Research Projects

The classical question of hearing the shape of a manifold without hearing its geometry can be answered mathematically by knowing the eigenvalues (noise frequencies) of the Laplace operator. This has coined the first question in inverse spectral theory, namely whether the spectrum determines a Riemannian manifold up to isometry. Spectra on functions determine the dimension, volume, and scalar curvature of M. However, the inverse spectrum problem is known to be negative, and counter-examples have been constructed by Milor, Vigneras, and Sunada. Mathematicians aim to investigate the inverse spectral problem in the arithmetical context of compact, locally symmetric manifolds, especially those arising from co-compact congruence lattices in an algebraic group G. Sunada showed that the spectrum on the Manifold side is related to the notion of spectrum in the field of representation theory. He constructed a relation between the spectrum of Laplace operator on smooth functions on locally symmetric space ? \ G/K and the equivalence of representations. To understand the spherical spectrum, proving the strong multiplicity one theorem for the spherical spectrum will lead to multiplicity one theorem in the context of compact, locally symmetric manifolds, especially those arising from cocompact congruence lattices in an algebraic group G. Rajan-Chandrasheels proved few spectral analogues of the strong multiplicity one theorem. Bhagwat and I proved an analogue of the strong multiplicity one theorem in the context of gama -spherical representations of the group G = SO(2, 1)0.