Modifying the theory of gravity by changing independent variables

We study some particular modifications of gravity in search for a natural way to unify the gravitational and electromagnetic interaction. The certain components of connection in the appearing variants of the theory can be identified with electromagnetic potential. The methods of adding matter in the form of scalar and spinor fields are studied. In particular, the expansion of the local symmetry group up to $GL(2,C)$ is explored, in which equations of Einstein, Maxwell and Dirac are reproduced for the theory with Weyl spinor.Comment: LaTeX, 6 pages, based on a talk given at the XXth International Seminar on High Energy Physics (QUARKS-2018), Valday, Russia, May 27 - June 2, 201

Some remarks on sign changing solutions of a quasilinear elliptic equation in two variables

We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.Comment: In v2, we discuss and fix a gap that was found in our method. However, our results are weaker after this modification. Can be considered part II of arXiv:1205.0785, with which it overlap

Changing noncommuting variables with free analytic functions

The talk concerns inequalities for functions having matrix variables. The functions are typically (noncommutative) polynomials or rational functions. A focus of much attention are the inequalities corresponding to convexity. Since systems problems seldom produce an LMI directly it is important to have a theory for changing variables to produce an LMI or a theory of convex hulls. While this is hopeless for classical polynomials in commuting variables, there is some chance of an informative theory for matrix variables. For noncommuting variables this produces a wide range of subsidiary problems which need to be solved. The talk will describe some of these and what is known about them. Most of the work is done jointly with Meric Augat, Igor Klep and Scott A. McCullough who also will be speaking at IWOTA so our talks will be somewhat coordinated