Speaker: Tomer Schlank (MIT)

Title: 'Etale Homotopy and Diophantine Equations

Abstract:
From the viewpoint of algebraic geometry
solutions to a Diophantine equation are just sections
of a corresponding map of schemes X -> S,
when schemes are usually considered as a certain type of "Spaces".
When considering sections of maps of spaces f: X -> S
in the realm of algebraic topology Bousfield and Kan developed
an Obstruction-Classification Theory using the cohomology of the S
with coefficients in the homotopy groups of the fiber of f.
In this talk we will describe a way to transfer Bousfield- Kan theory
to the realm of algebraic geometry.
Thus yielding a theory of homotopical obstructions for solutions of
Diophantine equations.
This would be achieved by generalizing the `etale homotopy type 
defined by Artin and Mazur to a relative setting X -> S.
In the case of Diophantine equation over a number field i.e.
when S is the spectrum of a number field, this theory can be used
to obtain a unified view of classical arithmetic obstructions 
such as the Brauer-Manin obstruction and descent obstructions.
If time permits I will present also applications to Galois theory.
This is a joint work with Yonatan Harpaz.