Caterina (Katia) Consani

My Research:

My research interests lie in noncommutative arithmetic geometry, with a particular focus on forging new connections between noncommutative geometry, algebraic geometry, and number theory. This work is motivated, in part, by the goal of developing a strategy to approach the Riemann Hypothesis.  

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 its zero-temperature vacuum states realize the global class field isomorphism for the rational number field Q.                                                                                                                                                                                                      In an extensive collaborative project with Alain Connes, we have uncovered new connections between the BC-system, p-adic analysis, and the theory of Witt vectors. This work has also led to the development of an archimedean counterpart to the theory of rings of periods in p-adic Hodge theory. Additionally, we have identified the arithmetic significance of the cyclic homology of schemes, which provides a framework 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 of Q has revealed a correspondence between its algebro-geometric structure and its original analytic definition (due to A. Connes). To establish a connection between this space and the Riemann zeta function, attention is restricted to a specific sector of the adele class space, defined by the maximal compact subgroup of the idele class group. The resulting algebro-geometric space, which describes this (noncommutative) double quotient, is a semi-ringed Grothendieck topos. Its foundational definition—the Arithmetic Site—is constructed as a topos over the Boolean B: the smallest idempotent semifield (also known as the smallest semifield of characteristic one). Our construction is synthesized in the Dialogue with Alain Connes, and in my talk at IHES 2015.  

Extending scalars from B to the multiplicative version of the tropical reals reveals a second significant topos: the Scaling Site. The points of this topos correspond one-to-one with the adele classes of the previously mentioned double quotient. Both the Arithmetic Site and the Scaling Site are defined over algebraic idempotent semi-structures. Consequently, my research also involves 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 2016-17, 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 Z and B, this research has recently branched to include a categorical framework of modules and algebras over the categorical sphere spectrum S. This approach, rooted in the theory of Segal's Gamma-sets, offers a characteristic-free, unified perspective on the geometry of the rational primes and the topos-theoretic description of the quotient of the adele class space of Q. This development is explained in the papers:On Absolute Algebraic Geometry the affine case, Segal’s Gamma rings and universal arithmetic, BC-system, absolute cyclotomy and the quantized calculus. 

This research has produced, in Arakelov geometry, a relevant advancement in the Riemann-Roch formula, following the idea that the compactification of the algebraic spectrum of the integers is a curve over the absolute ring S[+-1]. The results of this research are reported in the articles:  Riemann-Roch for the ring Z and On the metaphysics of F1.

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 Q to schemes related to Spec Z. The Scaling topos and its periodic orbits Cₚ, with lengths log p, provide a geometric framework for the well-established analogy between primes and knots. In this framework, the adele class space functions as the maximal abelian cover of the Scaling topos. The preimage of a periodic orbit Cₚ 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 Zₚ. This construction reveals the linking of the prime p with all other rational primes. The details of this development are contained in the recent preprint Primes, knots and the adele class space.

The overarching aim of this research in number theory is to adapt and extend some of A. Weil's ideas from his proof of the Riemann Hypothesis for function fields to the context of algebraic number fields, leveraging the tools and techniques of noncommutative geometry.  Several recent developments from this joint research with A. Connes (and lately also in collaboration with Henri Moscovici) are discussed in:

• Weil positivity and trace formula: the archimedean place 

• Quasi-inner functions and local factors 

• Spectral triples and zeta cycles 

• Zeta zeros and prolate wave operators: semilocal adelic operators  2025-Best Paper AOFA Award Winner (among 167 papers published in the years 2023 and 2024) in Annals of Functional Analysis

• On q-series and the moment problem associated to local factors