110,824 research outputs found
Congruences for Taylor expansions of quantum modular forms
Recently, a beautiful paper of Andrews and Sellers has established linear
congruences for the Fishburn numbers modulo an infinite set of primes. Since
then, a number of authors have proven refined results, for example, extending
all of these congruences to arbitrary powers of the primes involved. Here, we
take a different perspective and explain the general theory of such congruences
in the context of an important class of quantum modular forms. As one example,
we obtain an infinite series of combinatorial sequences connected to the
"half-derivatives" of the Andrews-Gordon functions and with Kashaev's invariant
on torus knots, and we prove conditions under which the sequences
satisfy linear congruences modulo at least of primes of primes
The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics
Prudent walks are special self-avoiding walks that never take a step towards
an already occupied site, and \emph{-sided prudent walks} (with )
are, in essence, only allowed to grow along directions. Prudent polygons
are prudent walks that return to a point adjacent to their starting point.
Prudent walks and polygons have been previously enumerated by length and
perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration
of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find
that the generating function is expressed in terms of a -hypergeometric
function, with an accumulation of poles towards the dominant singularity. This
expression reveals an unusual asymptotic structure of the number of polygons of
area , where the critical exponent is the transcendental number
and and the amplitude involves tiny oscillations. Based on numerical data, we
also expect similar phenomena to occur for 4-sided polygons. The asymptotic
methodology involves an original combination of Mellin transform techniques and
singularity analysis, which is of potential interest in a number of other
asymptotic enumeration problems.Comment: 27 pages, 6 figure
Jet Schemes of Locally Complete Intersection Canonical Singularities
We prove that if X is a locally complete intersection variety, then X has all
the jet schemes irreducible if and only if X has canonical singularities. After
embedding X in a smooth variety Y, we use motivic integration to express the
condition that X has irreducible jet schemes in terms of data coming from an
embedded resolution of X in Y. We show that this condition is equivalent with
having canonical singularities. In the appendix, this result is used to prove a
generalization of Kostant's freeness theorem to the setting of jet schemes.Comment: With an appendix by David Eisenbud and Edward Frenkel. Final version,
to appear in Inventiones Mathematica
Nilpotent bicone and characteristic submodule of a reductive Lie algebra
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g
is the subset of elements in g x g whose subspace generated by the components
is contained in the nilpotent cone of g. The main result of this note is that
the nilpotent bicone is a complete intersection. This affirmatively answers a
conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce
and study the characteristic submodule of g. The properties of the nilpotent
bicone and the characteristic submodule are known to be very important for the
understanding of the commuting variety and its ideal of definition. In order to
study the nilpotent bicone, we introduce another subvariety, the principal
bicone. The nilpotent bicone, as well as the principal bicone, are linked to
jet schemes. We study their dimensions using arguments from motivic
integration. Namely, we follow methods developed in
http://arxiv.org/abs/math/0008002v5 .Comment: 48 pages. Remark 8 has been modified; one sentence was not correct.
We thank Kari Vilonen for pointing out this erro
Bimodule deformations, Picard groups and contravariant connections
We study deformations of invertible bimodules and the behavior of Picard
groups under deformation quantization. While K_0-groups are known to be stable
under formal deformations of algebras, Picard groups may change drastically. We
identify the semiclassical limit of bimodule deformations as contravariant
connections and study the associated deformation quantization problem. Our main
focus is on formal deformation quantization of Poisson manifolds by star
products.Comment: 32 pages. Minor corrections in Sections 5 and 6, typos fixed. Revised
version to appear in K-theor
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