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Automorphic q-series and their application

Objective

This proposal aims to unravel mysteries at the frontier of number theory and other areas of mathematics and physics. The main focus will be to understand and exploit “modularity” of q-hypergeometric series. “Modular forms are functions on the complex plane that are inordinately symmetric.” (Mazur) The motivation comes from the wide-reaching applications of modularity in combinatorics, percolation, Lie theory, and physics (black holes).

The interplay between automorphic forms, q-series, and other areas of mathematics and physics is often two-sided. On the one hand, the other areas provide interesting examples of automorphic objects and predict their behavior. Sometimes these even motivate new classes of automorphic objects which have not been previously studied. On the other hand, knowing that certain generating functions are modular gives one access to deep theoretical tools to prove results in other areas. “Mathematics is a language, and we need that language to understand the physics of our universe.”(Ooguri) Understanding this interplay has attracted attention of researchers from a variety of areas. However, proofs of modularity of q-hypergeometric series currently fall far short of a comprehensive theory to describe the interplay between them and automorphic forms. A recent conjecture of W. Nahm relates the modularity of such series to K-theory. In this proposal I aim to fill this gap and provide a better understanding of this interplay by building a general structural framework enveloping these q-series. For this I will employ new kinds of automorphic objects and embed the functions of interest into bigger families

A successful outcome of the proposed research will open further horizons and also answer open questions, even those in other areas which were not addressed in this proposal; for example the new theory could be applied to better understand Donaldson invariants.

Field of science

  • /natural sciences/mathematics/pure mathematics/arithmetic
  • /natural sciences/mathematics
  • /humanities/languages and literature/languages - general
  • /natural sciences/physical sciences/astronomy/astrophysics/black hole

Call for proposal

ERC-2013-StG
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

UNIVERSITAET ZU KOELN
Address
Albertus Magnus Platz
50931 Koeln
Germany
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 240 500
Principal investigator
Kathrin Bringmann (Prof.)
Administrative Contact
Caren Wiegand (Mrs.)

Beneficiaries (1)

UNIVERSITAET ZU KOELN
Germany
EU contribution
€ 1 240 500
Address
Albertus Magnus Platz
50931 Koeln
Activity type
Higher or Secondary Education Establishments
Principal investigator
Kathrin Bringmann (Prof.)
Administrative Contact
Caren Wiegand (Mrs.)