Heat Kernel and Quantum Gravity, (Springer,  2000)

Amazon This book tackles quantum gravity via the so-called background field method and its effective action functional. The author presents an explicitly covariant and effective technique to calculate the De Witt coefficients and to analyze the Schwinger-De Witt asymptotic expansion of the effective action. He also investigates the ultraviolet behaviour of higher-derivative quantum gravity. The book addresses theoretical physicists, graduate students as well as researchers, but should also be of interest to physicists working in mathematical or elementary particle physics.

"This monograph rightly belongs to a series ‘Lecture notes in Physics’, as it represents a well-written review of main results by the author, who is a recognized expert on heat kernel techniques in quantum gravity. [...] The results exposed in this book reflect the major contributions of the author to differential geometry and the theory of differential operators. They have many applications in quantum field theory with background fields, and indeed, the book can be used as a text for a short graduate course in the heat kernel techniques and their quantum gravity." (Mathematical Reviews 2003a)

"Spectral theory for the heat equation represents one of the more exciting points of interaction between math and physics: It also serves as a deep link, via spectral theory, between geometry(math), and quantum gravity(physics). While the subject has roots far back, this lovely book presents some of the more exciting developments in the past decade. One of the success stories in interdisciplinary theoretical science! It is well written, and will be a great source for grad students. This very nice book further points toward the research trends of the future. Moreover, the results presented in the book are timeless. The book will be of value also ten years from now. Being an acknowledged authority in the subject, this author is in a unique position to write a book on the central themes and theories in the subject". (Palle Jorgensen,

Heat Kernel Method and Its Applications, (Birkhaeser, 2015)

Amazon 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 will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.  

“The main theme of the monograph is the development of a short-time expansion for the heat kernel. … The monograph is well organized; each chapter starts with an abstract and ends with a section of notes, that can be effectively used to navigate through the contents. Thanks to an extensive presentation of background material, the book is well suited for undergraduate and graduate students, but can be of interest also for applied mathematicians and physicists.” (Paolo Musolino, zbMATH 1342.35001, 2016)

Ivan Avramidi