Title: The algebraic K-theory of the sphere spectrum, the geometry of high-dimensional manifolds, and arithmetic

Abstract:
Quillen’s higher algebraic K-theory, applied to number fields,
captures information about the zeta function and L-functions.
Waldhausen’s generalization of K-theory to ring spectra
(multiplicative cohomology theories), applied to the “spherical group
ring” on the based loops of a manifold, captures information about
differential topology. In particular, K(S), the algebraic K-theory of
the sphere spectrum (corresponding to the cohomology theory stable
cohomotopy theory), captures information about BDiff of
highly-connected high dimensional manifolds.

This talk explains on-going work with Mike Mandell that amongst other
things provides a complete calculation of the homotopy groups of K(S)
in terms of the homotopy groups of K(Z), the sphere spectrum, and the
classifying space of a cobordism category. Time permitting, I will
also relate some open problems about K(S) to open problems in number
theory and Galois cohomology.

(No prior knowledge of spectra or algebraic K-theory will be assumed.)

Wednesday, Feb. 11, 4:30 – 5:30 p.m.
Mathematics 520
Tea will be served at 4:00 p.m.

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