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We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison–Singer problem. The first is Weaver's conjecture KS2, which is known to imply Kadison–Singer via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza et al. of Anderson's original paving conjecture(s), for which we are able to compute explicit paving bounds. The proof involves an analysis of the largest roots of a family of polynomials that we call the \"mixed characteristic polynomials\" of a collection of matrices.

We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves.

For a general Calderón—Zygmund operator T on ℝ N , it is shown that $\\normal{||Tf||_{L^{2}(w)} \\le C(T) \\cdot \\underset {Q}\\sup} \\LARGE{(} \\Huge{f}\\normal{_{Q^{w}} \\cdot} \\Huge{f}\\normal{_{Q^{w}}}\\small{^\\{-1}}\\LARGE{)}\\normal{\\cdot ||f||_{L^{2}(w)}}$ for all Muckenhoupt weights w ∈ A 2 . This optimal estimate was known as the A 2 conjecture. A recent result of Pérez—Treil—Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov—Treil—Volberg method of random dyadic systems with just one random system and completely without \"bad\" parts; (ii) a resulting representation of a general Calderón—Zygmund operator as an average of \"dyadic shifts;\" and (iii) improvements of the Lacey—Petermichl—Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

In this paper we prove the conjecture of Korevaar and Meyers: for each N ≥ c d t d , there exists a spherical t-design in the sphere S d consisting of N points, where c d is a constant depending only on d.

Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is one of the most fundamental questions about singularities. It is this long standing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities.

We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f(x) + Q(x, v) + g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to \"metrically regular\" problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to Q(x, y) = || JC — y||² and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with \"regular\" intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l¹ algorithm for compressive sensing and an application to rank reduction problems.

Alternating direction methods (ADMs) have been well studied in the literature, and they have found many efficient applications in various fields. In this note, we focus on the Douglas-Rachford ADM scheme proposed by Glowinski and Marrocco, and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number. The linearized version of this ADM scheme, which is known as the split inexact Uzawa method in the image processing literature, is also discussed.

We derive the sharp constants for the inequalities on the Heisenberg group ℍ n whose analogues on Euclidean space ℝ n are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. By considering limiting cases of these inequalities sharp constants for the analogues of the Onofri and log-Sobolev inequalities on ℍ n are obtained. The methodology is completely different from that used to obtain the ℝ n inequalities and can be (and has been) used to give a new, rearrangement free, proof of the HLS inequalities.

We show that the number of birational automorphisms of a variety of general type X is bounded by c • vol(X, K x ), where c is a constant that only depends on the dimension of X.

We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometries.