Noncommutative Geometry, Arithmetic, and Related Topics: Proceedings of the Twenty-First Meeting of the Japan-U.S. Mathematics InstituteCaterina Consani, Alain Connes This valuable collection of essays by some of the world’s leading scholars in mathematics presents innovative and field-defining 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 hyper-structures in algebra. Originally presented at the Twenty-First Meeting of the Japan-U.S. Mathematics Institute, these essays collectively provide mathematicians and physicists with a comprehensive resource on the topic. |
Contents
Nearby Cycles and Periodicity in Cyclic Homology | 1 |
Modular Index Invariants of Mumford Curves | 31 |
Characteristic 1 Entropy and the Absolute Point | 75 |
The GaussBonnet Theorem for the Noncommutative Two Torus | 141 |
Zeta Phenomenology | 159 |
A Case Study | 183 |
Absolute Modular Forms | 213 |
Absolute Zeta Functions and Absolute Tensor Products | 225 |
An Overview of Geometries over the Field with One Element | 241 |
Lectures on Algebraic Varieties over Fsub1 | 267 |
Transcendence of Values of Transcendental Functions at Algebraic Points Inaugural Monroe H Martin Lecture and Seminar | 279 |
The Hopf Algebraic Structure of Perturbative Quantum Gauge Theories | 297 |