Abstract: Since this year we shall be busy with curavture in noncommutative geometry, I thought I should start with the most fundamental classical incarnation of this notion: Gauss' theory of curvature for surfaces, and what it can teach us. All are welcome!

When Gauss, in his celebrated paper of 1827, {\it Disquisitiones generales circa superficies curvas} {(\it General investigations of curved surfaces)} after a long series of calculations

eventually showed that the extrinsically defined curvature of a

surface can be expressed entirely in terms of its intrinsic metric (= the

first fundamental form), he got so excited that he called the obvious corollary of this result Theorema Egregium (The Remarkable Theorem).

Gauss's formidable curvature formula, and the closely related {\it Gauss-Bonnet theorem} is the foundation stone for

all of differential geometry, as it was later shown by Riemann in 1859

that the curvature of higher dimensional manifolds can be understood

purely in terms of curvatures of its two dimensional submanifolds.

Theorema Egregium can also be regarded as the infinitesimal form of,

and in fact is equivalent to, the celebrated Gauss-Bonnet Theorem. This paper of Gauss is the single most important work in the entire history of differential geometry.

Abstract: In this talk, I will continue the lecture given by Masoud Khalkhali on our recent joint work on the Gauss-Bonnet theorem and scalar curvature for the noncommutative two torus, in the context of Alain Connes' noncommutative differential geometry. I will first construct the Connes-Tretkoff spectral triple encoding the metric information on this $C^*$-algebra so that we view it as a noncommutative Riemannian manifold equipped with a general metric. Then I will recall a spectral definition for its scalar curvature, and will illustrate the process of finding a local expression for the curvature by employing a special case of Connes' pseudodifferential calculus for $C^*$-dynamical systems by means of which one can pursue the heat kernel scheme of elliptic differential operators and index theory. I should mention that recently Connes and Moscovici also found precisely the same formula independently. At the end I will explain how this formula fits into our earlier work which extends the Gauss-Bonnet theorem of Connes and Tretkoff to general conformal structures on noncommutative two tori.