Research

Research Overview

My research focuses on the development of new geometric methods in arithmetic. A recurring theme throughout this work is the search for a geometric framework capable of revealing the underlying geometric structure of the integers and of placing arithmetic phenomena within a broader geometric context.

This research program did not originate from a predetermined plan. Instead, it evolved through a succession of mathematical questions, each emerging naturally from the insights gained at the previous stage. Over time, this process led from the study of the arithmetic of the integral model of the Bost--Connes system and the first investigations in geometry over $\mathbf{F}_1$, to the construction of the Arithmetic Site and the Scaling Site, and eventually to the search for a characteristic-free geometry of \(\operatorname{Spec}\mathbf Z\).

Progress toward answering this fundamental question has required the introduction of new mathematical structures and ideas, often revealing that previously accepted frameworks represented only partial manifestations of a deeper geometric picture.

This page offers a conceptual overview of this mathematical journey.

Noncommutative arithmetic geometry emerged in the early 1990s with the introduction of the Bost-Connes quantum statistical dynamical system (commonly known as the BC-system). This system features the Riemann zeta function as its partition function, while the zero-temperature vacuum states of the system realize the global class field isomorphism for the rational number field \(\mathbf Q\).
In a long-term collaboration with Alain Connes, we have uncovered new connections between the BC-system, p-adic analysis, and the theory of Witt vectors. This work also led to the development of an archimedean counterpart to the theory of rings of periods in p-adic Hodge theory.
Additionally, we identified the arithmetic significance of the cyclic homology of schemes, which provides a set-up to reinterpret Serre's archimedean factors of the Hasse-Weil L-function of a projective algebraic variety over a number field as regularized determinants.
The study of the adele class space \(\mathbf Q^\times\backslash\mathbb A_{\mathbf Q}\) revealed a remarkable correspondence between its original analytic definition and a new algebro-geometric interpretation. To relate this space to the Riemann zeta function, attention is restricted to a specific sector \(X_{\mathbf Q}:=\mathbf Q^\times\backslash\mathbb A_{\mathbf Q}/\widehat{\mathbf Z}^\times\) of the adele class space (called Riemann sector), defined as the quotient of the adele class space by the maximal compact subgroup of the idele class group. The algebro-geometric space, which describes set-theoretically this (noncommutative) double quotient, is the Grothendieck topos \(\widehat{\mathbf N^\times_0}\). Its original definition—the Arithmetic Site—was defined as a topos over the Boolean semifield \(\mathbf B\): the smallest idempotent semifield (also known as the smallest semifield of characteristic one). This construction is synthesized in the talk of Alain Connes at IHES , and in my talk at IHES 2015.
Extending scalars from \(\mathbf B\) to the multiplicative version of the tropical reals reveals a second significant topos: the Scaling Site \(\mathscr S\). The points of the topos \([0,\infty)\rtimes \mathbf N^\times\) underlying \(\mathscr S\) are in canonical bijection with the adele classes of the Riemann sector \(X_{\mathbf Q}\). The structure sheaf \(\mathcal O\) of the Scaling Site is a \(\mathbf N^\times\)-equivariant sheaf of idempotent semirings of continuous convex functions valued in \(\mathbf R_{\text{max}}\).
A particularly important subspace of \(\mathscr S\), in relation to the rational primes, is defined by the collection (disjoint union) of the periodic orbits \(C_p\) of the scaling flow. These are the points of \([0,\infty)\rtimes \mathbf N^\times\) over the image of \(\operatorname{Spec}\mathbf Z\) inside \(\widehat{\mathbf N^\times}\) determined by the the geometric morphism of toposes \[ \Theta: \operatorname{Spec}\mathbf Z \longrightarrow \widehat{\mathbf N^\times_0}\subset \mathscr S. \] For each prime $p$ the corresponding circle \(C_p\) of length $\log p$ within \(\mathscr S\) is endowed with a quasi-tropical structure which turns this orbit into a variant $C_p=\mathbf{R}_{+}^\times / p^{\mathbf{Z}}$ of the classical Jacobi description $\mathbf{C}^\times / q^{\mathbf{Z}}$ of an elliptic curve. The structure sheaf $\mathcal{O}_p$ of $C_p$ is obtained by restriction of $\mathcal{O}$ to $C_p$ and its sections are periodic functions $f(p \lambda)=f(\lambda), \lambda \in \mathbf{R}_{+}^\times$, which are convex, piecewise affine and whose derivatives take values in the group $H_p \subset \mathbf{R}$ of rational numbers with denominators a power of $p$.
