The research carried out at the University of Udine within the fields of Mathematical Physics and Theoretical Physics have been focused on peculiar aspects of geometric theories of spacetime, and on studying spontaneous symmetry breakings, by using algebraic methods.

A fascinating aspect of relativity is the fact that it provides a geometric description of spacetime. This feature of the theory offers countless opportunities for both conceptual and mathematical studies. At the University of Udine, a few fundamental aspects of the Lorentz invariance were studied, with particular attention for the possibilities of a violation of the same over short distances, that show up in a natural way in many quantum theories of gravitation, and to the appearance of pseudo-Riemannian structures of non-gravitational physics theories (analogue models). Moreover, both the formal and conceptual fundamentals of general relativity are studied in depth, as well as those of other theories of spacetime based on some metrics, and possible alternatives are studied, basing them on formalisms inspired by non-abelian gauge theories. Although this type of activities is about classical theories, the medium-term objective is to better understand their structure, with a view to a possible quantisation. In this context, another research thread is inserted, concerning the study of the quantum field theory in curved spacetimes, in particular the Hawking effect, both for black holes having a gravitational origin and for analogue models.

In addition to Einstein's Theory, a few gravitational theories were recently developed that are extensions of general relativity. These models were often stimulated by motives of a phenomenological nature: for instance, the naturality in the description of the accelerating expansion of the universe, both the early-time one (inflation), and the late-time one. At the same time, however, they also present interesting features also from a strictly theoretical/mathematical point of view: for instance, field equations can be over the second order (an atypical case in physics), and instabilities of several types (such as, for example, ghost 'instabilities') can be present, not always in an evident form. Within this context, the intention is to study in detail a few of these theories (for instance, f(R)-gravity, bigravity), both with regard to the technical aspects linked with their formulation, and in relation with any possible applications, predominantly (but not only) cosmological ones. In this field, too, the studies will start with the analysis of the "non-quantum" environment, with the idea to take into account the quantisation, too, at a later time. Moreover, it is expected that some of the problems mentioned above (for instance, the Hawking effect and the thermodynamics of black holes) can be reviewed within this context.

As to the second research thread, the geometric properties of the spaces of compact groups' orbits (finite and Lie groups) are studied, for the purpose of using them for studying the phase transitions in physical crystal systems, and for studying the spontaneous symmetry breakings in the unified theories of elementary particles. Within this context, studies were carried out about purely algebraic aspects concerning the theory of polynomial invariants of reflection groups.