In this lecture we will describe our recent joint work with Simon Brendle (, ) in which we give the differentiable classification of compact Riemannian manifolds with pointwise 1/4-pinched curvature. The proof employs the Ricci flow to deform a 1/4-pinched metric to one of constant curvature unless the initial metric was locally isometric to a rank 1 symmetric space. This problem has a long history which we describe below. A related notion which first arose in the context of second variation of minimal surfaces is the notion of positive curvature on isotropic planes (PIC). One of the key steps of our proof is to show that this condition is preserved under the flow. We use this to define two stronger conditions which are also preserved, and which are strong enough to prove convergence of the flow. To handle the borderline case of weakly 1/4-pinching, we develop a new strong maximum principle which we use to show that if the flow does not immediately move into the interior of our preserved cone, then the holonomy of the initial metric must have been restricted. From there we are able to show that the initial metric was locally isometric to a rank 1 symmetric space. One of the basic problems of Riemannian geometry is the classification of manifolds of positive sectional curvature. Recall that the Riemann curvature tensor may be used to define various notions of positivity. The sectional curvature of a manifold at a point attaches a number to each two dimensional subspace of the tangent space; that is, the Gauss curvature at the point of the surface gotten as the exponential image of the two plane. (The other classical notions of positivity arise from the traces of the curvature tensor. These are positive defintie Ricci curvature and positive scalar curvature.) For two dimensional manifolds, the Gauss-Bonnet theorem implies that M must have positive Euler characteristic if it has a metric of positive curvature, and therefore it must be either the two sphere or the real projective plane. In general dimensions the known examples of manifolds of positive sectional curvature include the spherical space forms (Sn and its constant curvature quotients) which carry constant curvature metrics and the rank 1 symmetric spaces whose canonical metrics have sectional curvatures at each point varying between 1 and 4. Remarkably these are the only known examples in dimension greater than 24. In 1951 H. E. Rauch  introduced the notion of curvature pinching for Riemannian manifolds and posed the question of whether a simply connected compact manifold Mn whose sectional curvatures all lie in the interval [1, 4) is necessarily diffeomorphic to the sphere Sn. We will use the terminology that pinching means strict pinching, and will use the term weak pinching when we allow the equality case. This question has been known as the Differentiable Sphere Theorem, and the purpose of this lecture is to describe its proof as well as substantially more general results. We will say that a manifold has (pointwise) 1/4-pinched curvature if all sectional curvatures are positive and for any point p∈M the ratio of the maximum to the minimum sectional curvature at that point is less than 4; more precisely, for any pair of two planes P1 and P2 contained in the tangent space TpM we have 0<K(P1)<4K(P2). This is also equivalent to the condition that there is a positive continuous function κ on M so that at each point p we have K(p)∈[1/4κ(p), κ(p)) where K(p) denotes any sectional curvature at p. Our first main result is the following.
A new method called “L∞-Energy Method” is introduced. Several basic tools for this method are prepared and a couple of typical ways of usage of this method are exemplified for quasilinear parabolic equations.
In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i. i. d Poisson random variables with mean λ:=p(n-1). Motivated by this fact, we introduce the Poisson cloning model GPC(n, p) for random graphs in which the degrees are i. i. d Poisson random variables with mean λ. Then, we first establish a theorem that shows the new model is equivalent to the classical model G(n, p) in an asymptotic sense. Next, we introduce a useful algorithm, called the cut-off line algorithm, to generate the random graph GPC(n, p). The Poisson cloning model GPC(n, p) equipped with the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the t-core of G(n, p). This new approach to the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning models for random uniform hypergraphs and random k-SAT problems. Then, the t-core problem for random uniform hypergraphs and the pure literal algorithm for random k-SAT problems are analyzed.