School of Mathematical Sciences
Raymond and Beverly Sackler distinguished lectures in Mathematics
Monday, December 30, 2013
Schreiber 006, 12:15
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