Le Séminaire Différentiel a vu le jour en 2003 à l’IMJ sous l’impulsion de Daniel Bertrand. Depuis 2018, il est co-organisé par Alin Bostan et Lucia Di Vizio. Les séances ont lieu en alternance entre Paris et Versailles, à raison d'un mardi par semestre et de 3 exposés par réunion, avec le soutien du Laboratoire de Mathématiques de Versailles, de l'INRIA, du GDR EFI et du projet ANR EAGLES.

Prochaine journée du séminaire différentiel  

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Date : 1er avril 2025
Lieu : LIP6, tour 25-26, salle 105, 4 place Jussieu, Paris

Programme de la journée:

10h45-11h45 Mahsa Shirmohammadi (CNRS, IRIF, U. Paris)
Differential Tree Automata

A rationally dynamically algebraic (RDA) power series is one that  arises as (a component of) the solution of a system of differential equations of the form $\boldsymbol{y}' = F(\boldsymbol y)$, where $F$ is a vector of rational functions that is defined at $\boldsymbol y(0)$. RDA power series subsume algebraic power series and are a proper subclass of differentially algebraic power series (those that satisfy a univariate polynomial-differential equation). We give a combinatorial characterisation of RDA power series in terms of  exponential generating functions of regular languages of labelled trees.  Motivated by this connection, we define the notion of a differential tree automaton.  Differential tree automata generalise weighted tree automata by allowing the transition weights to be rational functions of the tree size.  Our main result is that the ordinary generating functions of the formal tree series recognised by differential tree automata are exactly the differentially algebraic power series.   The proof of this result establishes a general form of recurrence satisfied by the sequence of coefficients of a differentially algebraic power series, generalising Reutenauer's matrix representation of polynomially recursive sequences. As a corollary we obtain a procedure for determining equality of differential tree automata.

12h-14h repas 

14h15-15h15 Pierre-Guy Plamondon (LMV, Université de Versailles Saint-Quentin-en-Yvelines)
Fans and polytopes arising from representation theory

The associahedron is a convex polytope that was introduced in the 1950's and 1960's by Tamari and Stasheff.  It encodes the combinatorics of many problems, such as triangulations of a polygon, Dyck paths, or parenthesizations of $n$ variables.  About twenty years ago, the associahedron was realized algebraically using representations of a quiver of Dynkin type $A$.  In this talk, I will give a basic introduction to representations of a quiver, and I will present how not only the associahedron, but also other polytopes such as the permutohedron, can be realized in various ways using representations of quivers.

15h30-16h30 Federico Pellarin (Università di Roma La Sapienza)
Non-commutative factorizations and multiple zeta values in positive characteristic

The classical factorization of the sine function can be used to prove Euler’s formula that $\zeta(n)$ is proportional to $\pi^n$ with a rational factor of proportionality if $2$ divides $n$, as well as similar other properties for multiple zeta values of a certain type. In 1935 Carlitz introduced his global function field variant of zeta values and proved analogue statements by constructing one of the simplest Drinfeld modules, Carlitz’s module, and inaugurating a new area of research. Later in 2004, Thakur also introduced multiple zeta values in this setting, generalizing Carlitz’s constructions. In this talk we review a joint work with Nathan Green where we attach certain sine functions to tensor powers of Carlitz’s module. While it seems difficult to construct explicit factorizations of such sine functions in the way Euler does for the classical sine function, it is instead possible to view our "higher sine functions’’ as elements of a non-commutative algebra of operators, where they can be factorized in a simple way. This process generates non-trivial relations among function field analogues of Thakur multiple zeta values.

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