Caterina (Katia) Consani

My Research:

My interests are in noncommutative arithmetic geometry, especially in developing new connections between the fields of noncommutative geometry and number-theory,  addressed, for instance,  toward building a strategy to attack the Riemann Hypothesis.                                                                                                                             Noncommutative arithmetic geometry originated in the early 90's, with the introduction of the Bost-Connes quantum statistical dynamical system (often referred to as the BC-system) whose partition function is the Riemann zeta function and with zero-temperature vacuum states implementing the global class field isomorphism for the field Q of the rational numbers.                                                                                                                                                                                                      In a long-running joint project with Alain Connes, we have determined new relations between the BC-system, p-adic analysis, and the theory of Witt vectors. This research has also produced the archimedean counterpart of the theory of rings of periods in p-adic Hodge theory, and the discovery of the arithmetic role played by cyclic homology of schemes to recast 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 the geometric description of this space and its analytic original definition. To connect this space with the Riemann zeta function, one restricts the attention to the sector of the adele class space of Q by the maximal compact subgroup of the idele class group. The geometric space that describes this (non-commutative) double quotient is a semi-ringed Grothendieck topos whose original definition--the Arithmetic Site--is given in terms of a topos over the Booleans B: the smallest idempotent semifield (aka of characteristic one). Our construction is synthesized in the Dialogue with Alain Connes, and in my talk at IHES 2015.  

By extending scalars from B to (the multiplicative version of) the tropical reals, one finds a second relevant topos: the Scaling Site, whose points are in one-to-one correspondence with the adele classes (of the afore-mentioned double quotient). These two toposes are defined over algebraic idempotent semi-structures and for this reason my research also includes the development of a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. The studied prototype being the category of modules over the Boolean semifield, understood as the replacement, in characteristic one, 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. 

Due to the absence of a direct connection between the ring Z of the integers and B, we turned our attention to the categorical concept of a module and an algebra over the categorical sphere spectrum S, as implemented in the theory of Segal's Gamma-sets, to provide a characteristic free, unifying description of the geometry of the rational primes and the topos 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: