Geometry & Model Theory Seminar – Tobias Kaiser – Periods, Power Series, and Integrated Algebraic Numbers

Date/Time
Date(s) - 24/01/2023
2:00 pm - 3:00 pm

Abstract. Periods are defined as integrals of semialgebraic functions defined over the rationals.
Periods form a countable ring not much is known about. Examples are given by
taking the antiderivative of a power series which is algebraic over the polynomial ring over
the rationals and evaluate it at a rational number. We follow this path and close these algebraic
power series under taking iterated antiderivatives and nearby algebraic and geometric
operations. We obtain a system of rings of power series whose coefficients form a countable
real closed field. Using techniques from o-minimality we are able to show that every period
belongs to this field. In the setting of o-minimality we define exponential integrated algebraic
numbers and show that exponential periods and the Euler constant are exponential
integrated algebraic number. Hence they are a good candiate for a natural number system
extending the period ring and containing important mathematical constants.

Dr. Tobias Kaiser is from the University of Passau, Germany

Location:  Fields Institute