## Section: New Results

### Tropical algebra, number theory and directed algebraic topology

#### An arithmetic site of Connes-Consani type for number fields with narrow class number 1

Participant : Aurélien Sagnier.

In 1995, A. Connes ( [65]) gave a spectral interpretation of the zeroes of the Riemann zeta function involving the action of ${\mathbb{R}}_{+}^{*}$ on the sector $X={\mathbb{Q}}_{+}^{\times}\setminus {\mathbb{A}}_{\mathbb{Q}}/{\widehat{\mathbb{Z}}}^{\times}$ of the adele class space ${\mathbb{A}}_{\mathbb{Q}}/{\mathbb{Q}}^{*}$ of the field of rational numbers. In [66], [68], the action of ${\mathbb{R}}_{+}^{*}$ on this sector $X$ was shown to have a natural interpretation in algebraic geometry. This interpretation requires the use of topos theory as well as of the key ingredient of characteristic one namely the semifield ${\mathbb{R}}_{max}$ familiar in tropical geometry. The automorphism group of this semifield is naturally isomorphic to ${\mathbb{R}}_{+}^{*}$ and plays the role of the Frobenius. As it turns out, its action on the points of a natural semiringed topos corresponds canonically to the above action on $X$. This semiringed topos is called the arithmetic site. In my PhD, I extended the construction of the arithmetic site, replacing the field of rational numbers by certain number fields. I considered the simplest complex case, namely that of imaginary quadratic fields on which we assume that the units are not reduced to $\pm 1$ that is when $K$ is either $\mathbb{Q}(\u0131)$ or $\mathbb{Q}(\u0131\sqrt{3})$. In particular, during this year, we showed that the semiring of convex polygons introduced for those cases satisfies a sublte arithmetical universal property. These results are presented in the accepted in *Journal of Number Theory* article [111]. In a further work, developed this year, I extended this construction,
dealing now with number fields $K$ with narrow class number 1, this generalization will rely on the universal property discovered this year and on the extensive use of Shintani's unit theorem. Here again tropical algebra play a crucial role in the geometrical constructions.

#### Duality between tropical modules and congruences

Participants : Stéphane Gaubert, Aurélien Sagnier.

In a joint work with Éric Goubault (LIX, École polytechnique), we establish a duality theorem between congruences and modules over tropical semifields.

#### Directed topological complexity and control

Participant : Aurélien Sagnier.

This is a joint work with Michael Farber and Eric Goubault.

The view we are taking here is that of topological complexity, as defined in [71], adapted to directed topological spaces.

Let us briefly motivate the interest of a directed topological complexity notion. It has been observed that the very important planification problem in robotics boils down to, mathematically speaking, finding a section to the path space fibration $\chi \phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}PX={X}^{I}\to X\times X$ with $\chi \left(p\right)=\left(p\right(0),p(1\left)\right)$. If this section is continuous, then the complexity is the lowest possible (equal to one), otherwise, the minimal number of discontinuities that would encode such a section would be what is called the topological complexity of $X$. This topological complexity is both understandable algorithmically, and topologically, e.g. as $s$ having a continuous section is equivalent to $X$ being contractible. More generally speaking, the topological complexity is defined as the Schwartz genus of the path space fibration, i.e. is the minimal cardinal of partitions of $X\times X$ into “nice” subspaces ${F}_{i}$ such that ${s}_{{F}_{i}}\phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}{F}_{i}\to PX$ is continuous.

This definition perfectly fits the planification problem in robotics where there are no constraints on the actual control that can be applied to the physical apparatus that is supposed to be moved from point $a$ to point $b$. In many applications, a physical apparatus may have dynamics that can be described as an ordinary differential equation in the state variables $x\in {\mathbb{R}}^{n}$ and in time $t$, parameterized by control parameters $u\in {\mathbb{R}}^{p}$, $\dot{x}\left(t\right)=f(t,x\left(t\right))$. These parameters are generally bounded within some set $U$, and, not knowing the precise control law (i.e. parameters $u$ as a function of time $t$) to be applied, the way the controlled system can evolve is as one of the solutions of the differential inclusion $\dot{x}\left(t\right)\in F(t,x\left(t\right))$ where $F(t,x(t\left)\right)$ is the set of all $f(t,x(t),u)$ with $u\in U$. Under some classical conditions, this differential inclusion can be proven to have solutions on at least a small interval of time, but we will not discuss this further here. Under the same conditions, the set of solutions of this differential inclusion naturally generates a dspace (a very general structure of directed space, where a preferred subset of paths is singled out, called directed paths, see e.g. [83]). Now, the planification problem in the presence of control constraints equates to finding sections to the analogues to the path space fibration (That would most probably not qualify for being called a fibration in the directed setting) taking a dipath to its end points. This notion is developed in this article, and we introduce a notion of directed homotopy equivalence that has precisely, and in a certain non technical sense, minimally, the right properties with respect to this directed version of topological complexity.

This notion of directed topological complexity also has applications in informatics where a directed space can be used to model the space of all possible executions of a concurrent process (ie when several running programs must share common limited ressources).

In the article [22], after defining the notion of directed topological complexity,this invariant (directed topological complexity) is studied for directed spheres and directed graphs.