Heat
Kernel and Quantum Gravity, (Springer,
2000)

This book tackles quantum gravity via the
socalled 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 SchwingerDe
Witt asymptotic expansion of the effective action.
He also investigates the ultraviolet behaviour of
higherderivative 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 wellwritten
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, Amazon.com) 


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


The heart of the book is the development of a
shorttime 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 shorttime 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) 

