Advances on bijections and surjections
Frédéric Patras, CNRS, Université de Nice Sopia Antipolis
Miércoles, 3 de diciembre, ICMAT, Aula Naranja, 13:30
Abstract.- During the last 20 years, it was realized that certain combinatorial objects (combinatorial Hopf algebras, to be precise) underly many mathematical theories. This phenomenon occurs in almost all fields of mathematics, from numerical analysis to theoretical physics or number theory and includes, for example, renormalization techniques in Hairer's theory of regularity structures for stochastic PDEs. The talk will survey some of these developments, and then focus largely on two of the most emblematic and universal such objects, namely the higher algebraic structures that can be constructed out of permutations (the so-called Malvenuto-Reutenauer Hopf algebra, or algebra of free quasi-symmetric functions), and out of surjections (word quasi-symmetric functions). Recent applications of these ideas and techniques (e.g. to Speicher's approach to free probabilities) will be featured.