Instructor: Matilde Marcolli

(Barbara Hepworth, Forms in Movement, 1942)

- Introduction and Physics Motivation
- Introduction to 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 NCG Standard Model
- Spectral Triples and Pati-Salam GUT
- Spectral Triples and Supersymmetry
- The spectral action and Cosmic Topology
- Spectral Action models of gravity and Packed Swiss Cheese Cosmology
- Spectral Action and Robertson-Walker Cosmologies, Part I
- Spectral Action and Robertson-Walker Cosmologies, 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

- January 5: Introduction and physics motivation: geometric models in physics; mathematical toolbox: manifolds, bundles, tensors, connections, curvature;action functionals in physics: classical mechanics, symmetries and conservation, Einstein-Hilbert action, gravity coupled to matter, Yang-Mills action; key idea of perturbative QFT, standard model of elementary particles
- January 7: Introduction to Noncommutative Geometry, equivalence relations as noncommutative spaces; tools from classical geometry: vector bundles as projective modules, connections; classical spin geometry and spectral triples
- January 12: finite spectral triples, finite metric spaces, bimodules, Morita equivalence, real structures, Krajewski diagrams, Barrett's random noncommutative geometries, random matrix theory
- January 14: random matrix theory, finite spectral triples and random matrix models, recursive Dyson-Schwinger equations, surface counting and topological recursion
- January 19: gauge networks, spin networks and gauge theories on a graph, quivers and category of finite spectral triples, configuration space, lattice field theory, Wilson action; finite spectral triples and the geometry of the Standard Model of particle physics
- January 21: finite spectral triples and the Standard Model of particle physics
- January 26: almost commutative geometries, the spectral action functional and the Standard Model Lagrangian
- January 28: Higgs mass in the NCG Standard Model, Pati-Salam grand unified theories
- February 2: Spectral action and Poisson summation formula, spheres and spherical space forms, tori and Bieberbach manifolds, inflation models with slow-roll potential from the spectral action, effect of different candidate cosmic topologies
- February 4: Packed Swiss Cheese cosmology model and multifractal cosmologies, Apollonian packings of spheres, spectral triples and Dirac operators of Apollonian packings, Dirac zeta function and poles and residues, zeta regularized terms in the spectral actiona nd log-periodic terms, effect on slow-roll inflation potentials
- February 9: Heat kernel expansion of the square Dirac operator on Robertson-Walker metrics, Feynman-Kac formula, Brownian motion and Brownian bridge, complete expansion in terms of Brownian bridge integrals and Bell polynomials
- February 11: Robertson-Walker metrics and multifractal structures (packed Swiss Cheese cosmology), heat kernel expansion, zeta regularized and log-periodic terms
- February 16: Robertson-Walker heat kernel expansion, algebraic change of variables and descriptio as period integrals, motives and periods, mixed Tate motives, proof that all terms in the expansion are periods of mixed Tate motives
- February 18: Bianchi IX metrics, Dirac operator, symbol, heat kernel expansion, algebraic change of variables, motives and periods, mixed Tate property, Bianchi IX gravitational instantons and Painleve VI equations, theta characteristics and parameterization of solutions
- February 23: noncommutative tori, Morita equivalences, real multiplication, noncommutative tori and elliptic curves; quantum statistical mechanical systems, KMS states, Q-lattices and commensurability, Bost-Connes system and 1-dimensional Q-lattices, convolution algebra of 2-dimensional Q-lattices
- February 25: GL2 quantum statistical mechanical system of 2-dimensional Q-lattices, representations, partition function, KMS states, modular field and arithmetic algebra, Galois symmetries, imaginary quadratic fields and maximal abelian extension, Hilbert's 12th problem and quantum statistical mechanics
- March 2: Quantum Hall effect, electrons in solids and Bloch theory, algebraic geometry of Fermi surfaces, discretization and Haper operator, group C* algebra and Brillouin torus, magnetic field and noncommutative torus
- March 4: Student presentations
- March 9: Student presentations

- J.C. Varilly, "An introduction to noncommutative geometry", European Mathematical Society, 2006.
- M. Khalkhali, "Basic noncommutative geometry", European Mathematical Society, 2013.
- W.D. van Suijlekom, "Noncommutative geometry and particle physics", Springer, 2015.
- A. Connes and M. Marcolli, "Noncommutative geometry, quantum fields and motives", American Mathematical Society, 2008.

- 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 E.Gesteau, "Renormalizing Yukawa interactions in the standard model with matrices and noncommutative geometry", J. Phys A Math. Theor. 54 (2021) 035203 (18pp)
- pdf M.Khalkhali, N.Pagliaroni, "Phase transition in random noncommutative geometries" arXiv:2006.02891
- pdf S.Azarfar, M.Khalkhali, "Random Finite Noncommutative Geometries and Topological Recursion", arXiv:1906.09362
- 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 F.Fathizadeh, Y.Kafkoulis, M.Marcolli, "Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson-Walker Cosmologies", Ann. Henri Poincare, 21 (2020), no. 4, 1329-1382
- pdf F.Fathizadeh, M.Marcolli, "Periods and motives in the spectral action of Robertson-Walker spacetimes", Comm. Math. Phys. 356 (2017), no. 2, 641-671
- 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

- March 4: Adam Artymowicz, Noncommutative geometry of the Integer Quantum Hall Effect
- March 4: Sunghyuk Park, Phase transition in random noncommutative geometries
- March 9: Aru Mukherjea, Heat kernel and zeta functions on fractals
- March 9: Dan Rostovtsev, Dirac spectra and cosmic topology
- March 9: Angus Gruen (asynchrounous), TBA