SPEAKER: Professor Alexandru Buium, University of New Mexico

Title:    Arithmetic differential equations

Abstract:
One can develop an arithmetic analogue of the theory of (ordinary/partial)  differential equations. In the (ordinary) arithmetic theory the ``independent variable'' t is replaced by a fixed prime integer p.
Smooth real functions, x(t), are replaced by integer numbers, a,  or, more generally, by integers in various (completions of) number fields. The derivative operator on functions x -> dx/dt
is  replaced by a ``Fermat quotient operator''  which, on integer numbers, acts as  a -> (a-a^p)/p. One can apply this theory to prove results in diophantine geometry.