This is the text of a series of three lectures on the recent solution of the square root problem for divergence form elliptic operators, a long-standing conjecture posed by Kato in the early 60’s. In this text, the motivations for this problem and its setting are given. The ideas from harmonic analysis on the T(1) theorem and T(b) theorem for square functions are described. In particular, an apparently new formulation of a local T(b) theorem for square functions is stated. The ideas of the full proof are presented.

In these notes we discuss some recent developments in the study of global solutions to dispersive equations, especially for low regularity data.

First, I want to present some questions on $L^2-$ harmonic forms on non-compact Riemannian manifolds. Second, I will present an answer to an old question of J. Dodziuk on $L^2-$ harmonic forms on manifolds with flat ends. In fact some of the analytical tools presented here apply in other situations (see [C4]).

The first two lectures are devoted to describing the basic concepts of scattering theory in a very compressed way. A detailed presentation of the abstract part can be found in [33] and numerous applications in [30] and [36]. The last two lectures are based on the recent research of the author.

Quantum Switch is one of most important elements of quantum networks, see, for instance [53, 54, 55]. In the present paper mathematical modeling of the Resonance Quantum Switch is reduced to the solution of the corresponding Scattering Problem on a network consisting of a quantum well and few quantum wires attached to it. In a simplest practical problem the roles of quantum wires are played by straight rectangular channels and the role of the quantum well is played by a circular domain, with the boundary having no flat pieces to match the rectangular channels. We suggest a procedure of matching the rectangular channels and a general convex domain with piecewise smooth boundary. The matching problem is reduced to some finite-dimensional equation with matrix-functions. Important spectral characteristics of the above scattering problem such as positions and life-times of resonances may be found numerically from the corresponding determinant condition or via asymptotic analysis — for sufficiently narrow channels.