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.

Machine learning approach to the extended Hubbard model

The 1d extended Hubbard model with soft-shoulder potential has proved itself to be very difficult to study due its non solvability and to competition between terms of the Hamiltonian. Given this, we tried to investigate its phase diagram for filling n=2/5 and range of soft-shoulder potential r=2 by using Machine Learning techniques. That led to a rich phase diagram; calling U, V the parameters associated to the Hubbard potential and the soft-shoulder potential respectively, we found that for V<5 and U>3 the system is always in Tomonaga Luttinger Liquid phase, then becomes a Cluster Luttinger Liquid for 57, with a quasi-perfect crystal in the U<3V/2 and U>5 region. Finally we found that for U<5 and V>2-3 the system shall maintain the Cluster Luttinger Liquid structure, with a residual in-block single particle mobility.