La symétrisation divisée est un opérateur linéaire sur les polynômes multivariés. Il a été introduit pour exprimer le volume des permutoèdres généralisés, et apparaît également dans le contexte du calcul de Schubert pour la variété de drapeaux. Nous expliquerons ces termes et décrirons divers aspects combinatoires et algébriques de la symétrisation divisée, notamment son action sur diverses familles de polynômes. Travail en commun avec V. Tewari (UPenn)

IRIF Cake^TM is an amazing opportunity to meet people while simultaneously eating cakes baked by your fellow colleagues! Join us every Thursday, at 5pm, in front of room 3052 (Sophie Germain 3rd floor) for a weekly feast. You can also express your cooking skills and volunteer to bake a cake by sending an email to cake@irif.fr.

We consider the problem of passive symbolic learning in the case of deterministic top-down tree transducers (DTOP). The passive learning problem deals with identifying a specific transducer in normal form from a finite set of behaviour examples. This problem is solved in word languages using the RPNI algorithm, that relies heavily on the Myhill-Nerode characterization of a minimal normal form on DFA. Its extensions to word transformations and tree languages follow the same pattern: first, a Myhill-Nerode theorem is identified, then the normal form it induces can be learnt from examples. To adapt this result in tree transducers, the Myhill-Nerode theorem requires that DTOP are considered with an inspection, i.e. an automaton that recognized the domain of the transformation. In its original form, the normalization (minimal earliest compatible normal form) and learning of DTOP is limited to deterministic top-down tree automata as inspections. In this talk, we show the challenges that an extension to regular inspections presents, and present two concurrent ways to deal with them:

1. first, by an extension of the Myhill-Nerode theorem on DTOP to the regular case, by defining a minimal *leftmost* earliest compatible normal form.
2. second, by reducing the problem to top-down domains, by using the regular inspection as a lookahead

The merits of these methods will be discussed for possible extensions of these methods to data trees.

A charted omega-category is like a strict omega-category, but instead of a globular set, it has an underlying regular polygraph: its cells have more complex pasting diagrams “charted” on their boundary. Several features of omega-categories generalise nicely, including joins and the monoidal biclosed structure of lax Gray products. I will detail some of the combinatorics involved, going deeper into the theory of globular posets than in my talk last July (which is not a prerequisite).