Speaker: Hans Schouten
Title: "Categorical Computability via Fermat's Last Theorem", 

Abstract:
Given two number fields, it is not hard to check (algorithmically) 
that they are isomorphic. The problem, however, becomes much harder 
if we consider non-algebraic or infinitely generated extensions. 
To capture this phenomenon, the notion of  categorical computability (cc) 
can be used: this is a computable (=recursive) field $F$ so that if $G$ 
is another computable field which is isomorphic as a field to $F$, 
then we can actually compute an isomorphism between them. 
The field of rationals, $Q$, is clearly cc, but imagine already 
the obstacles that arise when trying to compute an isomorphism 
between $Q(X)$ and an isomorphic copy $G$ of it: 
take any element from the first field that is not in $Q$, 
so it is transcendental over $Q$, and take a similar element in $G$, 
and map the first to the second. If you were lucky and picked in both cases 
the variable $X$ you're done. But what if the first element is 
not a transcendence basis, and you find out later (while computing 
your isomorphism) that the first element is in fact a square. 
So now you have to adjust your first choice by mapping it to some 
(transcendent) square in the other field. If you can do this after 
only finitely many times changing your mind, you're done. 
But what if we have infinite transcendence degree? 
Ershov proved that an ACF of finite transcendence degree is cc, 
but one of countable transcendence degree is not. 
It was expected that this is the general phenomenon. 
However, we (joint work with R. Miller) construct a field 
of infinite transcendence degree that is cc. 
The construction relies on the Mordell-Faltings fact 
that curves of high genus have only finitely many rational solutions, 
and by the positive solution of Fermat's conjecture, 
we have now a whole family for which we know the exact number of solutions.