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## A bit about myself

My research interests are in the fields of arithmetic geometry and algebraic geometry and the connections with noncommutative geometry. I work on the development of the notion of an `absolute geometry' (aka geometry over the `absolute point') and on a geometry in characteristic one.

The outcomes of this research have relevant applications in number-theory, in particular toward the Riemann Hypothesis.

Prior to joining the faculty at Johns Hopkins (2005), I worked at the University of Toronto (1999-2005) and at MIT (1996-1999).

I received the title of Dottore di Ricerca in Matematica from the Universities of Genoa and Turin in Italy (1993) and the Ph.D in Mathematics from the University of Chicago (1996).

I am a member of the Editorial Board of the Journal of the Institute of Mathematics of Jussieu , Journal
of Number Theory and of Rendiconti del Seminario Matematico della Universita' di Padova.
Some information about myself are reported in the Wikipedia page.

## Research

My current research interests are in noncommutative arithmetic geometry and focus on the development of new interconnections between the fields of noncommutative geometry and number-theory, also addressed toward the development of a strategy to attack the Riemann Hypothesis.

The field of noncommutative arithmetic geometry originated with the definition, in the early 90's, of the Bost-Connes quantum statistical mechanical system (BC-system) whose partition function is the Riemann zeta function and whose zero temperature vacuum states implement the global class field isomorphism for the field of the rational numbers. In the joint work with A. Connes, we have achieved new fundamental relations of the BC-system with p-adic analysis and the theory of Witt vectors. This research has also determined the development of 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.

My research comprehends the development of the notion of absolute geometry with relevant arithmetic contents. The joint development of the study of the adele class space of Q has brought us to establish a correspondence between an algebraic geometric description of this space and its analytic definition.
In fact, to link the geometry of this space to the study of the Riemann zeta function, we consider the quotient of the adele class space of the rationals by the maximal compact subgroup of the idele class group of Q. The algebraic geometric space that describes this double quotient is a semi-ringed (Grothendieck) topos whose original definition, the ``Arithmetic Site’’, is given in terms of a topos defined over B, the smallest (idempotent) semifield of characteristic one (our point of view on this construction is well synthesized in this Dialogue with Alain Connes, and in my talk at IHES 2015). After extending the scalars from B to R^max_+ (the multiplicative version of the tropical semifield of characteristic one R_max), one finds the ``Scaling Site’’, whose points are in one-to-one correspondence with the adele classes (of the double quotient). All these geometric constructions are defined over algebraic semi-structures in characteristic one and for this reason we have focussed our research on 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 basic Boolean semifield, understood as the replacement, in characteristic one, of the category of abelian groups. This construction is well described in the Course of Alain Connes at College de France 2016-17 , in my talk at the Conference in honor of Alain Connes 70th Birthday at Fundan University and in the second related talk given by A. Connes.

Due to the absence of a direct algebraic connection between the ring Z of the integers and B, we have turned to the categorical concept of a module and an algebra over the 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 our recent papers: ``On Absolute Algebraic Geometry the affine case'' and ``Segal’s Gamma rings and universal arithmetic''.

The ultimate goal is to transplant some of the ideas of A. Weil in number-theory, in his proof of the Riemann Hypothesis for function fields, to the case of algebraic number fields, using methods of noncommutative geometry. A very recent outcome of this research is explained in the paper Weil positivity and Trace formula: the archimedean place and is overviewed in the talk of Alain Connes at the Oberwolfach meeting (June 2020).

I have been the (co)organizer of several conferences, workshops and meetings dedicated to foster the research of some of the above topics, here are some samples:

- Cyclic Cohomology at 40s: achievements and future prospects. September 2021 (Organizers: A. Connes, C. Consani, M. Khalkhali, H. Moscovici).

- JAMI WORKSHOP: Riemann-Roch in characteristic one and related topics. October 18-20 2019 (Organizers: C. Consani, K. Mincheva, A. Sagnier).

- NCG 2017 Shanghai (in honor of Alain Connes 70th birthday). Fudan University (China), March-April 2017.

- NONCOMMUTATIVE GEOMETRY -- FESTIVAL (in honor of Henri Moscovici 70th birthday). University of Texas A&M (USA), May 2014.

- JAMI Conference ``Number Theory and Related Topics: in honor of Takashi Ono''. The Johns Hopkins University, April 2013.

- JAMI CONFERENCE ON ``NONCOMMUTATIVE GEOMETRY AND ARITHMETIC'' (Johns Hopkins University) March 22-25 2011 (Organizers: A. Connes, C. Consani, N. Kurokawa).

- SECOND WORKSHOP ON ``NONCOMMUTATIVE GEOMETRY AND GEOMETRY OVER THE FIELD WITH ONE ELEMENT'' (Johns Hopkins University) March 26-27 2009.

- JAMI PROGRAM ON NONCOMMUTATIVE GEOMETRY ARITHMETIC AND RELATED TOPICS (Johns Hopkins University, 2008/09) Final Conference: March 23-25 2009.

- THEMATIC PROGRAM ON ARITHMETIC GEOMETRY, HYPERBOLIC GEOMETRY AND RELATED TOPICS (Fields Institute Toronto) July-December 2008.

- FIRST WORKSHOP ON NONCOMMUTATIVE GEOMETRY AND GEOMETRY OVER THE FIELD WITH ONE ELEMENT (Vanderbilt University) May 15-18 2008.

- AMS SPECIAL SESSION in Noncommutative & Arithmetic Geometry (New Brunswick, New Jersey) October 6-7 2007.

### Teaching

Next academic year 2021-22 I will be on sabatical leave.

In the Spring semester 2021 I tought a course of Calculus II (Bio&Medicine) at the College.

Historically, since my coming at Johns Hopkins, I have taught a plethora of different courses both at the undergraduate and at the graduate level inclusive of:

Topics in Algebraic Geometry (AS 110.738), Topics in Algebraic Number-Theory (AS 110.734), Algebraic Geometry (AS 110.643-4), Number Theory (AS 110.617-8), Graduate Algebra II (AS 110.602), Advanced Algebra I&II (AS 110.401&2, Honors: 110.411&2), Honors Linear Algebra (AS 110.212), Honors Calculus III (AS 110.211), Linear Algebra (AS 110.201), Calculus II (Eng) (AS 110.109), Calculus I (AS 110.106).

My teaching work is also inclusive of supervising undergraduate and graduate theses and postdoctoral research.