A common theme studied in number theory are congruences between integers modulo prime numbers or modulo powers of prime numbers. A way to encode all those congruences at once is provided by a field that is called “the field of p-adic numbers”. Out of this field one can build interesting groups, called p-adic groups, which are number theoretic analogues of Lie groups, have a similar rich structure, and which play a central role in the Langlands program, for example. A key question that mathematicians ask is how one can represent these complicated-looking p-adic groups using more common complex matrix groups, in other words, using more traditional linear algebra. In this talk, I will introduce p-adic numbers and p-adic groups and then provide an overview of what we know about the representations of these groups including recent developments. This means I will explain how close we are to answering the key question above. I might also sketch applications to other questions in mathematics.

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