The Johns Hopkins Mathematics Department jointly with the Japan-U.S. Mathematics

Institute (JAMI) plan to organize a workshop on the weekend of October 18-20 2019.

The interactions between the fields of noncommutative arithmetic geometry, tropical geometry, the theory of toposes and mathematical logic, optimization and game theory have matured quite rapidly in the last few years and have produced very exciting results.

The goal of the workshop is thus that of coordinating these diverse research areas and with a common aim that is deeply related to the Riemann Hypothesis (RH).

The Noncommutative Geometry approach to RH has unveiled the exotic nature of the adele class space of the rationals, based on the theory of Grothendieck toposes on one hand and tropical geometry on the other. The search of a suitable Riemann-Roch formula on the square of the Scaling Site seems to require the development of algebraic geometry in characteristic one and of refinements of the tropical Riemann-Roch theory as developed so far and its connections to game and potential theory.

On the other hand, the theory of Segal's Gamma-rings provides a natural extension of the theory of rings and of semirings: the clear advantage of working with Gamma-rings is that this category forms the natural groundwork where cyclic homology is rooted (this means that de Rham theory is here naturally available). In fact, much more holds, since the simplicial version of modules over Gamma-rings (ie the natural set-up for homological algebra in this context) forms the core of the local structure of algebraic K-theory. In arithmetic geometry, the new Gamma-ring arising as the stalk of the structure sheaf of the Arakelov compactification of Spec

The goal of the workshop is thus that of coordinating these diverse research areas and with a common aim that is deeply related to the Riemann Hypothesis (RH).

The Noncommutative Geometry approach to RH has unveiled the exotic nature of the adele class space of the rationals, based on the theory of Grothendieck toposes on one hand and tropical geometry on the other. The search of a suitable Riemann-Roch formula on the square of the Scaling Site seems to require the development of algebraic geometry in characteristic one and of refinements of the tropical Riemann-Roch theory as developed so far and its connections to game and potential theory.

On the other hand, the theory of Segal's Gamma-rings provides a natural extension of the theory of rings and of semirings: the clear advantage of working with Gamma-rings is that this category forms the natural groundwork where cyclic homology is rooted (this means that de Rham theory is here naturally available). In fact, much more holds, since the simplicial version of modules over Gamma-rings (ie the natural set-up for homological algebra in this context) forms the core of the local structure of algebraic K-theory. In arithmetic geometry, the new Gamma-ring arising as the stalk of the structure sheaf of the Arakelov compactification of Spec

**Z**at the archimedean place, is intimately related to hyperbolic geometry and the Gromov norm. The theory of Gamma-rings in fact culminates with the development of a new theory of**ABSOLUTE ALGEBRAIC GEOMETRY**
All these developments are thus part of a general program for a workshop that brings together researchers from these different fields.

K. Consani