This book presents the relationship between classical theta
functions and knots. It is based on a novel idea of Razvan Gelca
and Alejandro Uribe, which converts Weil's representation of the
Heisenberg group on theta functions to a knot theoretical
framework, by giving a topological interpretation to a certain
induced representation. It also explains how the discrete Fourier
transform can be related to 3- and 4-dimensional topology.Theta
Functions and Knots can be read in two perspectives. Readers with
an interest in theta functions or knot theory can learn how the two
are related. Those interested in Chern-Simons theory will find here
an introduction using the simplest case, that of abelian
Chern-Simons theory. Moreover, the construction of abelian
Chern-Simons theory is based entirely on quantum mechanics and not
on quantum field theory as it is usually done.Both the theory of
theta functions and low dimensional topology are presented in
detail, in order to underline how deep the connection between these
two fundamental mathematical subjects is. Hence the book is
self-contained with a unified presentation. It is suitable for an
advanced graduate course, as well as for self-study.
General
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