The boundary of the unit disc in the plane may be identified with a family of positive harmonic functions: namely the Poisson kernels

The hyperbolic disc is the simplest example of a symmetric space of non-compact type. For such spaces, jointly with Guivarc'h (Université de Rennes) and Ji (university of Michigan), I have recently completed the study of the associated Martin compactifications that is to published as a research monograph with Birkhäuser. These compactifications are intimately related to particular cases of compactifications studied independently by Satake and Furstenberg in the 1960's. The main thrust of our current research is to understand the geometry of these compactifications and also the so-called Karpelevic compactification in terms of a type of compactification of euclidean space that has a combinatorial aspect.

This involves the polyhedral compactification, where for example this compactification for the decomposition of three dimensional Euclidean space induced by the eight octants is obtained by adding the boundary of a cube at infinity.

The discrete analogue of symmetric spaces are affine buildings and it is proposed to study the corresponding questions by using natural random walks. In the case of trees this has already been done, but the higher rank ones (for example produced by gluing together triangles rather than unit intervals) remain to be examined.

In addition, these questions lead into the corresponding questions for locally symmetric spaces where much remains to be clarified.

My research interests also include questions of stochastic calculus related to limiting behaviour.

- ( with Y. Guivarc'h and L.Ji)
*Compactifications of symmetric spaces*CRAS Paris 317(1993), 1103-1108. *Compactifications defined by a polyhedral cone decomposition of R^n*in Harmonic Analysis and Discrete Potential Theory, Proceedings of a conference at Frascati (Rome) 1991, edited by M.Picardello, Plenum Press, New York and London, 1992, pp 1-14.*Brownian motion on a symmetric space of non-compact type: asymptotic behaviour in polar coordinates*Canadian Journal of Mathematics 43(1991),1-21*The Martin compactification of a symmetric space of non-compact type at the bottom of the positive spectrum: an introduction*, in Potential Theory, Proceedings of the International Conference on Potential Theory, Nagoya 1990, edited by Masanori Kishi, Walter De Gruyter, Berlin New York 1992, pp 127-139.- (with A. Ancona) Some remarks on Widder's theorem and the uniqueness of isolated singularities for parabolic equations}in Partial Differential Equations with Minimal Smoothness and Applications, IMA volumes in Mathematics and its Applications, vol \# 42, edited by B.Dahlberg et al, Springer-Verlag, New York 1992, pp 15-23.

November 9, 1995