University of Sydney Algebra Seminar

James Borger (ANU)

Friday 21st March, 12:05-12:55pm, Place: 373

Boolean Witt vectors and total positivity

Rings of Witt vectors are certain exotic constructions that appear most often in local (i.e. p-adic) number theory. But they also have relations with global number theory, and so it is natural to hope that they interact with the "infinity-adic" numbers - that is, the real numbers. I will show that this is the case by extending the big Witt vector construction from rings to semirings. I will also explain how to calculate the big Witt vectors of, first, the Boolean semiring {0,1} and, second, the semiring of natural numbers. In the latter case, this turns out to be an integral analogue of the classical (but recently fashionable) Edrei-Thoma theorem, which classifies all totally positive formal power series with real coefficients. These calculations are joint work with Darij Grinberg.