Noncommutative geometry and basic physics springerlink. We discuss the possible relationship of homological mirror symmetry with deformation quantization. Developed by alain connes, noncummutative geometry is the set of tools and methods that makes. The noncommutativity arising from quantization is the main concern of noncommutative geometry. The theory of nonabelian vonneumann algebras is indeed. The very notion of noncommutative geometry comes from the identification. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields. Thisprocesstakesagivenalgebranotnecessarilycommutativeandyieldsanew algebra, referred to as a deformation of the original algebra. Fields medal in 1982, for his work on the theory of operator algebras. Examples of rcommutative algebras obtained by quantization of poisson algebras include the weyl algebra, noncommutative tori. Quantization, geometry and noncommutative structures in.
It was founded in the early eighties by the 1982 fields medalist alain connes on the basis of his fundamental works in operator algebras. The naive idea of quantization, the construction of a quantum mechanical theory from a. Rcommutative geometry and quantization of poisson algebras. Deformation quantization and noncommutative black holes. Allowing for the presence of symmetries requires working with principal fiber bundles in a noncommutative setup, where hopf algebras appear naturally. The witt construction in characteristic one and quantization.
Papers range from expository and historical surveys to original research articles. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. The notion of manifold in noncommutative geometry 598 5. The second annual spring institute on noncommutative geometry and operator algebras will be directed by alain connes. During the school part of the meeting several minicourses on a variety of topics from noncommutative geometry, operator algebras and related topics will be given by leading experts. Workshop schedule after 2020 theory of operator algebras and its applications, rims, april 2021march 2022. This is an excellent introductory volume for readers interested in learning about quantization as deformation, hopf algebras, and hodge structures in the framework of noncommutative algebraic geometry. Quantization, quantum groups, and operator algebras. Noncommutative differential geometry, quantization, and. Consani right after the end of the sixth annual spring schoolconference on noncommutative geometry and operator algebras, a second meeting took place at vanderbilt university, on may 1518. Mapping between classical and noncommutative geometry classical notion noncommutative generalization. This volume contains articles from the ams special session, operator algebras, quantization and noncommutative geometry.
As in the quantization of classical phasespace, coordinates are replaced by generators of the algebra 60. Except for a lot of evidence, the quantization scheme meets strong resistance in the case of noncompact reductive groups. In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings. And, the pure quantum states are rays in the hilbert space. The results obtained in the study of operator algebras are phrased in algebraic terms, while the techniques used are highly analytic.
Questions about quantization and noncommutative geometry. Positivity in hochschild cohomology and inequalities for the yangmills action 569 3. An introduction to noncommutative geometry joseph c. Spectral sequence and homology of currents for operator algebras pdf 625 kb. We will look at a uniqueness result for right lcm semigroup c algebras, and nd conditions under. Homological mirror symmetry, deformation quantization and. Seven papers on noncommutative geometry and operator algebras 1621 june 2014. The basic notions of differential geometry extend from commutative or supercommutative algebras to rcommutative algebras. Connes, a survey of foliations and operator algebras. Operator algebras, quantization, and noncommutative. Alain connes, noncommutative geometry academic press, 1994, the old testament.
Noncommutative geometry utilizes nonabelian or nonabelian methods for quantization of spaces through deformation to noncommutative spaces in fact noncommutative algebraic structures, or algebras of functions. Noncommutative geometry, quantum fields and motives pdf 6. Alain connes noncommutative geometry, started in 1982 0, widely develo ped in 1994 as expounded in his book at this date 0 it has grown meanwhile is a systematic quantization of mathematics. Noncommutative geometry, the spectral standpoint arxiv. Imia operator algebra and noncommutative geometry seminar university of wollongong title. Noncommutative geometry ncg is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions possibly in some generalized sense. In this book, we discuss several types of geometric objects in the usual sense of. Seven papers on noncommutative geometry and operator algebras. Deformation quantization and noncommutative manifolds 1. Editorial full text access editors preface for the topical issue on seven papers on noncommutative geometry and operator algebras. Noncommutative geometry and quantization mathematical. Noncommutative differential geometric structures are considered for a class of simple c algebras. Stanford libraries official online search tool for books, media, journals, databases, government documents and more.
Pdf noncommutative differential geometric structures are considered for a class of simple c algebras. Product of the continuum by the discrete and the symmetry breaking mechanism 574 4. They cover a broad spectrum of topics and applications, shedding light on the fruitful interactions between noncommutative geometry and a multitude of areas of contemporary research, such as operator algebras, ktheory, cyclic homology, index theory, spectral theory, geometry of groupoids and in particular of foliations. Graduate students and research mathematicians interested in noncommutative algebraic geometry, deformation quantization and operator. Operator algebras appeared as an outgrowth of quantum physics. We shall here deal with deformation quantization in the context of operator algebras and what falls under the description of strict deformation quantization. Istanbul workshop operator algebras, mathematical physics and low. The sixth annual spring institute on noncommutative geometry and operator algebras is a combination of spring school and international conference. Pdf noncommutative differential geometry, quantization. We update our year 2000 account of noncommutative geometry in 68. A noncommutative algebra is an associative algebra in. We speculate that after certain nonlinear twist the fukaya category becomes equivalent to the category of holonomic modules over a quantized algebra of functions. In some of these papers the author did the ground work for the theory, building up a database of results.
A basic library in noncommutative geometry should at least include. Pdf deformation quantization and noncommutative black holes. The purpose of this workshop is to link mathematicians working in operator theory and noncommutative calculus with mathematical physicists in the area of the spectral action principle. The thesis studies deformation quantization of c algebras, building on an established framework for deforming c algebras equipped with group actions. The metric aspect of noncommutative geometry 552 1. The subject of this thesis concerns deformation quantization for operator algebras, with considerations of some aspects of ktheory and certain concepts from noncommutative geometry. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. This short paper is based on the talk on the conference operator algebras and related topics held on july 2327, 2010, beijing.
This structure is defined in terms of smooth lie group actions on the c algebra in question together with a certain quantization mapping motivated directly by the known cohomological obstructions for the quantum mechanical quantization. Noncommutative geometry has proposed principles leading to interaction patterns close to the observed structure of the standard model of elementary particles, with its usual gauge group, but chirality is taken as an input. Noncommutative differential geometry is a novel approach to geometry, aimed in part at applications in physics. We will discuss semigroup c algebras, and those associated to a class of semigroups called right lcm. Noncommutative geometry noncommutative geometry and.
Alain connes and matilde marcolli, noncommutative geometry, quantum fields, and motives hindawiams, 2008, the new testament. Imia operator algebra and noncommutative geometry seminar. The meeting is a combination of spring school and international conference. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, like the commutative algebras associated to a ne algebraic varieties, di erentiable manifolds, topological spaces, and measure spaces.
Noncommutative differential geometry, quantization, and smooth symmetries of the c. Surveys in noncommutative geometry clay mathematics institute. Further development of noncommutative geometry gives generalizations of following classical geometric an topological notions. But gradually it was realized that they provide the natural framework for generalizing geometry, topology and measure theory in a fundamental noncommutative way of wide applicability.
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