Title: The Arithmetic Site
Speakers: Caterina Consani, Johns Hopkins University,
Alain Connes, Collège de France, IHES, The Ohio State University Distinguished Professor of Mathematics
Description: The first introductory lecture, on May 5, 2:00 - 3:30 in CH 240, will be given by Caterina Consani of The Johns Hopkins University. There will be a tea after this first talk at 3:30 - 4:30 in MW 724. Dr. Alain Connes will continue the mini-course, "The Arithmetic Site" as follows:
- May 6, 10:00 - 11:30
- May 7, 2:00 - 3:30
- May 8, 10:00 - 11:30
- May 12, 10:30 - 12:00, final lecture
all in CH240.
Abstract: We have uncovered, in our joint work with C. Consani, the "Arithmetic Site" : an object of algebraic geometry deeply related to the non-commutative geometric approach to RH. The set of points of the "Arithmetic Site" over the maximal compact subring of the tropical semifield coincides with the non-commutative space quotient of the adèle class space of the field of rational numbers by the action of the maximal compact subgroup of the idele class group. As we showed earlier this is the space that yields the correct counting function to obtain the complete Riemann zeta function as Hasse-Weil zeta function. The action of the Frobenius automorphisms of the tropical semifield on the above points corresponds to the action of the multiplicative group of positive real numbers on the adele class space that yields the expected counting function. The "Arithmetic Site" is an object of algebraic geometry, the underlying space is a topos and the structure sheaf is made by semirings of characteristic 1. The square of the arithmetic site is also well-defined and the Frobenius correspondences parametrized by positive real numbers, are interpreted as subvarieties of this square. I will also discuss the link between the arithmetic site and cyclic homology through its relation with the epicyclic site.
