# Etale Homotopy and Diophantine Equations

## Tomer Schlank (MIT)

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 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.

Tuesday, March 26th, at 4:00pm in HBH 227

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