1,607 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
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)
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
I give a conjectural generating function for the numbers of -nodal
curves in a linear system of dimension on an algebraic surface. It
reproduces the results of Vainsencher for the case and
Kleiman-Piene for the case . 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 . 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
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