- Operator algebras:
- 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 duality.

- The geometry and arithmetic of noncommutative tori:
- Moduli, noncommutative tori and elliptic curves, the noncommutative geometry at the boundary of modular curves, real multiplication.

- Cyclic co/homology:
- The Connes bicomplex, Chern character, index theorems, cyclic modules and the cyclic category.
- 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.

- 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 planar algebras"
- 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 and supergeometry"

- 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 mass terms
- 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

- 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