Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems added, and references expanded. Appears in "Advances in Noncommutative Geometry - on the occasion of Alain Connes' 70th birthday

Notes on triangular sets and triangulation-decomposition algorithms II: Differential Systems

This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently