What is the structure of space at the smallest scale?
What are the possible behaviors of matter at this scale?
If we assume relativity and forget about gravity, then our best answer is given by Quantum Field Theory or, more specifically, Conformal Field Theory because, at the smallest scales, the system becomes scale and conformal invariant. If we consider gravity, then our best answer is that spacetime is an emergent phenomena from the dynamics of some special quantum field theories with many degrees of freedom. Therefore, the space of conformal field theories hides many treasures. On the one hand, it contains all the possible (relativistic invariant) particle dynamics on a fixed continuum geometry, including the critical points of condensed matter systems and the universality classes of statistical physics. On the other hand, it also contains quantum theories of gravity.
Developing better (non-perturbative) methods to explore the space of quantum field theories is the main theme of our research activities.
Unsolved – A map of Quantum Field Theory is a short outreach video describing the Bootstrap approach to QFT done in collaboration with Alexander Monin and Francesco Riva.