Introduction to Noncommutative (Riemannian Spin) Geometry
Prof. Ludwik Dąbrowski
SISSA, 2011

Description:
These lectures concentrate on the latest layer of Noncommutative Geometry:
Riemannian and Spin. It is encoded in terms of a spectral triple
and its main ingredient, the Dirac operator.
The canonical spectral triple on a Riemannian and spin manifold will be
described starting with basic notions of multilinear algebra
and differential geometry. Its basic properties, and then additional
requirements that completely characterize this operator will be presented.
They are essential for a further fascinating generalization to
noncommutative spaces by A. Connes.
In the course of the account some previous levels of NCG will be mentioned
regarding the (differential) topology and calculus (like the equivalence
between (locally compact) topological spaces and C*algebras, and between
vector bundles and finite projective modules, projectors and K theory,
the Hochschild and cyclic cohomology, noncommutative integral,
pseudodifferential calculus and local index formula).
In the last part the concept of symmetries will be described in terms
of Hopf algebras and quantum groups and applied to equivariant spectral
triples. By concentrating the material in relatively few lectures some
well established topics (e.g. the index theory) will not be discussed.
Just an indispensable minimum from the well known theory of the (elliptic)
Laplace operator will be used. Such a selection among the wealth
of available material hopefully will lead us fast to some of the active
and interesting fields of current research.

Plan:
1. Introduction.
2. Exterior and Clifford algebras. Spin groups. Spinors.
3. Spin structures.
4. Dirac operator.
5. Some analytic properties. Spectral triple.
6. Other (seven) characteristic features:
 dimension (finite summability)
 regularity (smoothness)
 finiteness & projectivity
 reality
 first order
 orientation
 Poincare duality
7. Statement of the 'reconstruction theorem' of A. Connes.
8. Other notions in N.C. vein: bundles, connections,
pseudodifferential calculus, geodesic flows...
If time permits:
9. Examples: noncommutative tori, spheres, finite dimensional spectral triple
10. Symmetries
 Group actions (isometries, diffeomorphisms)
 Hopf algebras and equivariant spectral triples

References
1. A. Connes, Noncommutative Geometry, Academic Press, 1994.
2. A. Connes, Gravity coupled with matter and foundation of noncommutative
geometry, Commun. Math. Phys., 182 (1996) 155176.
3. A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields
and Motives http://www.alainconnes.org/en/downloads.php
4. L. Dabrowski, Group Actions on Spinors, Lecture Notes Bibliopolis,
Napoli, 1988
5. T. Friedrich, Dirac operators in Riemannian Geometry, Graduate Studies
in Mathematics, vol 25. AMS, Providence, Rhode Island, 2000
6. J. M. GraciaBondia, J. C. Varilly, H. Figueroa,
Elements of Noncommutative Geometry, Birkhauser Advanced Texts,
Birkhauser, Boston, MA, 2001.
7. P. Gilkey, Invariance Theory, the Heat Equation, and the AtiyahSinger
Index Theorem, CRC Press, Boca Raton, USA, 1995.
8. N. Higson, The residue index theorem (Lecture notes for the 2000
Clay Institute symposium on NCG); The local index formula
in noncommutative geometry (Trieste lecture notes);
http://www.math.psu.edu/higson/ResearchPapers.html.
9. G. Landi, An Introduction to Noncommutative Spaces and their Geometries,
Springer, Lecture Notes in Physics, 1997.
10. H. Lawson, M. Michelsohn, Spin geometry, Princeton University Press 1989.
11. R.J. Plymen, Strong Morita equivalence, spinors and symplectic spinors,
J. Oper. Theory 16