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This valuable collection of essays by some of the worlds leading scholars in mathematics presents innovative and fielddefining work at the intersection of noncommutative geometry and number theory.The interplay between these two fields centers on the study of the rich structure of the adele class space in noncommutative geometry, an important geometric space known to support and provide a geometric interpretation of the Riemann Weil explicit formulas in number theory. This space and the corresponding quantum statistical dynamical system are fundamental structures in the field of noncommutative geometry.Several papers in this volume focus on the field with one element subject, a new topic in arithmetic geometry; others highlight recent developments in noncommutative geometry, illustrating unexpected connections with tropical geometry, idempotent analysis, and the theory of hyperstructures in algebra.Originally presented at the TwentyFirst Meeting of the JapanU.S. Mathematics Institute, these essays collectively provide mathematicians and physicists with a comprehensive resource on the topic.
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