José Agapito

Intersection numbers of polygon spaces

We study the intersection ring of the space M(α 1 ,... , α m ) of polygons in R 3 . We find homology cycles dual to generators of this ring and prove a recursion relation in m (the number of edges) for their intersection numbers. This result is an analog of the recursion relation appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves and in the work of Weitsman on moduli spaces of flat connections on two-manifolds of genus g with m marked points. Based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples. Among others, we study the special case of equilateral polygon spaces (where all α i s are the same) and compare our results with the expressions for these particular spaces that have been determined by Kamiyama and Tezuka. Finally, we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of Yoshida, as well as with equivalent expressions for polygon spaces obtained by Takakura and Konno through different techniques.

A symbolic treatment of Riordan arrays

Abstract We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical Abelʼs identity for polynomials. In particular, we obtain a novel non-recursive formula for Riordan arrays and derive, from this new formula, some known recurrences and a new recurrence relation for Riordan arrays.