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At Oberwolfach in 2011

A bit about myself

My research interests are in the fields of arithmetic geometry and algebraic geometry and the connections with noncommutative geometry. I am working on the development of the notion of an `absolute geometry', aka geometry over the `absolute point', and a geometry in characteristic one.
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 Number Theory and of Rendiconti del Seminario Matematico della Universita' di Padova. Some information about myself are also reported at de.wikipedia.


Research

My current research interests are in noncommutative arithmetic geometry and are concentrated on the development of new interconnection between the fields of noncommutative geometry and number-theory. 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 recently 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 current study comprehends the development of a notion of absolute geometry with relevant arithmetic contents. The latest 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 topos whose original definition, the ``Arithmetic Site’’, is given in terms of a topos defined over B, the smallest (idempotent) semifield of characteristic one. 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.
Due to the absence of a direct algebraic connection between Z and B, we have turned to the categorical concept of 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.
The ultimate goal is to transplant 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.

In the recent past years I have been the (co)organizer of several conferences, workshops and meetings dedicated to foster the research of some of the above topics, here is some sample:

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

This academic year (2015-16) I will teach 2 graduate courses in number theory and a graduate course in algebraic geometry.

Historically, since my coming at Johns Hopkins, I have taught a plethora of different courses both at the undergraduate and graduate levels inclusive of:
Topics in Algebraic Geometry (AS 110.738), Topics in Algebraic Number-Theory (AS 110.734), Algebraic Geometry (AS 110.643), Number Theory (AS 110.617), Graduate Algebra II (AS 110.602), Advanced Algebra I&II (AS 110.401&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. At the moment I supervise the research of 1 graduate student.