Online: Homotopy and Stratification in Arithmetic Geometry

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

A major portion of the modern study of Diophantine Equations is guided trough the paradigm of  Arithmetic Geometry. That is, that the qualitative and quantitative behavior of rational points on an algebraic variety is to be understood as the result of the interaction of the absolute  Galois group of  $QQ$ with the geometry of said algebraic variety viewed over the algebraic closure.  Another indispensable idea is the local-to-global philosophy which suggests considering the place of rational points inside the more general adelic-points.  A beautiful example of these two philosophies combined Is the Brauer-Manin obstruction to the existence of rational points which when was first presented by Manin in 1971 and were able to account for all known examples of algebraic variety with adelic points but no rational points known at the time.
Later In 1999 Skoroboatov define the even finer etale-Brauer-Manin obstruction and used it to explain the absence of rational points on a variety which is not captured by the   Brauer-Manin obstruction. Later in 2008 Poonen gave an example of a 3-fold with no rational points which is not accounted for even by the etale-Brauer-Manin obstruction.

In this talk, I will suggest a perspective on  Poonen's 3-fold and other examples like it coming from homotopy theory.  I'll claim is that the Brauer-Manina and etale-Brauer-Manin obstructions are homotopical in nature. That is, the part of the geometry of the algebraic variety that they take into account is precisely its homotopy type. On the other hand, Poonen's 3-fold is can be accounted for by considering a more refined "stratified homotopy type".  I shall discuss some different ways that adding stratified information is exactly what's missing to obtain full account the absence of rational points.