**Title**: The Arithmetic Site

**Speakers**: Alain Connes, Collège de France, IHES, The Ohio State University Distinguished Professor of Mathematics,

Caterina Consani of The Johns Hopkins University

**Description**: Alain Connes of the Collège de France, IHES and as an OSU Distinguished Professor of Mathematics, will present a lecture course on *"The Arithmetic Site"* with the first introductory lecture on May 5, 2:00 - 3:30 in CH 240 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. Connes will continue the mini-course to be held:

- 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.