Local-global principles via combinatorial topology and rationality

Understanding algebraic structures such as Galois extensions, quadratic forms, division algebras and torsors for linear algebraic groups, can give important insights into the arithmetic of fields. In this talk, I will discuss recent work and work in progress showing ways in which parts of the arithmetic of function fields over complete fields may be encoded in the combinatorial topology of the special fiber. I will then describe how these observations lead to algebraic versions of Meyer-Vietoris sequences, the Seifert–van Kampen theorem, and examples and counterexamples to local-global principles.