Ma 191 c: Noncommutative Geometry, Part II

Last day of classes: June 1 (no class on June 3).

Brief Course Description

Prerequisites

Syllabus

• The Bost-Connes system: quantum statistical mechanics and class field theory, analytic and algebraic endomotives, Q-lattices modulo commensurability and scaling, the dual system and the Riemann zeta function, the Bost-Connes system and geometry over the "field with one element".
• Noncommutative Geometry and modular forms: limiting modular symbols and modular shadows, Connes-Moscovici modular Hecke algebras, 2-dimensional Q-lattices, complex multiplication and quantum statistical mechanics, function field case.
• The noncommutative geometry approach to the Riemann zeta function: the scaling Hamiltonian and prolate spheroidal wave functions, the Weil explicit formula as a trace formula, the Weil proof and the noncommutative geometry of the adele class space.

Outline of lectures

• Tuesday March 30: Bost-Connes algebra, generators and relations, time evolution, representations, symmetries, Q-lattices and commensurability, 1-dimensional Q-lattices.
• Thursday April 1: KMS states, model of the BC algebra over Z, BC endomotive, the field with one element, Soule's affine varieties over F1, the BC endomotive and extensions of F1, Borger's Lambda rings and F1, the BC algebra and Lambda rings
• Tuesday April 6 2-dimensional Q-lattices, coordinates, commensurability relation, groupoid, scaling action, convolution algebra, time evolution, symmetries, automorphisms and endomorphisms, modular field
• Thursday April 8 Modular field, generators and main properties, adelic description, noncommutative Shimura varieties, classification of KMS states of the GL(2) system
• Tuesday April 13 KMS states and measures, classification, arithmetic algebra of unbounded multipliers, action of symmetries on KMS states, intertwining with Galois action at zero-temperature, imaginary quadratic fields, 1-dimensional K-lattices, coordinates, commensurability, symmetries, the role of class number, classification of KMS states, comparison of systems
• Thursday April 20 Endomotives, Artin motives, provarieties, algebraic category of endomotives, Galois action, analytic category, state, Tomita-Takesaki theory and time evolution, low temperature KMS states and classical points, dual system, restriction map, abelian category of cyclic modules, morphisms, cokernel of the restriction map, cyclic homology.
• Thursday April 29 The dual system of the Bost-Connes endomotive, analytic properties and dual trace, the restriction map as an averaging map, the adelic form, the role of the adeles class space, the Riemann zeta function, counting primes and counting zeros, the Riemann asymptotic estimate, Siegel angular function, oscillatory term, analogy with Hamiltonian systems.
• Tuesday May 4 Scaling Hamiltonian, spectral projections, counting of quantum states and trace formula, principal values for local fields, counting of zeros, matching of cutoffs, restriction map (Hilbert space version), refined quantum system on the semilocal adele class space
• Thursday May 6 Modular curves and elliptic curves, invisible boundary, continued fraction expansion, shift map for finite index subgroups of the modular group, existence of a unique invariant measure, transfer operator and density as fixed point
• Tuesday May 11 Fixed point argument for the transfer operator and invariant measure, mixmaster cosmology models and geodesics on a modular curve, special solutions
• Thursday May 13 Lyapunov spectrum, limiting modular symbols, Perron-Frobenius and Ruelle operators, top eigenvalue and Lyapunov exponent, vanishing almost everywhere of limiting modular symbols, limiting modular symbols for quadratic irrationalities
• Thursday May 20 Level sets of the limiting modular symbols, modular forms on the boundary, Levy's lemma, identities of modular symbols and boundary integration, Ruelle transfer operator and the Selberg zeta function
• Tuesday May 25 Modular complex, relative homology of modular curves, Boundary NCG space, Pimsner six-terms exact sequence for K-theory of crossed products by groups acting on trees, K-theory and the modular complex; Noncommutative tori with real multiplication, solvgroups and 3-dimensional solvmanifolds, twisted group C* algebras, multipliers and SL(2) action, Shimizu L-function
• Thursday May 27 Homotopy quotiens in NCG (in the Baum-Connes sense), case of the 3-dimensional solvmanifold, Dirac operator, Connes-Landi deformations, induced Dirac operator on the fiber NC tori, Shimizu L-function, spectral triples and crossed products, Kronecker foliations and differential operators on NC tori
• Tuesday June 1 Unitary equivalences on spinors on the 3-dimensional solvmanifold, Shimizu L-function and eta function, Atiyah-Donnelly-Singer revisited, Lorentzian spectral triples, wave operator and norm on the real quadratic field, Krein spaces, Krein involution, arithmetic case and Galois involution, Lorentzian spectral triple and Krein spectral triple, arithmetic twisted group ring and Lorentzian spectral triple, Shimizu L-function revisited.

Seminar

• Thursday April 22: Branimir Cacic, Asymptotic spectral action on almost commutative geometries
• Wednesday May 19 (4:00-5:00): Ivan Dynov (York University), Type III von Neumann Algebras in the Theory of Infinite-Dimensional Groups
• Thursday May 20: Chris Estrada, The Standard Model in Noncommutative Geometry and Supersymmetry
• Thursday May 20 (extra talk 5:00-6:00): Snigdhayan Mahanta (Johns Hopkins University), Higher nonunital K-theory after Quillen
• Thursday May 27: Dori Bejleri, Geometry over the field with one element