### Inevitability in Mathematics

#### Posted by David Corfield

I’m just back from a conference in Nancy – From Practice to Results in Logic and Mathematics. As you see from the abstract, I was speaking about the inevitability of certain things appearing in mathematics: entities, facts, theories, ideas. We had a discussion of the related robustness a while ago.

Someone posed me the question of the inevitability of Lie groups. Forgetting worries about anachronism, what is our very best account of why the Lie group construct was going to be reached by a mathematics as sophisticated as ours? Can we do better than

Groups are interesting as models of symmetry, manifolds are interesting as models of smooth space, therefore group objects in the category of manifolds will be interesting as models of smoothly varying symmetry?

The other side of the coin is to think of constructs which are arbitrary and contingent. If someone had told me about hypergroups and hyperrings, where instead of a binary operation sending a pair of elements to a single element, it takes them to a nonempty set of elements, I would have imagined that they would have been good candidates for arbitrariness. When, however, you find that Alain Connes is using them in search of an absolute arithmetic, you naturally take notice:

We show that the trace formula interpretation of the explicit formulas expresses the counting function $N(q)$ of the hypothetical curve $C$ associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension $R^{convex}$ of the hyperfield $S$ of signs, which is the hyperfield analogue of the semifield $R_+^{max}$ of tropical geometry, admitting a one parameter group of automorphisms fixing $S$. Finally, we develop function theory over $Spec(S)$ and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over $Spec(S)$.

I wonder whether Durov’s generalized rings are related.

I see the Connes and Consani paper followed on from The hyperring of adèle classes:

We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space $\mathbb{H}_{\mathbb{K}} = \mathbb{A}_{\mathbb{K}} / \mathbb{K}^{\times}$ of a global field $\mathbb{K}$. After promoting $\mathbb{F}_1$ to a hyperfield

K, we prove that a hyperring of the form $R/G$ (where $R$ is a ring and $G \subset R^{\times}$ is a subgroup of its multiplicative group) is a hyperring extension ofKif and only if $G \union \{0\}$ is a subfield of $R$. This result applies to the adèle class space which thus inherits the structure of a hyperring extension $\mathbb{H}_{\mathbb{K}}$ ofK. We begin to investigate the content of an algebraic geometry overK. The category of commutative hyperring extensions ofKis inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field $\mathbb{K}$ of positive characteristic, the groupoid of the prime elements of the hyperring $\mathbb{H}_{\mathbb{K}}$ is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field $\mathbb{K}$.

Hyperstructures are at least 40 years old:

D. Stratigopoulos, Hyperanneaux non commutatifs: Hyperanneaux, hypercorps, hypermodules, hyperespaces vectoriels et leurs propriétés élémentaires. (French) C. R. Acad. Sci. Paris Sér. A-B 269 (1969) A489–A492.

## Re: Inevitability in Mathematics

I think it is inevitable that things we now consider inevitable will be seen as mere distractions by future generations.