Here we define and prove some properties of the semi-classical
wavefront set. We also define and study semi-classical Fourier
integral operators and prove a generalization of Egorov's theorem to
manifolds of different dimensions.

We introduce a natural Hopf algebra structure on the space of noncommutative
symmetric functions.
The bases for this algebra are indexed
by set partitions. We show that there exists a natural inclusion of the Hopf
algebra of noncommutative symmetric functions
in this larger space. We also consider this algebra as a subspace of
noncommutative polynomials and use it to
understand the structure of the spaces of harmonics and coinvariants
with respect to this collection of noncommutative polynomials and conclude
two analogues of Chevalley's theorem in the noncommutative setting.

The classical Hurwitz enumeration problem has a presentation in terms of
transitive factorizations in the symmetric group. This presentation suggests
a generalization from type~$A$ to other
finite reflection groups and, in particular, to type~$B$.
We study this generalization both from a combinatorial and a geometric
point of view, with the prospect of providing a means of understanding more
of the structure of the moduli spaces of maps with an $\gS_2$-symmetry.
The type~$A$ case has been well studied and connects Hurwitz numbers
to the moduli space of curves. We conjecture an analogous setting for the
type~$B$ case that is studied here.

This paper establishes general theorems which contain both moduli
of continuity and large incremental results for $l^\infty$-valued Gaussian
random fields indexed by a multidimensional parameter under explicit conditions.

We show that a characterization of scaling functions for
multiresolution analyses given by Hern\'{a}ndez and Weiss and that a
characterization of low-pass filters given by Gundy both hold for
multivariable multiresolution analyses.

We prove that for a topological operad $P$ the operad of oriented
cubical singular chains, $C^{\ord}_\ast(P)$, and the operad of
simplicial singular chains, $S_\ast(P)$, are weakly equivalent. As
a consequence, $C^{\ord}_\ast(P\nsemi\mathbb{Q})$ is formal if and only
if $S_\ast(P\nsemi\mathbb{Q})$ is formal, thus linking together some
formality results which are spread out in the literature. The proof
is based on an acyclic models theorem for monoidal functors. We
give different variants of the acyclic models theorem and apply
the contravariant case to study the cohomology theories for
simplicial sets defined by $R$-simplicial differential graded
algebras.

A commutative local Cohen--Macaulay ring $R$ of finite Cohen--Macaulay type is known to be an isolated
singularity; that is, $\Spec(R) \setminus \{ \mathfrak {m} \}$ is smooth.
This paper proves a non-commutative analogue. Namely, if $A$ is a
(non-commutative) graded Artin--Schelter \CM\ algebra which is fully
bounded Noetherian and
has finite Cohen--Macaulay type, then the non-commutative projective scheme determined by
$A$ is smooth.

In a recent paper, F. Zanello showed that level Artinian algebras in 3
variables can fail to have the Weak Lefschetz Property (WLP), and can
even fail to have unimodal Hilbert function. We show that the same is
true for the Artinian reduction of reduced, level sets of points in
projective 3-space. Our main goal is to begin an understanding of how
the geometry of a set of points can prevent its Artinian reduction
from having WLP, which in itself is a very algebraic notion. More
precisely, we produce level sets of points whose Artinian reductions
have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the
worst case), but fail to have WLP. We also produce a level set of
points whose Artinian reduction fails to have unimodal Hilbert
function; our example is based on Zanello's example. Finally, we show
that a level set of points can have Artinian reduction that has WLP
but fails to have the Strong Lefschetz Property. While our
constructions are all based on basic double G-linkage, the
implementations use very different methods.

On calcule les restrictions {\`a} l'alg{\`e}bre de Hecke sph{\'e}rique
des traces tordues compactes d'un ensemble de repr{\'e}sentations
explicitement construites du groupe $\GL(N, F)$, o{\`u} $F$ est
un corps $p$-adique. Ces calculs r\'esolve en particulier une
question pos{\'e}e dans un article pr\'ec\'edent du m\^eme auteur.

In this paper, we find equations that characterize locally
projectively flat Finsler metrics in the form $F = (\alpha +
\beta)^2/\alpha$, where $\alpha=\sqrt{a_{ij}y^iy^j}$ is a Riemannian
metric and $\beta= b_i y^i$ is a $1$-form. Then we completely
determine the local structure of those with constant flag curvature.

We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.