Keywords: Classical and quantum field theories, Gauge/Yang-Mills theories, Symmetries, Lie group fiber bundle theory, Canonical analysis.

My research interests focus on the interplay between geometry and mathematical physics in the context of classical and quantum field theories. Particularly illustrative from this point of view is the link between principal connections, on the geometrical side, and Yang-Mills theories, on the physical side.

Since the known fundamental interactions in physics are governed by gauge symmetries (and one might go as far as to say that physics on the whole is the study of symmetries), I like in particular to explore the possibility of constructing theories characterized by more general symmetries, different from the familiar ones.

Following this path brought me to study the geometric aspects of classical field theories, in other words of the calculus of variations on fiber bundles, using also variational sequences (sheaf theory), and as well to meet the mathematical formulation of gauge theories (through differential geometry, algebraic topology and in particular the theory of Lie groups and their representations), the gauge-natural formalism and the procedure to define PDE on diffeomorphism classes of smooth manifolds without added structures, like fixed metrics. This includes the study of well-posed Cauchy problems in Lagrangian field theories (canonical analysis), without forgetting topics looking more in the direction of theoretical physics such as spinor calculus, loop quantum gravity, supergravity and the AdS/CFT correspondence.

Recently, I have started to investigate if the notions of generalized principal bundle and generalized principal connection (Lie group fiber bundle theory and Aut(G)-structures), introduced by Castrillón López and Rodríguez Abella, are valuable in order to develop an instance of generalized Yang-Mills theories as a unifying language for Yang-Mills theories and general relativity.

Furthermore, and still in line with my interests, I am curious about the role that the theory of coalgebras and Hopf algebras has in the non-commutative analogue of gauge theories and about the implications in physics of the theory of pseudo-differential operators, of elliptic complexes and of knot theory.

Finally, I am also interested in keeping myself up-to-date with the latest developments in the foundational and epistemological aspects of contemporary mathematics and physics (philosophy of science).

•   Math. Dept.