Tomer Schlank (MIT)
Title: \'Etale Homotopy and Diophantine Equations
Abstract:From the view point of algebraic geometry solutions to
a Diophantine equation are just sections of
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 theory of homotopical obstructions for solutions
for 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 joint work with Yonatan Harpaz.
Some results and conjectures on the syzygies of higher