Keywords: Relativistic theories, Quantum gravity, Gauge theories.

My research interests smuggle methods back and forth from mathematics to theoretical physics. From the physical viewpoint I study relativistic theories, gravitational theories, and GR, which is, from the mathematical viewpoint, exactly the same thing as studying how to define PDEs on a class of diffeomorphisms of manifolds with no structure fixed on them. That includes how to discuss well-posed Cauchy problems (canonical analysis of Hamiltonian formulation of field theories), how to discuss conservation laws in a covariant setting, how to extend to systems with gauge symmetries (gauge natural theories) and spinor fields, how to account for analytical properties of the equations by studying geometrical properties of finite dimensional calculus (also known as geometric formulation of variational calculus).

Another research line is about how to define discrete quantum geometries (which corresponds to loop quantization of relativistic theories and gauge theories on a lattice). That involves Wilson loop Hilbert spaces on compact Lie groups and operators defined on them. More generally, that is studying gauge invariant functionals of a connections.

Finally, I have medium term researches about covariant positioning in a relativistic theory (also known as the theory of observables in a relativistic theory) and about covariant methods for perturbations in field theories (which is widely used in cosmology).

In the past I also studied symmetries in mechanics and field theory, covariant conservation laws in relativistic theories, field theories coupling gravity and spinor fields, black hole entropy and its relation with conservation laws, modified gravity, in particular Palatini f(\calR)-theories and conformal gravity and their applications to cosmology.

• Math. Dept.