# Characteristic 1 , entropy and the absolute point

@inproceedings{Connes1997Characteristic1, title={Characteristic 1 , entropy and the absolute point}, author={Alain Connes and Caterina Consani}, year={1997} }

We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point Spec F1. After introducing the notion of “perfect” semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in characteristic p > 1 to the limit case of characteristic one. This construction also unveils an interesting connection with entropy…

## 27 Citations

THE WITT CONSTRUCTION IN CHARACTERISTIC ONE AND QUANTIZATION

- Mathematics
- 2010

We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R; ) of a perfect semi-ring R of characteristic one and an element > 1 of R to real Banach…

Schemes over 𝔽1 and zeta functions

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- 2010

Abstract We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete Riemann…

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- 2010

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Projective geometry in characteristic one and the epicyclic category

- MathematicsNagoya Mathematical Journal
- 2015

Abstract We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic…

Lattices, spectral spaces, and closure operations on idempotent semirings

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- 2022

THE GEOMETRY OF BLUEPRINTS PART II: TITS–WEYL MODELS OF ALGEBRAIC GROUPS

- MathematicsForum of Mathematics, Sigma
- 2018

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$ , the…

N T ] 1 S ep 2 01 7 Counting and zeta functions over F 1 Abh

- Mathematics
- 2018

This paper is dedicated to counting functions and the corresponding zeta functions in F1-theory. The first F1-zeta function has been defined by Christophe Soulé in [Sou04] for schemes of finite type…

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- 2011

. The hyperring of adèle classes. I present here some recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) about the “characteristic 1 ” limit case. The main goal is to prove…

The relationship between difference and ratio and a proposal: Equivalence of temperature and time, and the first spontaneous symmetry breaking

- Physics
- 2010

The mathematical form of the energy-entropy relationship is that of the relationship between the difference and the ratio of any two 'entities' x and y creating a geometrical 3D projection x = y·z,…

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