MAS6396: NONCOMMUTATIVE GEOMETRY

Important announcements

• The LAST lecture of MAS6396 will take place on March 27.

• The first lecture of MAS6396 will take place on January 15 (instead of January 8).

Topics

• Hopf algebras and perturbative renormalization
• The Connes-Kreimer theory
• Feynman motives
• Equisingular connections and the Riemann-Hilbert correspondence
• Noncommutative spectral manifolds, real structures
• Finite geometries: moduli spaces
• The Standard Model of elementary particles
• The spectral action and the Standard Model
• Quantum Statistical Mechanics and KMS states
• The Bost-Connes system
• The GL(2)-system and modular functions
• Quantum Statistical Mechanics of Shimura varieties
• Quantum Statistical Mechanics and imaginary quadratic fields
• Quantum Statistical Mechanics over function fields
• Noncommutative tori with real multiplication
• The Riemann zeta function and quantum mechanics
• Adeles, ideles and gauge theory
• Endomotives
• The dual system and Quantum Statistical Mechanics
• Frobenius and scaling
• The spectral realization
• The Weil explicit formula as a trace formula
• The Weil proof and the adeles class space
• Quantum Gravity and Number Theory: analogies

Lectures

• Lecture 1: Jan 15, 2008 General overview
• Lecture 2: January 17, 2008 Feynman graphs and Feynman rules
• Lecture 3: January 22, 2008 Schwinger parameters, dimensional regularization, Feynman trick, graph hypersurfaces, BPHZ renormalization
• Lecture 4: January 24, 2008 Affine group schemes and Hopf algebras, the Connes-Kreimer Hopf algebra, Lie algebra
• Lectures 5 and 6: January 28 and 31, 2008 Birkhoff factorization in Lie groups, BPHZ as a Birkhoff factorization, the renormalization group and the beta function
• Lecture 7: February 5, 2008 The renormalization group, Gross-'t Hooft relations (counterterms and beta function), time-ordered exponential and differential equations, connections: gauge equivalence and Birkhoff factorization
• Lecture 8: February 7, 2008 From time ordered exponentials to differential systems, gauge equivalence and Birkhoff factorization, flat equisingular connections, flat equisingular vector bundles and W-equivalent connections, the universal group U* and the renormalization group
• Lecture 9: February 12, 2008 Tannakian category of flat equisingular vector bundle, equivalence with the category of representations of the universal affine group scheme U^*, generators of Lie(U) and beta functions
• Lecture 10: February 14, 2008 Introduction to spectral triples in noncommutative geometry, real structures, Morita equivalence and inner fluctuations, the Left-Right symmetric algebra
• Lecture 11: February 19, 2008 Odd bimodules, representations of the left-right symmetric algebra, generations and particles as basis elements, real structure and grading, KO-dimension and metric dimension
• Lecture 12: February 21, 2008 Dirac operator and the order-one condition: subalgebra, action of the subalgebra on the fermions, hypercharges of fermions, classification of Dirac operators
• Lecture 13: February 26, 2008 Moduli space of Dirac operators: CKM and PMNS matrices, Majorana mass terms for right handed neutrinos
• Lecture 14: February 28, 2008 Inner fluctuations and the SM bosons, the spectral action and the asymptotic expansion, bosonic part of the SM Lagrangian
• Lecture 15: March 4, 2008 The spectral action and YM coupled to gravity, the fermionic part of the action, gravity coupled to matter, normalization of kinetic terms and merging of the gauge coupling constants, mass relation at unification, top quark and Higgs masses via the renormalization group equation
• Lecture 16: March 6, 2008 Quantum Statistical Mechanics: algebra of observables, time evolution, Hamiltonian, states, equilibrium states, KMS condition
• March 10-14: Spring Break
• Lecture 17: March 18, 2008 Adeles and ideles, Q-lattices, commensurability relation, the Bost-Connes algebra and its time evolution
• Lecture 18 and 19: March 20 and 25, 2008 KMS states of the Bost-Connes system, 2-dimensional Q-lattices and their quantum statistical mechanics.
• Last lecture: March 27, 2008 The modular field, the arithmetic algebra of the GL(2)-system, KMS states at zero temperature and embeddings of the modular field in C; endomotives: algebraic and analytic version, Galois action, states and time evolution, classical points, dual system and the scaling action, the restriction map, cokernel and cyclic homology, scaling action and the spectral realization of zeros of zeta.

Other

• Photo (taken by Ahmad Zainy al-Yasry, Feb 28, 2008)

Research Seminar

• Thursday Jan 24, LOV200, 5pm: Bryan J. Fields (FSU Physics) Computing Feynman Diagrams and One loop quark-quark-gluon vertex corrections
• Thursday Jan 31, LOV104, 2pm: Bram Mesland (MPI) Limit sets and noncommutative geometry
• Wednesday Feb 6, LOV104, 3:35 pm: Bram Mesland (MPI) Limit sets and noncommutative geometry II
• Wednesday Feb 13, LOV104, 3:35pm: Bram Mesland (MPI) Limit sets and noncommutative geometry III
• Thursday Feb 21, LOV201 (NOTICE: CHANGE OF ROOM!) 2pm: Rafael Torres-Ruiz (MPI) A lenient introduction to smooth 4-manifolds
• Thursday Feb 28, LOV104 (NOTICE: CHANGE OF ROOM!), 2pm: Rafael Torres-Ruiz (MPI) Exotic R^4's (An overview of the main features/distinctions of exotic R^4's)
• Thursday March 6, 2pm LOV104: Ahmed Zainy al-Yasry (MPI) Khovanov homology
• Tuesday March 18, 2pm LOV104: Ahmed Zainy al-Yasry (MPI) Khovanov homology, II
• Wednesday March 19, 3:35pm LOV104: Ivan Dynov (MPI) Representations of infinite dimensional nilpotent groups and von Neumann algebras
• Thursday March 20, 2pm LOV104: Snigdhayan Mahanta (University of Toronto) KK-theory and a Representation Theoretic Noncommutative Correspondence Category
• Tuesday March 25, 2pm LOV104: Ahmed Zainy al-Yasry (MPI) Khovanov homology, III
• Wednesday March 26, 3:35pm LOV104: Bram Mesland (MPI) The noncommutative geometry of SL(2,Z)

Textbook

* A.Connes, M.Marcolli: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, Colloquium Publications Vol.55, January 2008.

Survey Articles

• A.Zee, "Quantum Field Theory in a nutshell" Princeton University Press.

• Claude Itzykson and Jean-Bernard Zuber "Quantum Field Theory", Dover.
• James D Bjorken and Sidney D. Drell "Relativistic Quantum Mechanics" and "Relativistic Quantum Fields", McGraw-Hill.

• W.C.Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, Springer Verlag, 1979.

• J.Collins, Renormalization, Cambridge University Press, 1984.