Geometrisation of p-adic representations of p-adic Lie groups

The proposed research is in Pure Mathematics, more precisely, in an area of intersection of Algebraic Number Theory and Representation Theory. One of the main goals of the first subject is to solve Diophantine problems that ask for whole number solutions to polynomial equations in several variables. A very famous example of such a Diophantine problem is due to the French mathematician Fermat, and asks whether it is possible for the sum of two integer n-th powers to be again the n-th power of an integer. When the parameter n is equal to two, it has been known since ancient times that there are infinitely many solutions to this problem, such as 9 + 16 = 25 and 25 + 144 = 169. Fermat guessed way back in the seventeenth century that his problem has no interesting solutions whatsoever for any higher value of n, but a rigorous proof of this guess, the so-called Fermat's Last Theorem, was only obtained around 20 years ago by the British mathematician Andrew Wiles. The second subject, Representation Theory, tries to determine all the possible ways in which symmetries can occur in nature, and has numerous applications in several neighbouring academic disciplines such as Crystallography and Theoretical Physics. A typical problem in Representation Theory asks for the classification of the basic building blocks, or atoms, of the theory --- known as the irreducible representations. To mention an example: according to the Standard Model of Theoretical Physics, nearly everything around us in the visible universe is composed of certain elementary particles called Quarks. The classification and behaviour of these quarks is in turn explained by the classification and behaviour of the irreducible representations of the three-dimensional special unitary group SU(3). The proposed research will develop new geometric tools and methods in order to classify the irreducible representations in a particular sub-field of Representation Theory. This classification project would then be used to build a conceptual bridge called the p-adic local Langlands correspondence between Algebraic Number Theory and Representation Theory, which would in turn be used to make progress in both areas by transferring information from one to the other.

This research proposal is concerned with very theoretical aspects of pure mathematics. As is very often the case in this pure mathematics, a significant part of the total impact of this research will primarily be of academic type. This research proposal is intradisciplinary, being situated on the interface of at least two branches of pure mathematics: number theory and non-commutative algebra. Consequently there are at least two distinct pathways to real societal impact many years later down the line: the path starting with number theory may lead to applications in cryptography and financial security, whereas the path through non-commutative algebra could find lead to applications in high-energy physics, and eventually to advances in 21st century technology. Academics working in neighbouring areas in both number theory and non-commutative algebra will benefit from the introduction of new methods and the development of new results. The fundamental nature of this research will contribute to a deeper understanding of appropriate parts of both disciplines, and could conceivably open up totally new areas of research for exploration. In addition to the impact this research will have to global knowledge, it will also have significant local impact to people in the Mathematical Institute, for example through the training of highly skilled PhD students and post-docs who are keenly in demand in the modern economy.