Theta functions are one of the most important tools for the construction and study of modular forms, with numerous applications in number theory (representations of integers by quadratic forms), algebraic geometry (theory of Abelian varieties), coding theory and more. The first part of this year's lecture focused on the classical theory of holomorphic theta functions associated with positive definite quadratic forms, while the second part, which will be continued in the 2005-2006 lecture, was devoted to the study of theta functions associated with indefinite quadratic forms, a much less developed field of investigation but potentially just as rich in applications.
Lecture