This book is aimed at theoretical as well as primarily physicists graduate students in field working quantum theory, quantum gravity, theories, gauge to sdme and and, it is not extent, general relativity cosmology. Although aimed at a I that it also be of level, hope in mathematically rigorous may terest to mathematical and mathematicians in physicists working spectral of differential mani geometry, spectral asymptotics on operators, analysis differential and mathematical methods in folds, geometry quantum theory. Thisbook will be considered too abstract some but certainly by physicists, not...

The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented.

The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that...