebras; and their super-extensions. We discuss their applications to the construction of $W$-gravity and $W$-string theories.

We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1. Intersection numbers 2. The Kontsevich integral 2.1. The main theorem 2.2 Expansion of Z on characters and Schur functions 2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4. Matrix Airy equation and Virasoro highest weight conditions 5. Genus expansion 6. Singular behaviour and Painlev'e equation. 7. Generalization to higher degree potentials

The space of all non degenerate bilinear structures on a manifold $M$ carries a one parameter family of pseudo Riemannian metrics. We determine the geodesic equation, covariant derivative, curvature, and we solve the geodesic equation explicitly. Each space of pseudo Riemannian metrics with fixed signature is a geodesically closed submanifold. The space of non degenerate 2-forms is also a geodesically closed submanifold. Then we show that, if we fix a distribution on $M$, the space of all Riemannia metrics splits as the product of three spaces which are everywhere mutually orthogonal, for the usual metric. We investigate this situation in detail.

We construct new coordinates for the Teichm\"uller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_\mu$ that depend holomorphically on a parameter $\mu$ varying in a simply connected domain in $\bold{C}$. They describe the geometry of the hyperbolic manifold $\bold{H}^3/G_\mu$; they reflect exactly the visual patterns one sees in the limit sets of the groups $G_\mu$; and they are directly computable from the generators of $G_\mu$.

To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much investigation, but this activity has not been exposed much to the wider mathematical community. This article attempts to fill this gap and explain various aspects of the recent work on generalized trees. The subject is very appealing for it mixes very na\"{\i}ve geometric considerations with the very sophisticated geometric and algebraic structures. In fact, part of the drama of the subject is guessing what type of techniques will be appropriate for a given investigation: Will it be direct and simple notions related to schematic drawings of trees or will it be notions from the deepest parts of algebraic group theory, ergodic theory, or commutative algebra which must be brought to bear? Part of the beauty of the subject is that the na\"{\i}ve tree considerations have an impact on these more sophisticated topics and that in addition, trees form a bridge between these disparate subjects.

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the $u^5$-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.

Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular points of energy minimizing maps. Unfortunately these are not unique, even for generic boundary conditions. Examples are discussed which have isolated singularities with a continuum of distinct tangent maps.

The authors announce the following theorem. Theorem 1. If $G=A*_H B$ is an amalgamated product where $A$ and $B$ are finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity. If $G=A*_H$ is an HNN-extension where $A$ is finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity.

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of "traceless" vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed.