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 dimensional varieties