# Séminaire par Krzysztof Ziemianski: «Directed paths on cubical complexes»

**Speaker:**Krzysztof Ziemianski

**Location:**Salle Marcel-Paul Schützenberger

**Date:**Lun. 28 mai. 2018, 15h00-16h00

Pour le prochain séminaire de l'équipe Cosynus, nous aurons le plaisir d'accueillir Krzysztof Ziemianski qui nous parlera de chemins dans des espaces topologiques dirigés.

**Résumé:** *Let K be a semi-cubical set that satisfies certain mild conditions. I will present a construction of a CW-complex that is homotopy equivalent to the space of directed paths on the geometric realization of K from two fixed vertices of K. This construction satisfies certain minimality condition which makes it useful for direct calculations. Furthermore, this CW-complex carries a "permutahedral" structure - its cells can be identified with products of permutohedra and attaching maps are inclusions of faces. In the case when K is a Euclidean complex this model can be reduced further using Discrete Morse Theory. In some cases, for example if K is the space of states of a PV-program using only one resource, this leads to the optimal construction: the cells of the constructed CW-complex are in 1-1 correspondence with the generators of homology of the directed path space.*