هندسه غیرقابل جابجایی حسابی (سری سخنرانی‌های دانشگاهی)

Arithmetic noncommutative geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties. Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry. With a foreword written by Yuri Manin and a brief introduction to noncommutative geometry, this book offers a comprehensive account of the cross fertilization between two important areas, noncommutative geometry and number theory. It is suitable for graduate students and researchers interested in these areas.

Arithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at archimedean places of arithmetic surfaces and varieties. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable ''fibers at infinity''), by adding boundaries that are invisible to algebraic geometry, such as degenerations of elliptic curves to noncommutative tori. The text of the book is organized around series of invited lectures delivered by the author at various universities, and the results presented are based on work of the author in collaboration with Alain Connes, Katia Consani, Yuri Manin, and Niranjan Ramachandran.

Marcolli works from her invited lectures delivered at several universities to address questions and reinterpret results and constructions from number theory and arithmetric algebraic geometry, generally is they are applied to the geometry and arithmetic of modular curves and to the fibers of archimedean places of arithmetic surfaces and varieties. One of the results is to refine the boundary structure of certain classes of spaces, such as moduli spaces (like modular curves) or arithmetric varieties completed by suitable fibers at infinity by adding boundaries that are not visible within algebraic geometry. Marcolli defines the noncommutative spaces and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois theory, and noncommutative geometry at arithmetric infinity.