Though educated as a theoretical physicist (Master and PhD) I soon developed a taste for the mathematical axiomatization of physical theories, so, as a researcher, I naturally leaned toward questions in mathematical physics. All my investigations are strongly motivated by fundamental questions from physics. In my work I make attempts at seeing the 'whole picture' and, for this reason, I often prefer to study general abstract frameworks rather than specific problems and solutions.
I am interested in the following topics
General relativity: Einstein's gravitational theory is a vast field, as possible modifications of its geometrical assumptions and governing equations have been explored in several directions. GR is possibly the most beautiful and elegant among physical theories. Furthermore, it has a deep philosophical content that challenges our common sense. Would you have ever imagined that clocks and hence time are influenced by gravity?
Lorentzian geometry/Mathematical relativity: Mathematical relativity is basically General Relativity approached with all the suitable technical tools developed by mathematicians. Of these, I am particularly interested in the geometrical aspects, hence in Lorentzian geometry, though I also use analysis and geometric measure theory when needed.
Causality theory: This is a subfield of Lorentzian geometry in which one is primarily interested in the distribution of light cones and in the many causal pathologies that might be present on spacetime (naked singularities, regions of chronology violation and grand-father paradox). The study of geodesic singularities and hence the famous Hawking and Penrose singularity theorems belong to this subject. It is the topic to which I contributed most in my work.
Low regularity relativity: We all know that one of the most important open problems in physics is the unification of general relativity with quantum mechanics. Observables in the latter theory are naturally discrete in some circumstances. Might the universe itself be discrete or non-smooth in some regions? If so, before any attempt at unification, we would need to generalize our gravitational theory to this regime. We prepare the ground by studying low-regularity versions of Lorentzian geometry. This is done using tools from the mathematics of cone structures, metric geometry, optimal transport, topological ordered spaces, the chosen approach depending on the postulated roughness of spacetime.
Topological ordered spaces: I studied pretty in deep this mathematical subject, as several years ago I realized that it could be used to understand better the spacetime structure and its unification with quantum theories. It provides the best mathematical tools to study time functions and their continuity. It is not suited to deal with differentiability, but the differentiability of spacetime would supposedly be lost anyway at high energy/small scales.
Screw theory/ D-module geometry: I teach analytical mechanics to students in engineering, and, along with many authors before me, I rediscovered in my teaching Screw Theory. This approach allows one to unify several results in mechanics connected to rigid body motions and explain certain dualities between forces and angular velocities (e.g., the central axis is analogous to the instantaneous axis of rotation). I enjoy working in this theory and I developed D-module geometry, showing that screw theory is really vector geometry over the ring of dual numbers D rather than the usual reals. In other words, in three dimensions, the affine world is obtained from the vectorial world by introducing a unit whose square is zero.
Not all works are available at the preprint archive, e.g. my review of Lorentzian causality theory
Should you not have access to some paper, feel free to contact me to request a pdf copy.
In the last three years I collaborated with the following researchers (alphabetic order): Ivan Costa e Silva, Alfonso Garcia-Parrado, Sebastian Gurriaran, Jacob Hedicke, Raymond Hounnonkpé, Martin Lesourd, Yufeng Lu, Shin-ichi Ohta, Benedict Schinnerl, Roland Steinbauer, Stefan Suhr.
(joint work with Sebastian Gurriaran) Surface gravity of compact non-degenerate horizons under the dominant energy condition, Commun. Math. Phys. 395 (2022) 679–713.
This work contributed to the solution (in the affirmative) of the non-degenerate case of the Isenberg-Moncrief conjecture. This conjecture, inspired by the cosmic censorship conjecture, states that, under reasonable energy conditions, spacetimes admitting a compact horizon should have symmetries (with the Killing vector being tangent to the null hypersurface). In particular, in this paper we proved that if the horizon admits an incomplete generator then the so called surface gravity is constant.
2. A gravitational collapse singularity theorem consistent with black hole evaporation, Lett. Math. Phys. 110 (2020) 2383–2396
Since the early seventies it was believed (e.g. Hawking and Ellis 1973) that predictability, namely the condition of global hyperbolicity, was necessary for the validity of Penrose’s singularity theorem. Attempts at weakening it were unsuccessful. Somewhat surprisingly I showed in this work that, in fact, global hyperbolicity can be almost completely removed, a fact that helps reconciling Quantum Field Theory, with its implications for black hole evaporation, and General Relativity. The theorem presented in this work can be considered as the strongest version of Penrose’s singularity theorem so far obtained.
3. Lorentzian causality theory, Living Reviews in Relativity 22 (2019) 3 (202 pages)
This work presents in a systematic way the theory of causality over Lorentzian manifolds and includes many new results particularly for what concerns singularity theorems.
4. Causality theory for closed cone structures with applications, Rev. Math. Phys., 31 (2019) 1930001 (139 pages)
This work generalizes causality theory to the anistropic case (Lorentz-Finsler theories) and to low differentiability assumptions (cone distributions that are continuous or even only upper semicontinuous). There it is shown that much of the results of causality theory are preserved. Through the new methods presented in this work we solve two well known open problems: (a) we prove the validity of the “Lorentzian distance formula” conjectured by Parfionov and Zapatrin (2000), which is of interest for Connes’ non-commutative program of unification of fundamental forces, and (b) we characterize the spacetimes that are Lorentzian submanifolds of (flat) Minkowski spacetime thus providing the Lorentzian version of Nash’s theorem.
5. Light cones in Finsler spacetime. Commun. Math. Phys. 334 (2014) 1529–1551
In Lorentz-Finsler geometry the metric g_v(X, Y ) is defined through the Hessian with respect to the velocities of the Finsler Lagrangian. For the physical interpretation of the theory one would want to prove that on each tangent space TpM the set {v : gv(v, v) < 0} consists of two sharp cones, which then would play the role of past and future timelike cones. In 1971 John Beem found 2-dimensional examples of manifolds for which the number of cones is larger than two. This work proves that, actually, for more than two dimensions, and hence for the physical four dimensional case, the number of cones is precisely two as desired. This result paves the way for the development of Finslerian generalizations of general relativity.
6. Time functions as utilities, Commun. Math. Phys. 298 (2010) 855-868
This work points out that time functions are nothing but utility functions for the Seifert’s relation, the concept of utility being originally developed in the field of microeconomics. This correspondence allows one to import theorems developed in the economics field to prove new results in causality theory. For instance, it is clarified that the Seifert relation (and not the causal relation) can be recovered from the sets of time functions allowed by the spacetime. This work is related to the field of dynamical systems in which Lyapunov functions can also be regarded as utilities.