MAT1314HS:Introduction to Noncommutative Geometry

Instructor: Matilde Marcolli

(Georges Mathieu, "Vaires", 1965)

Brief Course Description

Slides of Classes

• Introduction to NCG and Physics Applications
• Introduction: What is Noncommutative Geometry?
• Finite Spectral Triples
• Models Based on Finite Spectral Triples
• The Spectral Geometry of the Standard Model
• The Spectral Action and the Standard Model
• The Higgs Mass in the Standard Model
• Spectral Triples and Pati-Salam GUT
• Spetral Triples and Supersymmetry
• The Spectral Action and Cosmic Topology
• Spectral Action and Packed Swiss Cheese Cosmology
• Spectral Action for Robertson-Walker metrics: Part I
• Spectral Action for Robertson-Walker metrics: Part II
• Arithmetic Structures in Spectral Models of Gravity
• Spectral Action of Bianchi IX Gravitational Instantons
• Noncommutative spaces: geometry and dynamics
• Noncommutative Geometry, Quantum Statistical Mechanics, and Number Theory
• Noncommutative Geometry and the Quantum Hall Effect
• Solvmanifolds and noncommutative tori with real multiplication

Reading Material

• pdf A.H.Chamseddine, A.Connes, "The Spectral Action Principle", Commun.Math.Phys.186:731-750,1997
• pdf B.Cacic, "A reconstruction theorem for almost-commutative spectral triples", Lett. Math. Phys. 100 (2012), no. 2, 181-202
• pdf W.D.van Suijlekom, "Renormalizability conditions for almost-commutative geometries", Ann. Henri Poincare' 15 (2014), no. 5, 985-1011
• pdf B.Iochum, C.Levy, "Spectral triples and manifolds with boundary", J. Funct. Anal. 260 (2011), no. 1, 117-134
• pdf M.A.Kurkov, F.Lizzi, M.Sakellariadou, A.Watcharangkool, "Spectral action with zeta function regularization", Phys.Rev. D91 (2015) 6, 065013
• pdf W.Nelson, J.Ochoa, M.Sakellariadou "Gravitational Waves in the Spectral Action of Noncommutative Geometry" Phys.Rev.D82:085021,2010
• pdf A.H.Chamseddine, A.Connes, "The Uncanny Precision of the Spectral Action", Commun.Math.Phys.293:867-897,2010
• pdf W.D. van Suijlekom, "Perturbations and operator trace functions", J.Funct.Anal. 260 (2011), no. 8, 2483-2496
• pdf F.Fathizadeh, A.Ghorbanpour, M.Khalkhali, "Rationality of Spectral Action for Robertson-Walker Metrics", JHEP 12 (2014) 064
• pdf W.Kalau, M.Walze, "Gravity, Non-Commutative Geometry and the Wodzicki Residue", J.Geom.Phys. 16 (1995) 327-344
• pdf F.Fathizadeh, M.Khalkhali, "The Algebra of Formal Twisted Pseudodifferential Symbols and a Noncommutative Residue", Lett.Math.Phys. 94 N.1 (2010) 41-61
• pdf J.W.Barrett, "Matrix geometries and fuzzy spaces as finite spectral triples", arXiv:1502.05383
• pdf G.V.Dunne, "Heat Kernels and Zeta Functions on Fractals", J.Phys. A45 (2012) 374016
• pdf S.Brain, B.Mesland, W.D.van Suijlekom, "Gauge Theory for Spectral Triples and the Unbounded Kasparov Product", arXiv:1306.1951
• pdf T. Krajewski, "Classification of Finite Spectral Triples", J.Geom.Phys. 28 (1998) 1-30
• pdf C.Connes, M.Marcolli, N.Ramachandran, "KMS states and complex multiplication", Selecta Math. (N.S.) 11 (2005), no. 3-4, 325-347
• pdf M.Laca, N.Larsen, S.Neshveyev, "On Bost-Connes type systems for number fields", J. Number Theory 129 (2009), no. 2, 325-338
• pdf C.Consani, M.Marcolli, "Quantum statistical mechanics over function fields" J. Number Theory 123 (2007), no. 2, 487-528
• pdf A.Connes, C.Consani, M.Marcolli, "Noncommutative geometry and motives: the thermodynamics of endomotives". Adv. Math. 214 (2007), no. 2, 761-831
• pdf A.Connes, C.Consani, M.Marcolli, "The Weil proof and the geometry of the adeles class space", in Algebra, Arithmetic, and Geometry: in honor of Yu. I. Manin. Vol. I, 339-405, Progr. Math., 269, Birkhaeuser, 2009
• pdf Yu.I.Manin, M.Marcolli, "Continued fractions, modular symbols, and noncommutative geometry", Selecta Math. (N.S.) 8 (2002), no. 3, 475-521
• pdf Yu.I.Manin, "Real Multiplication and noncommutative geometry", in The legacy of Niels Henrik Abel, 685-727, Springer, 2004

Schedule of Presentations

• Wednesday April 3
  • Matthew Sunohara "Obstructing Extensions of the Functor Spec to Noncommutative Rings"
  • Petr Kosenko "Noncommutative Gelfand transforms"
• Tuesday April 9
  • Daniel Nackan, "Spectral triples and manifolds with boundary"
  • Yung-Chuan Huang, "Formal Twisted Pseudodifferential Symbols and Noncommutative Residue"
  • Calder Morton-Ferguson, "A reconstruction theorem for almost-commutative spectral triples"
  • Yucong Jiang, "Noncommutative Poisson Algebras"
• Wednesday April 10
  • Anubhav Mathur, "Gravitational Waves in the Spectral Action"
  • David Urbanik, "KMS States and Complex Multiplication"