Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Measurement of the inclusive and differential t/t cross sections at 13 TeV in the all-jets boosted regime with CMS

The t/t production cross section is measured with the CMS detector in the all-jets channel in $pp$ collisions at the centre-of-mass energy of 13 TeV. The analysis is based on the study of t/t events in the boosted topology, namely events in which decay products of the quark top have a high Lorentz boost and are thus reconstructed in the detector as a single, wide jet. The data sample used in this analysis corresponds to an integrated luminosity of 2.53 fb-1. The inclusive cross section is found to be sigma(t/t) = 727 +- 46 (stat.) +115-112 (sys.) +- 20~(lumi.) pb, a value which is consistent with the theoretical predictions. The differential, detector-level cross section is measured as a function of the transverse momentum of the leading jet and compared to the QCD theoretical predictions. Finally, the differential, parton-level cross section is reported, measured as a function of the transverse momentum of the leading parton, extrapolated to the full phase space and compared to the QCD predictions.