Exponential motives


Javier Fresan
2016-12-15  09:00 - 10:30
Room 103, Mathematics Research Center Building (ori. New Math. Bldg.)

Exponential periods form a class of complex numbers containing the special values of the gamma and the Bessel functions, the Euler-Mascheroni constant and other interesting numbers which are not expected to be periods in the usual sense of algebraic geometry. However, we can interpret them as coefficients of the comparison isomorphism between two cohomology theories associated to varieties with a potential: the de Rham cohomology of a connection with irregular singularities and the so-called ``rapid decay'' cohomology. In this series of lectures, I will explain how this point of view allows one to construct a Tannakian category of exponential motives over a subfield of the complex numbers and to produce Galois groups which conjecturally govern all algebraic relations among these numbers. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstracts aspects of the theory. The following topics will be covered: rapid decay cohomology, Nori motives, the comparison isomorphism, the perverse realisation, the determinant of periods, explicit computation of Galois groups and relation to transcendence theory. This is a joint work with Peter Jossen (ETHZ).