Martin Westerholt-Raum: Modular forms and generating series

This talk will be split into two parts, the first of which is dedicated to explaining the basic cocept of modular forms at a level suitable for PhD students. In the second part, we given an overview of some recent developments in the field, including applications to algebraic geometry, arithmetic geometry, and combinatorics.

Modular forms were thought up as a formal concept much more than 100 years ago. The proof of Fermat's Last Theorem was the field's greatest achievement, but many more success stories can be told. But what is a modular form? It can be thought of as holomorphic function that is "inordinately symmetric". The consequences of this symmetry penetrate the theory.

Modern development in the field comprise the study of more general symmetries, of modular forms that are not holomorphic but satisfy specific differential equations inspired by geometry, and modular forms with allowed singularities. Each of these extensions of the concept "modular form" has applications: General symmetries can be connected to the study of cycles in algebraic geometry.  When dropping the holomorphicity assumption, one can tackle arithmetic geometry. Allowing for singularities makes it possible to treat additional problems in combinatorics. We sketch these three applications and give details on the now established Modularity Conjecture by Kudla, which addresses rational equivalences of cycles on Shirmura varities.