On the p-adic Langlands program

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Number theory is the study of whole numbers and the solvability of equations in whole numbers. It is one of the oldest branches of mathematics. One particularly famous problem is Fermat's Last Theorem from 1637, which says that the sum of two n-th powers of non-zero whole numbers cannot be the n-th power of a non-zero whole number, as soon as n is at least 3. It was only solved around 1994 by Wiles and Taylor who established a deep connection between elliptic curves (objects of geometry) and modular forms (objects of analysis and the theory of symmetries). This deep connection is a special instance of the Langlands Program, which consists of a number of very general and interlinked conjectures that, in particular, help to explain many phenomena in number theory.******My work concerns a p-adic generalisation of the Langlands program which has been attracting a lot of interest in recent years. So far, it has only been properly understood in the simplest possible case (dimension 2). This alone has led to the solution of problems in number theory that previously seemed out of reach. In the proposed work I aim to shed light on the n-dimensional case of the p-adic Langlands program for n > 2. In particular, I propose to study the global candidate representations that arise in the cohomology of Shimura varieties, as well as their locally analytic vectors.