Research

Evolution of the Research Program

In search of the geometry of the prime numbers

Mathematics often evolves through a succession of questions, where each new construction arises from the limitations of the preceding one. This page illustrates the conceptual evolution of the research program (joint with A. Connes) described in the Research Overview.

In the diagram below each node represents a mathematical idea or construction that marked a significant stage in the development of this program. The arrows indicate how one question naturally led to the next, revealing progressively richer geometric structures.

Conceptual Roadmap

Background Bost--Connes system early 1990s

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2008–2014 Integral model of the BC-system Geometry and arithmetic The role of the Riemann zeta function, as partition function of the BC-system over $\mathbf{C}$ is replaced, in the $p$-adic case, by $p$-adic $L$-functions and polylogarithms whose values at roots of unity encode the KMS states.

2009–2014 Geometries over $ \mathbf F_1 $ Zeta function of $C=\overline{\operatorname{Spec}\mathbf Z}$ over $ \mathbf F_1 $ Complete Riemann-zeta function is recast as the limit $q\to 1$ of the Hasse-Weil zeta function of a real-valued continuous function $N_C(q)$, $q\in [1, \infty)$ satisfying a polynomial bound.

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2014–2016 Arithmetic Site Points of the arithmetic topos realize the finite part of the Riemann sector $X_{\mathbf Q}$ of the adele class space.

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2016–2017 Scaling Site Incorporating the archimedean component of the Riemann sector and the periodic orbits $C_p$ (of length $\log p$) associated to the primes.

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2017–2019 Homological algebra in characteristic one The non-abelian category of modules over the Boolean semifield.

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2020-2025 Characteristic-Free Geometry The absolute base $\mathbf F_1$ Riemann-Roch for $\overline{\operatorname{Spec}\mathbf Z}$ over $\mathbf F_1$. Knots, primes and class field theory: functor mapping finite abelian extensions of $\mathbf{Q}$ to finite covers of the Riemann sector $X_{\mathbf{Q}}$, with the monodromy of periodic orbits $C_p$ under the scaling action corresponding to the Galois action of the Frobenius at the prime p. The sheaf $\mathscr F=\mathscr{O} \rtimes \mathbf{G}_m$ of algebras on $\overline{\operatorname{Spec}\mathbf Z}$: the root of the link between $\operatorname{Spec}\mathbf Z$ and analysis

2019–2021 Weil positivity and trace formula: the archimedean place Weil's distribution vs. trace of scaling action on Sonin's space and prolate spheroidal wave functions.

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2021-2025 Spectral triples and $\zeta$-cycles Semi-local Weil quadratic form. Description of the BC-system as the universal Witt ring (K-theory of endomorphisms) of the "algebraic closure" of the absolute base $\mathbf F_1$. Zeta zeros and prolate wave operators: semilocal adelic operators: introduction of a semilocal analogue of the prolate wave operator (with A. Connes and H. Moscovici).

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2026 Jacobian of $ \operatorname{Spec}\mathbf Z $ A bridge between algebro-geometric and adelic-analytic structures.

The $\mathbf F_1$-arithmetic curve $(\operatorname{Spec}\mathbf Z)_{\mathbf F_1} $ Common geometric origin for structures that had appeared unrelated: p-adic Hodge theory (Fargues-Fontaine curve), complex analytic geometry, and of the adelic scaling site

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Current Directions Maximal abelian cover of $(\operatorname{Spec}\mathbf Z)_{\mathbf F_1}$.

Current Directions Zeta spectral triples: combining the infrared and ultraviolet regimes (with A. Connes and H. Moscovici).