# London Algebra Colloquium abstract

## Multiplicity in difference geometry

### Konstantin Ardakov (Queen Mary), 6th December 2012

#### Abstract

Classical commutative rigid analytic spaces were introduced by John Tate in the 1960s
with number-theoretic applications in mind. They are defined in terms of a particular
*p*-adic completion of the polynomial algebra, called the Tate algebra. When we
replace the polynomial algebra by the Weyl algebra and perform the same construction,
we obtain the so-called Tate–Weyl algebra. Using these rings, we build a sheaf *D^*
of non-commutative rings on any smooth rigid analytic space *X* that can be viewed
as a quantisation of the cotangent bundle of *X*. If time permits, I’ll
explain the connection between *D^*-modules and the *p*-adic representation
theory of compact *p*-adic Lie groups.

This is joint work with Simon Wadsley.