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New connections in number theory and physics

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

8th August 2022 to 30th September 2022
Daniel Persson
Daniel Bump
Solomon Friedberg
Michael Green
Axel Kleinschmidt
Katrin Wendland

Programme theme

During the 20th century, interactions between geometers and theoretical physicists were revolutionary in both areas, leading to such unexpected developments as mirror symmetry and the impact of topological field theory on invariants of 4-manifolds. Another momentum is now developing around new connections between number theory and physics, mainly but not exclusively through the theory of automorphic forms. We have reached a time when contacts between number theorists and physicists have the potential to generate another surge of developments. 

The purpose of this interdisciplinary programme is to investigate connections between various fields of theoretical physics, in particular string theory and statistical models of crystals, and the theory of automorphic forms, mock modular forms and beyond. The non-perturbative structure of string theory has led to new uses of automorphic forms that have already appeared in number theory, as well as new exotic automorphic objects that have not yet been encountered.

This is an emergent research area with a lot of potential for cross-disciplinary collaborations with great benefit to both areas. Recent work has revealed, on the one hand, a profound significance of automorphic forms and automorphic representations for string scattering amplitudes and, on the other hand, an unexpected appearance of mock theta functions in answering key questions pertaining to quantum properties of black holes. In addition, solvable lattice models have recently appeared in connection with Eisenstein series on metaplectic groups, calling for some underlying structure that remains to be understood.

The program will begin with a one-week school, with the purpose of bridging the gap between physicists and mathematicians and to ignite the seeds for collaborations and discussions during the remainder of the program. Following the school, there will be on-going lecture series and research talks, focusing on developing the core themes of the program. There will also be a second workshop in the middle of the program period to highlight recent research developments and to describe potential future research directions.  This workshop will provide a foundation for the second half of the program.

The core themes of the program are: 

- String amplitudes and automorphic representations 

- Modular graph functions and multiple zeta values

- Mock modular forms, moonshine and black holes

- Metaplectic Eisenstein series and quantum groups

- Automorphic forms on Kac-Moody groups

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons