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Matt Kerr: Higher normal functions and irrationality proofs

Wednesday Stockholm Zoom Seminar (SPECIAL TIME)

Time: Wed 2020-06-17 15.00 - 16.00

Location: Zoom, meeting ID: 689 4480 8319

Participating: Matt Kerr, Washington University in St. Louis

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R. Apéry’s 1978 proof of the irrationality of ζ(3) relied upon two sequences of rational numbers whose ratio limits to ζ(3) very quickly. Beukers and Peters discovered in 1984 that the generating function of the first sequence was a period of a family of K3 surfaces. The corresponding algebro-geometric interpretations for the second generating function and the limit, however, have been missing until recently.

Normal functions are certain “well-behaved’’ sections of complex torus bundles, first studied by Poincaré and Lefschetz. They arise in particular from algebraic cycles (formal sums of subvarieties) on families of complex algebraic manifolds. A more general notion of cycles, due to Bloch and Beilinson and closely related to algebraic K-theory and motivic cohomology, leads to generalizations called “higher normal functions”. Both sorts are found lurking beneath many an arithmetic or functional property of periods.

In this talk, we offer a brief tour of their unexpected role in Apéry’s proof, and in a more general circle of objects surrounding it, including motivic Gamma functions, Feynman integrals, and Apéry constants of Fano varieties. (No knowledge of algebraic cycles will be assumed.)