BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//CERN//INDICO//EN
BEGIN:VEVENT
SUMMARY:Counting lattice walks confined to cones
DTSTART;VALUE=DATE-TIME:20190503T080000Z
DTEND;VALUE=DATE-TIME:20190503T103000Z
DTSTAMP;VALUE=DATE-TIME:20190618T153225Z
UID:indico-event-3372@indico.math.cnrs.fr
DESCRIPTION:The study of lattice walks confined to cones is a very lively
topic in combinatorics and in probability theory\, which has witnessed ric
h developments in the past 20 years. In a typical problem\, one is given a
finite set of allowed steps S in Z^d\, and a cone C in R^d. Clearly\, the
re are |S|^n walks of length n that start from the origin and take their s
teps in S. But how many of them remain the the cone C?\nOne of the motivat
ions for studying such questions is that lattice walks are ubiquitous in v
arious mathematical fields\, where they encode important classes of object
s: in discrete mathematics (permutations\, trees\, words...)\, in statisti
cal physics (polymers...)\, in probability theory (urns\, branching proces
ses\, systems of queues)\, among other fields.\nThe systematic study of th
ese counting problems started about 20 years ago. Beforehand\, only sporad
ic cases had been solved\, with the exception of walks with small steps co
nfined to a Weyl chamber\, for which a general reflection principle had be
en developed. Since then\, several approaches have been combined to unders
tand how the choice of the steps and of the cone influence the nature of t
he counting sequence a(n)\, or of the the associated series A(t)=\\sum a(n
) t^n. For instance\, if C is the first quadrant of the plane and S only c
onsists of "small" steps\, it is now understood when A(t) is rational\, al
gebraic\, or when it satisfies a linear\, or a non-linear\, differential e
quation. Even in this simple case\, the classification involves tools comi
ng from an attractive variety of fields: algebra on formal power series\,
complex analysis\, computer algebra\, differential Galois theory\, to cite
just a few. And much remains to be done\, for other cones and sets of ste
ps.\n\n \n\nAfter the break\, the second talk will be on:\n\nWalks in the
quadrant: Tutte's invariant method\n\nThe common starting point to all ap
proaches in the (exact) enumeration of quadrant walks is a functional equa
tion that defines a three variable power series\, counting these walks acc
ording to their length and the coordinates of their endpoint. Similar equa
tions appeared in the seventies in the work of William Tutte on the enumer
ation of properly coloured planar maps. In particular\, Tutte devoted ten
years (and as many papers) to the enumeration of properly coloured triangu
lations\, for which he established\, at the end\, a non-linear differentia
l equation for the main generating function.\nIn this talk\, we will show
how to apply (and then to extend to a more analytic framework) Tutte's ide
as to treat in the uniform manner two kinds of quadrant problems:\n- the 4
that have an algebraic generating function\n- the 9 that have a different
ially algebraic (but not differentially finite) generating function.\nTutt
e's notion of "invariants" is central to this solution.\nThese results hav
e been obtained in collaboration with Olivier Bernardi and Kilian Raschel.
\n\n \n\nhttps://indico.math.cnrs.fr/event/3372/
LOCATION:ENS de Lyon\, site Monod Amphi B
URL:https://indico.math.cnrs.fr/event/3372/
END:VEVENT
END:VCALENDAR