Periodic functions are extremely important mathematical objects. The study of these functions is not only one of the fundamental cornerstones of (applied) mathematics, but has also revolutionized different ares of the sciences including physics, chemistry, biology, medicine, economics etc. Most of us are acquainted with the notion of periodic functions over real or complex spaces. However, the notion of periodic functions can be generalized to more "symmetric" topological spaces too, where the periodicity inherits the symmetric structure of the topological space. Such functions are naively what are known as "automorphic forms".
The theory of automorphic forms has been vital in modern mathematics. In the past century, automorphic forms have gone from objects (in a linguistic sense) firmly in the realm of harmonic analysis to objects that bridge harmonic analysis, number theory and geometry. Automorphic forms and L-functions have played a definitive role in solving challenging problems, such as the Modularity theorem (which proves the Taniyama-Shimura-Weil conjecture, and consequentially proved Fermat's last theorem), and are central to challenging problems in mathematics such as the Langlands program.
In physics, automorphic forms have many applications too, mostly in string theory and quantum field theory. Some of these applications include values of Feynman integrals, statistical mechanics of string theoretic black holes, arithmetic quantum chaos, scattering amplitudes in string theory etc. String theory in this mathematical sense is a rich source of various symmetric spaces which are related by dualities, with each symmetric space being a source for new kinds of automorphic properties. During my stay, I plan to study the spectral properties of a few simple and duality related symmetric spaces that appear in string theory and low dimensional quantum gravity, and understand what physics can be learned from their spectral properties, and vice versa.
Duration of first stay: 14th September - 14th December 2023
Duration of second stay: 1st - 30th April 2024
|Abhiram Mamandur Kidambi
|Max Planck Institute for Mathematics in the Sciences