Arakelov theory and estimates of automorphic forms.

Abstract : In this talk, we describe my research accomplishments, and the direction of my future research. The main focus of my research is Analytic aspects of Arakelov theory and estimates of cusp forms. In my PhD thesis, we have derived optimal estimates of Arakelovian invariants like certain canonical Green's function and the hyperbolic Green function. We explain the significance of these estimates in arithmetic geometry and number theory. Since then, we have employed heat kernel techniques to study estimates of cusp forms. We describe our geometric approach from theory of heat kernels to derive to estimates of cusp forms.