Topological viewpoints on local-global principles 
-- Daniel Krashen, March 23, 2018

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Understanding algebraic structures such as Galois extensions, quadratic forms and division algebras, can give important insights into the arithmetic of fields. In this talk, I will discuss recent work 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 versions of Meyer-Vietoris sequences, the Seifert-van Kampen theorem, and examples and counterexamples to local-global principles.