This proposal aims to discover new geometric and analytic structures in number theory and in automorphic forms. It is of a highly intra-disciplinary nature and involves several areas of mathematics. It is based on infinite-dimensional analysis and geometry, a field lying at the front-line of modern mathematics, and extremely promising for applications ranging from number theory and quantum physics, to quantum information, polymer models and economics.
The first step in handling an infinite-dimensional object is to represent it as a limit of finite-dimensional ones, and then to study the finite-dimensional counterparts. However, the most important objects appear to have purely infinite-dimensional nature, without relevant finite-dimensional counterparts. This leads to consideration of models of the infinite-dimensional objects, depending on their various properties: algebraic, topological, analytic, etc. As an analogy one may recall the famous fable about six blind men, describing what an elephant look like. The six blind men touch six different parts: the leg, the stomach, the ear, the tail, the tusks and the snout, and each blind man gives a different description of the elephant. Something similar usually happens in the infinite-dimensional analysis: to get the whole picture of an object one has to put together different descriptions, and one cannot drop any of the existing models without losing the whole picture.
One of the key purely infinite-dimensional objects in number theory is Euler's Gamma-function, and the problem of finding its relevant model had been open for more than a century. The Gamma-function plays a crucial role in the study of analytic properties of the famous Riemann's zeta-function and in the Langlands programme. One can use Langlands correspondence to uncover hidden symmetries of infinite-dimensional objects and predict an excitingly profound and delicate relation between finite and infinite dimensional objects. The local Langlands correspondence identifies two analytic objects, given by a product of Gamma-functions (or its substitutes): the first object is constructed from a finite-dimensional object in number theory, and the other is from an infinite-dimensional object in representation theory. The finite-dimensional object is not a counterpart of the infinite-dimensional one, and a priori, they are of different nature, living in different areas of mathematics. Nevertheless there is a reason for this relationship, which is still far from being well understood.
Similar phenomena also occur in quantum physics, when a finite-dimensional object may have infinite-dimensional symmetries, and vice versa. Recently, I have discovered a very promising representation for the Gamma-function, motivated by quantum physics, which sheds new light on the Langlands programme. In particular, I have shown that the Mirror Symmetry in string theory turns out to coincide with the Archimedean Langlands correspondence. This result yields a novel geometric setting for number theoryand arithmetic geometry via infinite-dimensional symplectic geometry.
My project is a groundbreaking work linking the two most influential contemporary research streams, the Langlands programme and Mirror Symmetry, on the basis of new analytic and geometric constructions from string theory and integrable systems. The initial part of the project contains a comparison of my symplectic geometric model with various known algebraic models for the infinite geometry behind the Langlands programme. This will create numerous synergies among various fields of quantum physics and modern mathematics and will indicate potential areas for further progress. My proposal will connect apparently remote areas of Physics and Mathematics in a new way through the realisation that they are facets of the same underlying structure. Such connections between Quantum Physics and Number Theory are very promising for the development of both fields in the 21st century.