MAS6396: NONCOMMUTATIVE GEOMETRY
Spring 2008, FSU Math Department, TuesdayThursday 3:35pm, LOV200
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 ConnesKreimer theory
 Feynman motives
 Equisingular connections and the RiemannHilbert 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 BostConnes 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 ConnesKreimer 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), timeordered 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 Wequivalent 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 LeftRight symmetric algebra

Lecture 11: February 19, 2008
Odd bimodules, representations of the leftright symmetric algebra,
generations and particles as basis elements, real structure and grading,
KOdimension and metric dimension

Lecture 12: February 21, 2008
Dirac operator and the orderone 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 1014: Spring Break

Lecture 17: March 18, 2008 Adeles and ideles, Qlattices,
commensurability relation, the BostConnes algebra and its time
evolution

Lecture 18 and 19: March 20 and 25, 2008 KMS states of the
BostConnes system, 2dimensional Qlattices 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 alYasry, Feb 28, 2008)
Research Seminar
The course will be accompanied by a Research Seminar:
 Thursday Jan 24, LOV200, 5pm: Bryan J. Fields (FSU Physics)
Computing Feynman Diagrams and One loop quarkquarkgluon 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 TorresRuiz (MPI)
A lenient introduction to smooth 4manifolds
 Thursday Feb 28, LOV104 (NOTICE: CHANGE OF ROOM!),
2pm: Rafael TorresRuiz (MPI) Exotic R^4's
(An overview of the main features/distinctions of exotic R^4's)
 Thursday March 6, 2pm LOV104: Ahmed Zainy alYasry (MPI)
Khovanov homology
 Tuesday March 18, 2pm LOV104: Ahmed Zainy alYasry (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) KKtheory and a Representation Theoretic Noncommutative
Correspondence Category
 Tuesday March 25, 2pm LOV104: Ahmed Zainy alYasry (MPI)
Khovanov homology, III
 Wednesday March 26, 3:35pm LOV104: Bram Mesland (MPI)
The noncommutative geometry of SL(2,Z)
Textbook
The class will follow the book:
*
A.Connes, M.Marcolli: Noncommutative Geometry, Quantum Fields and Motives.
American Mathematical Society, Colloquium Publications Vol.55, January 2008.
Additional reading material used in the class will be listed below.
Survey Articles
Feynman motives and graph hypersurfaces:
Motives associated to graphs (Spencer Bloch)
Renormalization and motives:
Quantum Fields and Motives
(ConnesMarcolli)
Hopf algebras in physics:
Hopf algebra structures in
particle physics (Stefan Weinzierl)
General Bibliographical References
Quantum Field Theory:
An introductory reading:
 A.Zee, "Quantum Field Theory in a nutshell" Princeton University Press.
More detailed books on QFT (if you are serious about it):
 Claude Itzykson and JeanBernard Zuber "Quantum Field Theory", Dover.
 James D Bjorken and Sidney D. Drell "Relativistic Quantum Mechanics" and "Relativistic Quantum Fields", McGrawHill.
Affine group schemes and commutative Hopf algebras
 W.C.Waterhouse, Introduction to affine group schemes,
Graduate Texts in Mathematics, Springer Verlag, 1979.
Renormalization
 J.Collins, Renormalization, Cambridge University Press, 1984.