A BRIEF REFLECTION ON WORK OF PROF. S.MINAKSHISUNDARAM

By Prof. Floyd L.Williams.

Some fifty five years have passed now since the appearance of some very fundamental work of Prof.S.Minakshisundaram (SMS) on Eigenvalues of Laplace operators on Riemannian manifolds. One can obtain a quick comprehension of the level of importance of this work by noting that it has captured the attention and considerable interest of such world-class mathematicians as Weyl, Seeley, Mckean, Singer, Patodi, Bott, Ativah, Milnor and Gilkey- and of course many other. Moreover, generalizations of initial ideas of SMS now find common -place applications in various areas of theoretical physics- areas ranging from quantum field theory and quantum gravity to Bose-Einstein condensation, and string and M-theory in cosmology.

My personal interests have centered around explicating meromorphic continuations of SMS spectral zeta functions in the particular case when the underlying Riemannain manifold is a rank one symmetric space. This is possible in this nice setting because of the availability of certain trace formulas, and it also has important implications for physics (regarding the regularization of Casimir energies , for example, or the calculation of various anomalies.) I have also been influenced by work of SMS in studies where I consider a discrete group of isometries acting on hyperbolic space where the fundamental domain for that action is non finite. One needs to consider such infinite volume domains to study 3-dimensional black holes, for example.

Two particular measures of the value of one’s work and of its fundamental essence are :

1. the off-spring effect it has towards the production of many other works based on it, and
2. Its (often unexpected ) relevance for many other areas-many perhaps outside its initially envisioned sphere.By these later, the value of the work of SMS is outstanding. It is a gem and a treasure. Although he was taken from the earth too soon, his influence will continue to abide for quite many years to come.

By
Prof.Floyd.L.Williams.
Department of Mathematics
University of Massachusetts.
Amherst-MA,USA.