Serre-type conjectures and mod p Langlands correspondences

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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, which concerns the insolvability in the integers of certain simple equations. It was only solved some 15 years ago, by establishing 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 some aspects of the Langlands program that link modular forms (and their generalisations) to Galois representations. In the last decade there has been a lot of interest in the so-called p-adic Langlands program. So far, it has only been properly understood in the simplest possible case. This alone has led to the solution of problems in number theory that previously seemed out of reach. In my work I aim to shed light on the "next" case of the p-adic Langlands program.