1,607 research outputs found

    Ten Conferences WORDS: Open Problems and Conjectures

    Full text link
    In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

    On the Surjectivity of Engel Words on PSL(2,q)

    Full text link
    We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q).Comment: v2: 25 pages, minor changes, accepted to the Journal of Groups, Geometry and Dynamic

    A conjectural generating function for numbers of curves on surfaces

    Full text link
    I give a conjectural generating function for the numbers of δ\delta-nodal curves in a linear system of dimension δ\delta on an algebraic surface. It reproduces the results of Vainsencher for the case δ≤6\delta\le 6 and Kleiman-Piene for the case δ≤8\delta\le 8. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso-Harris for the Severi degrees in ¶2\P_2. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.Comment: amslatex 13 page
    • …
    corecore