To participate in this seminar, please register here: https://www.msri.org/seminars/25206

 I will give an introduction to two of my specific research areas, both of which hinge on an application of ideas from topology to Diophantine problems. A key notion is that of a torsor, a more algebraic version of principal bundles from topology. A torsor in this context is a set with an action of a finite or algebraic group along with an auxiliary action of the absolute Galois group of the field in question. I will explain in a simple case how Kummer theory may be phrased in terms of torsors. Then I will discuss how they apply in two of my research projects: one about the Brauer-Manin and etale homotopy obstructions and their limits, another about Kim's non-abelian Chabauty's method for provably finding the (finite) set of integral points on certain varieties.