Elden Elmanto, Harvard University

Abstract: In 1957, Grothendieck defined the K-groups of vector bundles on an algebraic variety to formulate his Riemann-Roch theorem, bringing an algebraic perspective to a theorem in complex analysis. The higher K-groups of varieties were discovered in 1973 by Quillen and have remained a source of both mystery and inspiration in mathematics, and interacts with many vastly different subjects.

In this talk, I will provide a panoramic tour on K-theory and its cousins - cycles and cobordisms. In particular I will explain how the birational geometry of Hilbert schemes (joint with Bachmann) and p-adic Hodge theory (joint with Morrow) can be used to shed new light on these creatures.