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**ABSTRACT:**This is a survey of recent results on manifolds with positive curvature from a series of lecture given in Guanajuato, Mexico in 2010. It also contains some hitsorical comments.10/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, is strictly positive; and this can be done with an arbitrarily small lower bound on the distance between the planes considered. Although they have positive Ricci curvature, these metrics do not have nonnegative sectional curvature. Such metrics also have positive biorthogonal curvature, meaning that the average of sectional curvatures of any two orthogonal planes is positive.09/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**Manifolds admitting positive sectional curvature are conjectured to have a very rigid homotopical structure and, in particular, comparatively small Euler charateristics. However, this structure is still highly speculative and best results in this direction are known under the assumption of large isometric torus actions. In this article, we obtain upper bounds for Euler characteristics of closed manifolds that admit metrics with positive curvature and isometric torus actions. We apply our results to prove obstructions to symmetric spaces and manifold products and connected sums admitting positively curved metrics with symmetry, providing evidence for a conjecture of Hopf. We also derive vanishing properties of the elliptic genus of positively curved manifolds.02/2013;

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