000			02170cam a2200241 i 4500
001			138305
003			ISI Library, Kolkata
005			20180502164617.0
008			161208s2017 riu b 001 0 eng
020			_a9781470436230 (alk. paper)
040			_aISI Library
082	0	4	_a510MS _223 _bAm512
100	1		_aBuium, Alexandru, _eauthor
245	1	0	_aFoundations of arithmetic differential geometry / _cAlexandru Buium.
260			_aProvidence : _bAmerican Mathematical Society, _c©2017.
300			_ax, 344 pages ; _c27 cm.
490	0		_aMathematical surveys and monographs ; _vv 222.
504			_aIncludes bibliographical references and index.
505	0		_aIntroduction -- 1. Algebraic background -- 2. Classical differential geometry revisited -- 3. Arithmetic differential geometry : generalities -- 4. Arithmetic differential geometry : the case of GLn -- 5. Curvature and Galois groups of Ehresmann connections -- 6. Curvature of Chern connections -- 7. Curvature of Levi-Civitáa connections -- 8. Curvature of Lax connections -- 9. Open problems.
520			_aThe aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is 'intrinsically curved'; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
650		0	_aDifferential geometry.
942			_2ddc _cBK
999			_c424430 _d424430