Tel-Aviv University
School of Mathematical Sciences

Department Colloquium

Raymond and Beverly Sackler distinguished lectures in Mathematics

Monday, December 30, 2013

Schreiber 006, 12:15

Tomer Schlank


Étale homotopy, Sections and Diophantine Equations

From the view point 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 for  Diophantine equations . This would be achieved by  a generalizing the étale homotopy type  defined by Artin and Mazur to a relative setting X → S . In the case of  Diophantine equation  over a number filed 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 Glaios theory.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006