Ma 140 b: Noncommutative Geometry, Part I
Winter 2010, Caltech Math Department, SLN 153, Tuesday-Thursday 1:00-2:30 pm.
Brief Course Description
This will be a first introductory course
on noncommutative geometry, which will
include a quick review of the needed
notions from operator algebra and a detailed
discussion of the main tools such as cyclic
cohomology, spectral triples, KMS states,
and with a focus on examples of noncommutative
spaces of relevance to physics.
140a is recommended, but
a course in quantum mechanics also constitutes
a valid prerequisite: both mathematics and
physics students are encouraged to attend.
- C* algebras, Gelfand-Naimark, topological noncommutative
spaces, crossed product C* algebras and noncommutative
quotients, K-theory of C* algebras, Hilbert modules,
Fredholm modules and KK-theory, strong Morita equivalence.
- von Neumann algebras, an outline of the theory of factors,
quantum statistical mechanics and KMS states, Connes-Takesaki
The geometry and arithmetic of noncommutative tori:
- Moduli, noncommutative tori and elliptic curves,
the noncommutative geometry at the boundary of
modular curves, real multiplication.
- The Connes bicomplex, Chern
character, index theorems, cyclic modules and the
- The Connes-Chern character and noncommutative
geometry models of the quantum Hall effect,
quasicrystals and aperiodic solids.
Metric noncommutative geometry:
- Spectral triples, dimension spectrum, finite
and theta summability, zeta functions, the
spectral action functional, asymptotic expansion.
- Examples of spectral triples: manifolds, fractals,
quantum groups, noncommutative tori, particle
physics models, cosmological applications,
spectral triples in quantum gravity.
Noncommutative spaces and dynamical systems,
with applications to arithmetic geometry.
The course is accompanied by a research seminar dedicated to
various aspects of noncommutative geometry. The seminar meets Thursdays 3:30-4:30pm in SLN 151.
- Thursday January 14: Organizational meeting
- Thursday January 21: Vasiliy Dolgushev (UCR) "Hochschild
complexes and noncommutative Cartan calculus"
- Thursday January 28: Dapeng Zhang "Twisted K-theory and Chern character"
- Thursday February 4: Ozgur Ceyhan (Amsterdam) "Open string theory and
- Tuesday Feb 9 (special time: 1:00pm in 153 SLN) Dapeng Zhang
"Twisted K-theory and Chern character, Part II"
- Thursday Feb 11 (special time: 1:00pm in 153 SLN) Rafael Torres
"An introduction to spin geometry"
- Thursday Feb 18: Kevin Teh "The spectral action on SU(2)/Q8"
- Thursday Feb 25: Domenic Denicola "Spectral action with torsion"
- Thursday March 4: Branimir Cacic "Almost commutative geometries
Outline of classes
- Tuesday January 5 : Introduction to NCG, NCG and quantum mechanics,
NCG spaces as algebras and categories, commutative spaces,
the Gelfand-Naimark theorem.
- Thursday January 7 : Gelfand-Naimark theorem (continued),
quotient spaces in NCG, group C* algebras, Pontrjagin duality and NCG,
maximal and reduced norm, C* dynamical systems,
crossed product algebras.
- Tuesday January 12 : Crossed product algebras, groupoid
algebras, the noncommutative torus as a crossed product,
noncommutative tori and elliptic curves, semigroup rings.
- Thursday January 14: creation/annihilation operators,
Toeplitz and Cuntz algebras, graphs as noncommutative spaces,
semigroupoids, algebras from categories; states, the space
of states, pure states, GNS representation, irreducible representations.
- Tuesday January 19: Time evolutions, Gibbs states, KMS condition, equilibrium states, KMS states on Toeplitz and Cuntz algebras, ground states and zero temperature KMS states, symmetries.
- Thursday January 21: von Neumann algebras, double commutant theorem, abelian case and measure spaces, factors, projections and von Neumann dimension, types, Tomita theory, a sketch of Connes' classification
- Tuesday January 26: Hilbert modules, adjointable maps, endomorphisms, inite rank and compact endomorphisms, finite projective modules, Serre-Swan theorem
- Thursday January 28: bimodules, strong Morita equivalence, finite projective modules on noncommutative tori.
- Tuesday February 2: Morita equivalences on noncommutative tori,
AF algebras, embedding of noncommutative tori in AF algebras, range
of the trace.
- Thursday February 4: Noncommutative geometry models of the Quantum Hall Effect.
- Tuesday February 16: Hochschild complex and cohomology, cyclic cohomology,
cyclic homology, group cohomology and Hochshild cohomology, Connes exact sequence, the b-B bicomplex, the case of manifolds and noncommutative tori, Chern characters
- Thursday February 18: K-theory, pairing with cyclic cohomology,
Chern character, even and odd case, periodicity map, periodic cyclic
cohomology, K-homology and Fredholm modules, pairing with cyclic homology,
Chern character, index theorem, spectral triples, notions of dimension
- Tuesday February 23: Spectral triples for spin manifolds, distance,
product geometries, differential forms, spectral triples on noncommutative
tori, isospectral deformations
- Thursday February 25:
Local index formula, Morita equivalences and spectral triples,
connections, self-Morita equivalences and inner fluctuations, the spectral
action functional, asymptotic expansion, gauge transformations
- Tuesday March 2:
Real structures, KO dimension on spectral triples, real subalgebra,
order one condition, finite noncommutative geometries,
left-right symmetric algebra, bimodules, odd bimodules
- Tuesday March 2:
Representation, grading and real structure for the left-right symmetric
algebra, generation, matter/antimatter, fermions, Dirac operators on the finite geometry and the order one condition
- Thursday March 4:
Left-right symmetry breaking, subalgebra and order one condition,
hypercharges and gauge groups
- Thursday March 4:
classification of Dirac operators, moduli space and Yukawa parameters, Majorana
- Tuesday March 9:
Product geometry, inner fluctuations, gauge bosons, Higgs boson, hypercharges,
spectral action functional on the product geometry
Thursday March 11: asymptotic expansion, gravitational terms,
fermionic part, Grassmann variables and Pfaffian, running of
coefficients witht the renormalization group, cosmological applications
Suggested reading material
- M.Khalkhali, "Basic Noncommutative Geometry", EMS 2009
- J.Varilly, "An introduction to noncommutative geometry", EMS 2007
- A.Connes, "Noncommutative Geometry", Academic Press, 1994
- A.Connes, M.Marcolli, "Noncommutative geometry, quantum fields,
and motives" AMS 2008
- K.Davidson, "C* algebras by example", Fields Institute, 1996
- P.Fillmore, "A user's guide to operator algebras", Wiley, 1996