The geometric structure of the periodic orbits \(C_p\) is very rich: they admit a theory of theta functions and Cartier divisors. One remarkable feature is that the degree of a divisor is a real number. For a divisor $D$ on $C_p$ there is a corresponding Riemann-Roch problem with solution space $H^0(D)=H^0\left(C_p, \mathcal{O}(D)\right)$. We introduce the notion of continuous dimension $\operatorname{Dim}_{\mathbf{R}}\left(H^0(D)\right)$ of this $\mathbf{R}_{\text {max}}$-module using a limit of normalized topological dimensions and find that $\operatorname{Dim}_{\mathbf{R}}\left(H^0(D)\right)$ is a real number. The topological dimension used here is the Lebesgue covering dimension. The appearance of arbitrary positive real numbers as continuous dimensions of $\operatorname{Dim}_{\mathbf{R}}\left(H^0(D)\right)$ is due to the density in $\mathbf{R}$ of the subgroup $H_p \subset \mathbf{Q}$ and the fact that continuous dimensions are defined as limits of normalized dimensions $p^{-n} \operatorname{dim}_{\text {top }}\left(H^0(D)^{p^n}\right)$ where $H^0(D)^{p^n}$ is a natural filtration of $H^0(D)$ involving the $p$-adic norms of the derivatives. This is the characteristic 1 counterpart of the statement for matroid $C^*$-algebras and the type II normalized traces. The continuous dimensions which affect arbitrary positive real values appear when passing to the von Neumann algebra of type II obtained as the weak closure of the $C^*$-algebra using the trace to perform the completion.
Combining these results we obtain a Riemann-Roch formula (with real values) for the periodic orbits \(C_p\) that takes the form: \[ \operatorname{Dim}_{\mathbf{R}}\left(H^0(D)\right)-\operatorname{Dim}_{\mathbf{R}}\left(H^0(-D)\right)=\operatorname{deg}(D). \] Both the Arithmetic Site and the Scaling Site were originally defined over algebraic idempotent semi-structures. Consequently, this research also involved developing a general theory of homological algebra for categories such as the category of sheaves of idempotent modules over a topos. A key example is the category of modules over the Boolean semifield, viewed as the characteristic-one analog of the category of abelian groups. This research is described in the Course of Alain Connes at College de France 2015-16, and in my talk at the Conference in honor of Alain Connes 70th Birthday at Fundan University.
Given the lack of a direct link between the ring of the integers \(\mathbf Z\) and \(\mathbf B\), this research naturally expanded to include the study of a categorical framework of modules and algebras over a basic categorical object. In modern algebraic K-theory and topological Hochschild homology, it is a well-established paradigm that the fundamental base ring for arithmetic is not \(\mathbf Z\), rather the sphere spectrum \(\mathbb{S}\). However, to construct a concrete spatial object—one possessing well-defined local points, stalks, and moduli spaces—one requires a strict, combinatorial incarnation of this stable homotopy theory that can be integrated into the framework of Grothendieck toposes. We write this basic functorial object as \(\mathbf F_1\). Our proposed base \(\mathbf F_1\) is therefore the spherical algebra of the pointed monoid \(\{0,1\}\), serving as the strict combinatorial model for the sphere spectrum, and our \(\mathbf F_1\)-algebras correspond to Segal's \(\Gamma\)-rings. This approach, rooted in the theory of Segal's \(\Gamma\)-sets and offers the important advantage of providing a characteristic-free (a.k.a. absolute), unified perspective on the geometry of the rational primes and the topos-theoretic description of the sector \(X_{\mathbf Q}\).
