Time: 10:30
Speaker: Ali Fathi (Western)
Title: "Gauss-Bonnet Formula for Hypersurfaces"
Room: MC 108

Abstract: Gauss–Bonnet theorem or Gauss–Bonnet formula is one of the star attractions of modern differential geometry. It states that for a compact oriented manifold M, the "curvatura integra" over M is equal to a multiple of Euler Characteristic of M.

We shall give an "extrinsic" proof for M as an embedded submanifold (actually an even dimensional hyper surface)of a Euclidean space. The proof is heavily based on the Poincare-Hopf index theorem which states that the sum of indexes of a smooth vector field over M is equal to the Euler characteristic of M.

Time: 14:30
Speaker: Mohab Safey El Din (Universite Pierre et Marie Curie (Paris 6), INRIA Paris-Rocquencourt Research Centre)
Title: "Polynomial System Solving over the Reals: Algorithms, Complexity, Implementations and Applications"
Room: MC 320

Abstract: Solving non-linear algebraic problems is one of the major challenges in scientific computing. In several areas of engineering sciences, algebraic problems encode geometric conditions on variables taking their values over the reals. Thus, most of the time, one aims to obtain some informations on the real solution set of polynomial systems. The resolution of these problems often has a complexity which is exponential in the number of variables.

In this talk, I will review some geometric and algebraic techniques which enable to obtain fast practical algorithms meeting the best known complexity bounds. These algorithms are implemented in the maple package (RAGlib: The Real Algebraic Geometry library) which has the feature to provide algorithms of asymptotically optimal algorithms in real geometry. Its practical performances will be discussed and some applications will be presented.

This talk is based on joint work with J.C. Faugere, A. Greuet, E. Schost, PJ Spaenlehauer and L. Zhi.