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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.

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 show that the Kervaire invariant one elements $\\theta _j \\epsilon \\pi _{2^{j+1}-2}S^0$ exist only for j ≤ 6. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras.

We endow the set of lattices in ${\\mathrm{\\mathbb{Q}}}_{\\mathrm{p}}^{\\mathrm{n}}$ with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in ℝ3 is at least 2π2. We prove this conjecture using the min-max theory of minimal surfaces.

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.

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.

In this paper, we prove that a set of N points in R2 has at least $c\\frac{N}{logN}$ distinct distances, thus obtaining the sharp exponent in a problem of Erdős. We follow the setup of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of point-line incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decomposition using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results or, alternatively, the points lie on the walls of the cells, in which case they are in the zero-set of a polynomial of surprisingly low degree, and we may apply the algebraic method. In order to control points incident to only two lines, we use the flecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.

The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.