This proposal aims to discover new geometric and analytic structures in number theory and in automorphic forms. It is of a highly intradisciplinary nature and involves several areas of mathematics. It is based on infinitedimensional analysis and geometry, a field lying at the frontline 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 infinitedimensional object is to represent it as a limit of finitedimensional ones, and then to study the finitedimensional counterparts. However, the most important objects appear to have purely infinitedimensional nature, without relevant finitedimensional counterparts. This leads to consideration of models of the infinitedimensional 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 infinitedimensional 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 infinitedimensional objects in number theory is Euler's Gammafunction, and the problem of finding its relevant model had been open for more than a century. The Gammafunction plays a crucial role in the study of analytic properties of the famous Riemann's zetafunction and in the Langlands programme. One can use Langlands correspondence to uncover hidden symmetries of infinitedimensional 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 Gammafunctions (or its substitutes): the first object is constructed from a finitedimensional object in number theory, and the other is from an infinitedimensional object in representation theory. The finitedimensional object is not a counterpart of the infinitedimensional 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 finitedimensional object may have infinitedimensional symmetries, and vice versa. Recently, I have discovered a very promising representation for the Gammafunction, 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 infinitedimensional 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.
