{"product_id":"9789814663861","title":"Nonabelian Multiplicative Integration On Surfaces","description":"\u003cp\u003eNonabelian multiplicative integration on curves is a classical theory. This volume is about the 2-dimensional case, which is much more difficult. In our construction, the setup is a Lie crossed module: there is a Lie group \u003ci\u003eH\u003c\/i\u003e, together with an action on it by another Lie group \u003ci\u003eG\u003c\/i\u003e. The multiplicative integral is an element of \u003ci\u003eH\u003c\/i\u003e, and it is the limit of Riemann products. Each Riemann product involves a fractal decomposition of the surface into kites (triangles with strings connecting them to the base point). There is a twisting of the integrand, that comes from a 1-dimensional multiplicative integral along the strings, with values in the group \u003ci\u003eG\u003c\/i\u003e.\u003c\/p\u003e\u003cp\u003eThe main result of this work is the 3-dimensional nonabelian Stokes theorem. This result is new; only a special case of it was predicted (without proof) in papers in mathematical physics. Our constructions and proofs are of a straightforward nature. There are plenty of illustrations to clarify the geometric constructions.\u003c\/p\u003e\u003cp\u003eOur volume touches on some of the central issues (e.g., descent for nonabelian gerbes) in an unusually down-to-earth manner, involving analysis, differential geometry, combinatorics and Lie theory — instead of the 2-categories and 2-functors that other authors prefer.\u003c\/p\u003e\u003cb\u003eContents:\u003c\/b\u003e\u003cul\u003e\n\u003cli\u003eIntroduction\u003c\/li\u003e\n\u003cli\u003ePolyhedra and Piecewise Smooth Geometry\u003c\/li\u003e\n\u003cli\u003eEstimates for the Nonabelian Exponential Map\u003c\/li\u003e\n\u003cli\u003eMultiplicative Integration in Dimension One\u003c\/li\u003e\n\u003cli\u003eMultiplicative Integration in Dimension Two\u003c\/li\u003e\n\u003cli\u003eQuasi Crossed Modules and Additive Feedback\u003c\/li\u003e\n\u003cli\u003eStokes Theorem in Dimension Two\u003c\/li\u003e\n\u003cli\u003eSquare Puzzles\u003c\/li\u003e\n\u003cli\u003eStokes Theorem in Dimension Three\u003c\/li\u003e\n\u003cli\u003eMultiplicative Integration on Triangular Kites\u003c\/li\u003e\n\u003c\/ul\u003e\u003cbr\u003e\u003cb\u003eReadership:\u003c\/b\u003e Graduate students, algebraic topologists, mathematical physicists and theoretical physicists.\u003cbr\u003e\u003cb\u003eKey Features:\u003c\/b\u003e\u003cul\u003e\n\u003cli\u003eThe author is an expert on algebraic geometry and deformation quantization. In this volume, he takes a journey into another area of mathematics (differential geometry), and the result of this journey is a collection of original and deep constructions and theorems\u003c\/li\u003e\n\u003cli\u003eThe volume contains an unusually detailed study of the nonabelian exponential map and of piecewise smooth differential forms\u003c\/li\u003e\n\u003cli\u003eThe volume includes an edited version of lecture notes on the subject, that provides a quick and lucid overview of the main features\u003c\/li\u003e\n\u003c\/ul\u003e","brand":"World Scientific Publishing Company, Incorporated","offers":[{"title":"Default Title","offer_id":47185305436400,"sku":"9789814663861","price":28.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0737\/7593\/9824\/files\/9789814663861_p0.jpg?v=1763691377","url":"https:\/\/shop-qa.barnesandnoble.com\/products\/9789814663861","provider":"Barnes \u0026 Noble (DEV)","version":"1.0","type":"link"}