Abstract: I will continue last week's talk by listing the axioms for the topological index, and then sketching how these axioms can be used to prove the index theorem (with emphasis on the sketching).

Abstract: I shall first recall the classical theory of Ray-Singer analytic torsion, and conformal anomaly, for families of elliptic operators. I will mostly focus on families of elliptic operators on Riemann surfaces. The methods used here are based on ideas of spectral geometry and hence stand a chance of extension to a noncommutative setting. The extensions, when possible, are however quite nontrivial and involve many new elements and difficult computations. I shall then look at some known examples of noncommutative Riemann surfaces, the noncommutative elliptic curves equipped with curved metrics, and sketch the progress made in the last few years in understanding their conformal and spectral geometry. Scalar curvature can be defined by study of special values of spectral zeta functions. In particular I shall explain a formula for scalar curvature obtained in my joint work with F. Fathizadeh (and independently by Connes and Moscovici). This formula plays an important role for further study of noncommutative spectral geometry of noncommutative tori.