Dominique Manchon a récemment mis en évidence le rôle des algèbres de Novikov pour la théorie des équations aux dérivées partielles stochastiques. L'intérêt pour la théorie s'est rapidement développé, à la fois d'un point de vue algébrique et d'un point de vue probabiliste. L'exposé, basé sur un travail en cours avec Ruggero Bandiera (Sapienza Univ. Roma), abordera certains aspects...
The global dimension is an important homological invariant of an algebra, often serving as a good analogue of the dimension of a smooth affine algebraic variety. However, there are examples where the global dimension does not align with geometric intuition. This often leads to consider the Hochschild cohomological dimension rather than the global dimension. It is thus a natural question to...
Kontsevich celebrated universal deformation quantization formula involves certain coefficients, that are periods. Banks, Panzer and Pym have shown that these coefficients are linear combinations of multiple zeta values (MZVs). We explain a generalization to the setting of deformation quantization in the presence of branes (in the sense of Cattaneo-Felder), where MZVs are replaced by coloured MZVs.
In this talk, we will review some recent applications of operad theory to singular SPDEs. It is an essential tool for characterising the chain rule symmetry in the full subcritical regime which leads to renormalising quasilinear SPDEs with local counterterms. Also, it provides negative results about the existence of another combinatorial set lying between multi-indices and decorated trees for...
The Magnus expansion was introduced by Wilhelm Magnus in his 1954 paper “On the Exponential Solution of Differential Equations for a Linear Operator” (CPAM 7 (1954) 649), where he addressed a central problem in applied mathematics: computing the logarithm of the operator- or matrix-valued solution of a linear initial value problem. Since then, the Magnus expansion has evolved into a versatile...
We present a construction of pre-Lie on rooted trees whose edges and vertices are decorated, with a grafting product twisted by an action of a map acting on both edges and vertices. We show that this construction indeed gives a pre-Lie algebra if, and only if, a certain commutation relation is satisfied. Then, this pre-Lie algebra can be extended as a post-Lie algebra through a semi-direct...
Shuffle permutations appear in many different contexts, such as Hopf algebras, shift registers in coding theory and kryptology, symbolic dynamics for chaotic dynamical systems, card tricks, and in the design of efficient permutation networks for parallel computing.
I will give a leisurely discussion of shuffles in various contexts, and in particular discuss the problem of designing...
In the 1990s, two Hopf algebras were defined in terms of shuffles: shuffles of permutations for Malvenuto-Reutenauer Hopf algebras and shuffles of rooted binary trees for Loday-Ronco Hopf algebras. Permutations and binary trees label the vertices of the permutohedron and the associahedron respectively. The 1-skeletons of these polytopes correspond moreover to two well-known posets: the weak...
In this talk, I will focus on the Zariski-Closure Conjecture for coadjoint orbits of exponential solvable Lie groups, a central open question in the orbit method and the deformation theory of unitary representations. This emphasizes the interplay between representation theory, quantization, and Poisson geometry. The aim is to introduce a new dequantization approach that links the theory of...
