We introduce the concept of N-differential graded algebras ($N$-dga), and study the moduli space of deformations of the differential of a $N$-dga. We prove that it is controlled by what we call the $N$-Maurer-Cartan equation. We provide geometric examples such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives. We also consider deformations of the differential of a $q$-differential graded algebra. We prove that it is controlled by a generalized Maurer-Cartan equation. We find explicit formulae for the coefficients involved in that equation. Deformations of the $3$-differential of $3$-differential graded algebras are controlled by the $(3,N)$ Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of $N$-differential graded algebras, and use these results to study $N$-Lie algebroids. We study higher depth algebras, and work towards the construction of the concept of $A^N_ \infty$-algebras.[-]

We show that the spectrum of fundamental particles of matter and their symmetries can be encoded in a finite quantum geometry equipped with a supplementary structure connected with the quark-lepton symmetry. The occurrence of the exceptional quantum geometry for the description of the standard model with 3 generations is suggested. We discuss the field theoretical aspect of this approach taking into account the theory of connections on the corresponding Jordan modules.[-]

Liouville CFT is a conformal field theory developped in the early 80s in physics, it describes random surfaces and more precisely random Riemannian metrics on surfaces. We will explain, using the Gaussian multiplicative chaos, how to associate to each surface $\Sigma$ with boundary an amplitude, which is an $L^2$ function on the space of fields on the boundary of $\Sigma$ (i.e. the Sobolev space $H^{-s}(\mathbb{S}^1)^b$ equipped with a Gaussian measure, if the boundary of $\Sigma$ has $b$ connected components), and then how these amplitudes compose under gluing of surfaces along their boundary (the so-called Segal axioms).
This allows us to give formulas for all partition and correlation functions of the Liouville CFT in terms of $3$ point correlation functions on the Riemann sphere (DOZZ formula) and the conformal blocks, which are holomorphic functions of the moduli of the space of Riemann surfaces with marked points. This gives a link between the probabilistic approach and the representation theory approach for CFTs, and a mathematical construction and resolution of an important non-rational conformal field theory.
This is joint work with A. Kupiainen, R. Rhodes and V. Vargas. [-]