An Introduction to Noncommutative Spaces and their Geometries : Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment

These Lecture Notes Are An Introduction To Several Ideas And Applications Of Noncommutative Geometry. It Starts With A Not Necessarily Commutative But Associative Algebra Which Is Thought Of As The Algebra Of Functions On Some 'virtual Noncommutative Space'. Attention Is Switched From Spaces, Which In General Do Not Even Exist, To Algebras Of Functions. In These Notes, Particular Emphasis Is Put On Seeing Noncommutative Spaces As Concrete Spaces, Namely As A Collection Of Points With A Topology. The Necessary Mathematical Tools Are Presented In A Systematic And Accessible Way And Include Among Other Things, C'*-algebras, Module Theory And K-theory, Spectral Calculus, Forms And Connection Theory. Application To Yang--mills, Fermionic, And Gravity Models Are Described. Also The Spectral Action And The Related Invariance Under Automorphism Of The Algebra Is Illustrated. Some Recent Work On Noncommutative Lattices Is Presented. These Lattices Arose As Topologically Nontrivial Approximations To 'contuinuum' Topological Spaces. They Have Been Used To Construct Quantum-mechanical And Field-theory Models, Alternative Models To Lattice Gauge Theory, With Nontrivial Topological Content. This Book Will Be Essential To Physicists And Mathematicians With An Interest In Noncommutative Geometry And Its Uses In Physics. Introduction -- Noncommutative Spaces And Algebras Of Functions -- Projective Systems Of Noncommutative Lattices -- Modules As Bundles -- A Few Elements Of K-theory -- The Spectral Calculus -- Noncommutative Differential Forms -- Connections On Modules -- Field Theories On Modules -- Gravity Models -- Quantum Mechanical Models On Noncommutative Lattices -- Basic Notions Of Topology -- The Gel'fand-naimark- Segal Construction -- Hilbert Modules -- Strong Morita Equivalence -- Partially Ordered Sets -- Pseudodifferential Operators. Giovanni Landi. Includes Bibliographical References (p. [189]-195) And Index. Introduction....Pages 1-5 Noncommutative Spaces and Algebras of Functions....Pages 7-20 Projective Systems of Noncommutative Lattices....Pages 21-58 Modules as Bundles....Pages 59-68 A Few Elements of K -Theory....Pages 69-81 The Spectral Calculus....Pages 83-103 Noncommutative Differential Forms....Pages 105-121 Connections on Modules....Pages 123-132 Field Theories on Modules....Pages 133-150 Gravity Models....Pages 151-163 Quantum Mechanical Models on Noncommutative Lattices....Pages 165-169 These notes are from a series of introductory seminars on Connes' noncommutative geometry, and they emphasize noncommutative spaces seen as concrete spaces. Examples of noncommutative spaces are provided by noncommutative lattices.