This development has produced in Arakelov geometry a relevant advancement, following the idea that the (Arakelov) compactification of \(\operatorname{Spec}\mathbf Z\) is a curve over the absolute ring \(\mathbf F_1[\pm 1]\). We obtained the Riemann-Roch formula ($D=$ Arakelov divisor on $\overline{\operatorname{Spec} \mathbf{Z}}$): \[ \operatorname{dim}_{\mathbf{F}_1} H^0(D)-\operatorname{dim}_{\mathbf F_1} H^1(D)=\left\lceil\operatorname{deg}_2 D\right\rceil^{\prime}+1 . \] (\(\lceil x\rceil^{\prime}\) denotes the right continuous function which agrees with the function ceiling \((x)\) for \(x>0\) non-integer, and with-ceiling \((-x)\) for \(x<0\) non-integer). The proof views \(\mathbf Z\) as a ring of polynomials over \(\mathbf F_1\) with generator \(-2\). This Riemann-Roch formula for the absolute curve $\overline{\operatorname{Spec} \mathbf{Z}}$ displays a perfect analogy with the Riemann-Roch formula holding for curves of genus \(0\). The canonical divisor \(K=-2\{2\}\) has integral degree \(\operatorname{deg}_2(K)=-2\). The results of this research are reported in the articles: Riemann-Roch for the ring \(\mathbf Z\), and On the metaphysics of \(\mathbf F_1\).
Just as Grothendieck's theory of the étale fundamental group extends Galois theory to schemes, the adele class space, viewed as a covering of the Scaling topos, provides a corresponding extension of the class field isomorphism for \(\mathbf Q\) to schemes related to Spec \(\mathbf Z\). The Scaling topos and its periodic orbits \(C_p\) provide a geometric framework for the well-established analogy between primes and knots. In this framework, the adele class space naturally functions as the maximal abelian cover of the Scaling topos. The preimage of a periodic orbit \(C_p\) in this cover is canonically isomorphic to the mapping torus of the the multiplication by the geometric Frobenius at p in the abelianized étale fundamental group of the spectrum of the local ring \(\mathbf Z_{(p)}\). This construction reveals how the prime $p$ is linked to every other rational prime. The details of this development are contained in the paper Knots, primes and class field theory.
In a natural continuation of the geometric interpretation of the adele class space of the rationals, the recent preprint On the Jacobian of \(\operatorname{Spec}\mathbf Z\) develops a Picard–Jacobian formalism for \(\overline{\operatorname{Spec} \mathbf{Z}}\) within the adelic framework. In this construction, elements of the adele class space, and specifically its Riemann sector \(X_{\mathbf Q}\), are viewed in terms of torsion-free rank-one abelian groups equipped with additional rigidifying data that extend the notion of metrized line bundles in Arakelov geometry. This point of view leads to the definition of a Picard monoid associated with the completed arithmetic curve and to a corresponding arithmetic Jacobian obtained by quotienting by the natural scaling action of the positive real numbers.
Within this framework the product of adeles acquires a direct geometric meaning through the tensor product of these rank-one groups, while the Riemann sector \(X_{\mathbf Q}\) appears naturally as a distinguished component of the resulting geometric structure. This approach also provides a conceptual interpretation of the geometric strata arising in the spectral realization of L-functions and further clarifies the geometric role of the adele class space in the study of the Riemann zeta function.
The Riemann sector $X_{\mathbf{Q}}$, when viewed through the lens of functional analysis, is best understood by focusing on a finite subset of places at a time. Specifically, this involves considering a finite subset $S \subset \Sigma_{\mathbf{Q}}$ of places of $\mathbf{Q}$, which includes the archimedean place $\infty$. This approach stems from the fact that the function space associated with $X_{\mathbf{Q}}$ is expressed as a cross product and comprises functions defined on the adeles and more specifically belonging to the Bruhat-Schwartz space characterized by their non-trivial dependence on only finitely many places. The Bruhat-Schwartz functions $f$ are finite linear combinations of the products $f=\otimes_v f_v$ over places $v$ of $\mathbf{Q}$, where each $f_v$ is a Bruhat-Schwartz function on the local field $\mathbf{Q}_v$ and $f_v=\mathbf{1}_{\mathcal{O}_v}$ is the characteristic function on the ring of integers $\mathcal{O}_v$ for all but finitely many $v$. Except for a finite set $S$ of places, the functions reduce to the characteristic functions of the maximal compact subring of the restricted infinite product of local fields $\prod_{v \notin S}^{\prime} \mathbf{Q}_v$. Functional analysis focuses on the locally compact rings $\mathbb{A}_S=\prod_{v \in S} \mathbf{Q}_v$, moreover, the group involved in the cross product, which globally corresponds to $\mathbf{Q}^{\times}$, naturally restricts to a subgroup $\Gamma_S:=\mathbf{Z}_S^{\times} \subset \mathbf{Q}^{\times}$that depends on the finite set $S$. These semilocal algebras form a sheaf \(\mathscr O\) of algebras over $\overline{\operatorname{Spec}\mathbf{Z}}$. The stalk of the sheaf \(\mathscr O\) at the generic point $\eta \in \operatorname{Spec} \mathbf{Z}$ is the Bruhat-Schwartz algebra of functions on $\mathbb{A}_{\mathbf{Q}}$, whereas the global sections form the Schwartz space $\mathcal{S}(\mathbf{R})=\mathcal{S}\left(\mathbb{A}_{\{\infty\}}\right)$: the Bruhat-Schwartz space of functions on the (semi)local adeles at \(\infty\). The identification $\Gamma_S=\mathbf{G}_m\left(S^c\right)$ supplies the conceptual meaning of the groups $\Gamma_S$ as sections of the sheaf in groups $\mathbf{G}_m$ on Spec $\mathbf{Z}$. The algebraic cross product $\mathscr F:=\mathscr{O} \rtimes \mathbf{G}_m$ defines an interesting sheaf of algebras on $\overline{\operatorname{Spec} \mathbf{Z}}$ such that $$ \mathscr F\left(S^c\right)=\mathcal{S}\left(\mathbb{A}_S\right) \rtimes \mathbf{Z}_S^{\times} . $$ The stalk of \(\mathscr F\) at the generic point $\eta \in \operatorname{Spec} \mathbf{Z}$ is the global cross product $\mathcal{S}\left(\mathbb{A}_{\mathbf{Q}}\right) \rtimes \mathbf{Q}^{\times}$, while the global sections form the cross product $\mathcal{S}(\mathbf{R}) \rtimes\{ \pm 1\}$.
Building on the Picard–Jacobian formalism, the very recent preprint On the absolute geometry of \(\operatorname{Spec}\mathbf Z\) pushes the same circle of ideas below the ring \(\mathbf Z\), constructing the absolute \(\mathbf F_1\)-arithmetic curve: \[ (\mathrm{Spec}\,\mathbf Z)_{\mathbf F_1} =(\operatorname{Spec}\mathbf Z, \Theta^{-1}(\mathbf F_1[T])) \] by pulling back the \(\mathbf F_1\)-structure sheaf of the arithmetic topos \(\widehat{\mathbf N^\times_0}\) to \(\mathrm{Spec}(\mathbf Z)\). Because it is defined below \(\mathbf Z\), this absolute curve is insensitive to the characteristic of the fields over which it is evaluated, and it furnishes a common geometric origin for structures that had appeared unrelated: those of \(p\)-adic Hodge theory, of complex analytic geometry, and of the adelic Scaling Site \(\mathscr S\). Evaluated over a perfectoid field, the local stalk of this absolute curve behaves as a universal tilting functor: it recovers the tilt intrinsically and identifies the moduli space of its points, taken modulo the natural symmetries of the completed stalk, with the closed points of the Fargues–Fontaine curve. This yields at once a concrete realization and a broad generalization of Scholze's heuristic description of the points of \(\mathrm{Spec}\,\mathbf Z\) over a perfectoid field. Evaluated instead over the complex numbers, the same absolute geometry produces the complex Tate curve, which decomposes canonically as the product of the adelic periodic orbit \(C_p=\mathbf R^\times_+/p^{\mathbf Z}\) — carrying the entire dependence on the prime \(p\) — and a \(p\)-independent factor that emerges as a real analogue of the Fargues–Fontaine curve. A single absolute object thus brings to light the universal geometric principles underlying results that live, a priori, in very different worlds, and connects them further to the characteristic-1 geometry of the scaling site through a canonical tropicalization.

The accompanying page, Evolution of a Research Program, summarizes this whole development through a conceptual diagram whose nodes represent the principal ideas and constructions.

Geometric focus

The present website focuses on the geometric side of the program. Analytic developments connected with the Riemann Hypothesis, trace formulas, functional analysis, and noncommutative geometry will be mentioned where they directly illuminate a geometric construction, but they are not developed here as an independent narrative.