The Johns Hopkins Mathematics Department jointly with the Japan-U.S. Mathematics Institute (JAMI) organizes a year-long program on Noncommutative Geometry, Arithmetic and Related Topics during the period September 2008-May 2009. The Johns Hopkins Mathematics Department has cooperated fruitfully, since 1988 with JAMI to foster collaboration in mathematical research through broadly based programs in mathematics and to promote in general, interaction among mathematicians. This program plans to investigate, with the organization of several mini-courses, lecture series, seminars, a proceedings book and a final conference on March 2009 a number of topics pertaining to the rich interconnection between the fields of Noncommutative Geometry, Number Theory and Mathematical Physics. The interaction between the aforementioned fields is a quite new area of mathematics, which has matured rapidly in the past few years and has produced very exciting results. It seems very timely at this point to organize an extended program on these topics with the aim to pursue a wider exploration of this area of research and also with the goal to analyze the major advances that have been obtained in Noncommutative Geometry in the recent past. In view of very recent developments obtained as a joint project by the PI's, particular attention will be given, during the whole program and at the final conference, to the topic ``Noncommutative geometry, the Riemann zeta-function, motives and the field with one element.'' In the weekly NCGA seminars (noncommutative geometry and arithmetic seminars) and at the final conference main emphasis will be given to describe the link that has recently emerged connecting a well-known quantum statistical dynamical system in noncommutative geometry (the BC-system), the corresponding noncommutative motive (BC-endomotive) and a newly developed algebraic-geometric theory of schemes over the absolute point.
The direct impact of NSF funding is that of supporting and training a significant number of junior U.S. researchers (junior faculty, postdoctoral fellows and graduate students), who will gain the opportunity to participate in the program. Mentoring and training of junior faculty, postdoctoral fellows, and graduate students is a central part of every JAMI program, as is helping these individuals develop networks that include the world's leading researchers. To accomplish this, the PI's have recruited a broad and diverse participant group, paying particular attention to women, minorities, and persons with disabilities. In light of several very recent results, which reveal surprising new connections between the fields of Number Theory and Noncommutative Geometry, the PIs expect that the research broadcast by this program will have a major impact on the development of these fields, under a unified methodology. The rapidly increasing interest in the area of ``Noncommutative Arithmetic Geometry'' has led the PI's to the organization of several recent workshops, which generated collaborative work between theoretical mathematicians and physicists and have attracted the interest of an impressive number of graduate students and young researchers across different fields.
This project has investigated with the organization of several internal seminars, dedicated lecture series and an international (Jami) conference at Johns Hopkins University on March 2011 a number of topics and open problems pertaining to the rich interconnection between the fields of noncommutative geometry, number theory and mathematical physics. The interaction between these fields is a quite new area of mathematics, which has matured rapidly in the past few years and has produced very exciting results. Several main results in noncommutative arithmetic geometry have been recently obtained in the on-going research program of the two P.I.'s A. Connes and C. Consani. This research pursues the viewpoint that the study of the adele class space and of a certain quantum statistical dynamical system (BC-system) naturally associated to this noncommutative space should originate in a basic algebro-geometric framework that we designate as "absolute". The most recent results describe several new arithmetic properties of the BC-system, showing new and interesting connections with the theory of Witt vectors over the algebraic closure of finite fields and with p-adic analysis. The broad impact of this project is to develop research topics in which the fields of number-theory and noncommutative geometry interact under a unified methodology. The main goal of the meetings had been that to provide a larger platform of communication by bringing together researchers from different fields to present and discuss together the fundamental properties and the main outcomes of an "absolute arithmetic theory". The final goal of this project is eventually that to define an analogue, for number-fields, of the geometry underlying the arithmetic theory of function fields and to transplant the ideas of A. Weil, for his proof of Riemann Hypothesis for function fields, to the case of algebraic number fields.
