This review is based on two lectures given at the 2000 TMR school in Torino (TMR school
on contemporary String Theory and Brane Physics, 26 January-2 February 2000, Torino …

LECTURES ON GRADED DIFFERENTIAL ALGEBRAS AND NONCOMMUTATIVE GEOMETRY
Page 1 LECTURES ON GRADED DIFFERENTIAL ALGEBRAS AND NONCOMMUTATIVE …

We develop the formalism for noncommutative differential geometry and Riemmannian
geometry to take full account of the∗-algebra structure on the (possibly noncommutative) …

Recently Connes1 has proposed a definition of spectral triples, which is supposed to extend
the notion of the Riemannian geometry to a noncommutative setup. This could be a natural …

Spectral triples from bimodule connections and Chern connections Page 1 J. Noncommut. Geom.
11 (2017), 669–701 DOI 10.4171/JNCG/11-2-7 Journal of Noncommutative Geometry © …

We systematically study noncommutative and nonassociative algebras A and their
bimodules as algebras and bimodules internal to the representation category of a …

A general definition of a bimodule connection in noncommutative geometry has been
recently proposed. For a given algebra this definition is compared with the ordinary …

We show that the bicovariant first-order differential calculi on a factorisable quantum group
with the Peter-Weyl decomposition property are in 1− 1 correspondence with irreducible …

We review quantum gravity model building using the new formalism ofquantum Riemannian
geometry'to construct this on finite discrete spaces and on fuzzy ones such as matrix …

We show that if g Γ is the quantum tangent space (or quantum Lie algebra in the sense of
Woronowicz) of a bicovariant first-order differential calculus over a co-quasitriangular Hopf …