Abstract:
We review the basics of the two most widely used approaches to Lorentz violation - the Stardard Model Extension and Noncommutative Field Theory - and discuss in some detail the example of the modified spectrum of the synchrotron radiation. Motivated by touching upon such a fundamental issue as Lorentz symmetry, we ask three questions: What is behind the search for Lorentz violation? Is String Theory a physical theory? Is there an alternative to Supersymmetry?

Abstract:
A reliable method to construct Supersymmetric Noether currents is presented. As the most important application the central charge of the N=2 Supersymmetric Yang-Mills effective theory, known as Seiberg-Witten (SW) theory, is computed. The analisys is carried out in the SW low energy U(1) effective Sector, as well as in the SW high energy SU(2) effective Sector.

Abstract:
The central charge for the Seiberg-Witten low-energy effective Action is computed using Noether supercharges. A reliable method to construct supersymmetric Noether currents is presented.

Abstract:
The conformal invariance of the low energy limit theory governing the electronic properties of graphene is explored. In particular, it is noted that the massless Dirac theory in point enjoys local Weyl symmetry, a very large symmetry. Exploiting this symmetry in the two spatial dimensions and in the associated three dimensional spacetime, we find the geometric constraints that correspond to specific shapes of the graphene sheet for which the electronic density of states is the same as that for planar graphene, provided the measurements are made in accordance to the inner reference frame of the electronic system. These results rely on the (surprising) general relativistic-like behavior of the graphene system arising from the combination of its well known special relativistic-like behavior with the less explored Weyl symmetry. Mathematical structures, such as the Virasoro algebra and the Liouville equation, naturally arise in this three-dimensional context and can be related to specific profiles of the graphene sheet. Speculations on possible applications of three-dimensional gravity are also proposed.

Abstract:
In the first attempt to introduce gauge theories in physics, Hermann Weyl, around the 1920s, proposed certain scale transformations to be a fundamental symmetry of nature. Despite the intense use of Weyl symmetry that has been made over the decades, in various theoretical settings, this idea never found its way to the laboratory. Recently, building-up from work by Lochlainn O'Raifeartaigh and collaborators on the Weyl-gauge symmetry, applications of Weyl-symmetry to the electronic properties of graphene have been put forward, first, in a theoretical setting, and later, in an experimental proposal. Here I review those results, by enlarging and deepening the discussion of certain aspects, and by pointing to the steps necessary to make graphene a testing ground of fundamental ideas.

Abstract:
A recently proposed step-by-step procedure, to merge the low-energy physics of the $\pi$-bonds electrons of graphene, and quantum field theory on curved spacetimes, is recalled. The last step there is the proposal of an experiment to test a Hawking-Unruh effect, emerging from the model, that manifests itself as an exact (within the model) prediction for the electronic local density of states, in the ideal case of the graphene membrane shaped as a Beltrami pseudosphere. A discussion about one particular attempt to experimentally test the model on molecular graphene is presented, and it is taken as an excuse to solve some basic issues that will help future experiments. In particular, it is stated that the effect should be visible on generic surfaces of constant negative Gaussian curvature, that are infinite in number.

Abstract:
Two recent investigations are reviewed: quantum effects for DNA aggregates and scars formation on virus capsids. The possibility that scars could explain certain data recently obtained by Sundquist's group in electron cryotomography of immature HIV-1 virions is also briefly addressed. Furthermore, a bottom-up reflection is presented on the need to invent new physics to pave the way to a rigorous physical theory of biological phenomena. Our experience in the two researches presented here and our personal interpretation of Schroedinger's vision are behind the latter request.

Abstract:
We discuss the r\^ole of quantum deformation of Weyl-Heisenberg algebra in dissipative systems and finite temperature systems. We express the time evolution generator of the damped harmonic oscillator and the generator of thermal Bogolubov transformations in terms of operators of the quantum Weyl-Heisenberg algebra. The quantum parameter acts as a label for the unitarily inequivalent representations of the canonical commutation relations in which the space of the states splits in the infinite volume limit.

Abstract:
We find that, for a very specific shape of a monolayer graphene sample, a general relativistic-like description of a back-ground spacetime for graphene's conductivity electrons is very natural. The corresponding electronic local density of states is of finite temperature. This is a Hawking-Unruh effect that we propose to detect through an experiment with a Scanning Tunneling Microscope.

Abstract:
We study the canonical quantization of the damped harmonic oscillator by resorting to the realization of the q-deformation of the Weyl-Heisenberg algebra (q-WH) in terms of finite difference operators. We relate the damped oscillator hamiltonian to the q-WH algebra and to the squeezing generator of coherent states theory. We also show that the q-WH algebra is the natural candidate to study thermal field theory. The well known splitting, in the infinite volume limit, of the space of physical states into unitarily inequivalent representations of the canonical commutation relations is briefly commented upon in relation with the von Neumann theorem in quantum mechanics and with q-WH algebra.