Our main results are the proofs of celebrated conjectures that were made by physicists on the Liouville Conformal Field Theory (LCFT):
1. An exact expression due to Dorn, Otto, Zamoldchicov and Zamoldchicov for the three-point correlation function of LCFT on the two-sphere, the DOZZ formula.
2. The spectrum of critical exponents of LCFT.
3. Using 1 and 2 to verify conformal bootstrap axioms and to obtain explicit expressions for LCFT correction functions on all Riemann surfaces.
These are major breakthroughs in the field.
Other major results of the project are:
-A probabilistic construction of the Virasoro algebra symmetry of LCFT by constructing a a new family of Markovian dynamics associated to holomorphic vector fields defined in the disk. Using this we have also determined explicitly the scattering matrix of the theory.
-Solution of the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian Multiplicative Chaos measures.
-Solution of the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature and computation of the critical perimeter exponents
-Use of singular stochastic PDE methods and variational and stochastic control method to proving sInvariance of phi^4 QFT measure under nonlinear wave and Schrödinger equations on the plane, stochastic quantization of Sinh-Gordon QFT, construction of phi^4 QFT model in infinite volume and construction of invariant Gibbs measure for Anderson Non Linear Wave equation
-Regularity results for singular SPDES such as Stochastic Schauder-Tychonoff type theorems, stability and moment estimates and Sobolev regularity of occupation measures and paths
- Self similar long time asymptotic for coagulation equations with applications to atmospheric physics and population dynamics- proving self similar behaviour, long time asymptotic and also the existence of anomalous self similarity. Franco also applied renewal equations to population dynamics of physiologically structured populations.
The results of the project have been disseminated in journal articles and conference talks. They were also featured in a Quanta magazine article and selected for the main mathematics breakthroughs in 2022. The proof of the DOZZ formula was awarded the Georg Polya prize by SIAM.
