1. CJM 2016 (vol 68 pp. 445)
 Martins, Luciana de Fátima; Saji, Kentaro

Geometric Invariants of Cuspidal Edges
We give a normal form of the cuspidal edge
which uses only diffeomorphisms on the source
and isometries on the target.
Using this normal form, we study differential
geometric invariants of
cuspidal edges which determine them up to order three.
We also
clarify relations between these invariants.
Keywords:cuspidal edge, curvature, wave fronts Categories:57R45, 53A05, 53A55 

2. CJM 2015 (vol 68 pp. 241)
 Allermann, Lars; Hampe, Simon; Rau, Johannes

On Rational Equivalence in Tropical Geometry
This article discusses the concept of rational equivalence
in tropical
geometry
(and replaces an older and imperfect version).
We give the basic definitions in the context of tropical varieties
without boundary points and prove some basic properties.
We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing
that they are isomorphic
to the group of fan cycles. The main step in the proof is of
independent interest:
We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated)
fan cycles. This also
proves that the intersection ring of tropical cycles is generated
in codimension 1 (by hypersurfaces).
Keywords:tropical geometry, rational equivalence Category:14T05 

3. CJM Online first
 Levinson, Jake

Onedimensional Schubert problems with respect to osculating flags
We consider Schubert problems with respect to flags osculating
the rational normal curve. These problems are of special interest
when the osculation points are all real  in this case, for
zerodimensional Schubert problems, the solutions are "as real
as possible". Recent work by Speyer has extended the theory
to the moduli space
$
\overline{\mathcal{M}_{0,r}}
$,
allowing the points to collide.
These give rise to smooth covers of
$
\overline{\mathcal{M}_{0,r}}
(\mathbb{R})
$, with structure
and monodromy described by Young tableaux and jeu de taquin.
In this paper, we give analogous results on onedimensional Schubert
problems over
$
\overline{\mathcal{M}_{0,r}}
$.
Their (real) geometry turns out to
be described by orbits of SchÃ¼tzenberger promotion and a
related operation involving tableau evacuation. Over
$\mathcal{M}_{0,r}$,
our results show that the real points of the solution curves
are smooth.
We also find a new identity involving "firstorder" Ktheoretic
LittlewoodRichardson coefficients, for which there does not
appear to be a known combinatorial proof.
Keywords:Schubert calculus, stable curves, ShapiroShapiro Conjecture, jeu de taquin, growth diagram, promotion Categories:14N15, 05E99 

4. CJM Online first
 Hartz, Michael

On the isomorphism problem for multiplier algebras of NevanlinnaPick spaces
We continue the investigation of the isomorphism problem for
multiplier algebras of reproducing kernel
Hilbert spaces with the complete NevanlinnaPick property.
In contrast to previous work in this area,
we do not study these spaces by identifying them with restrictions
of a universal space, namely the DruryArveson space.
Instead, we work directly with the Hilbert spaces and their
reproducing kernels. In particular,
we show that two multiplier algebras of NevanlinnaPick spaces
on the same set are equal if and only if the Hilbert
spaces are equal. Most of the article is devoted to the study
of a special class of
complete NevanlinnaPick spaces on homogeneous varieties. We
provide a complete
answer to the question of when two multiplier algebras of spaces
of this type
are algebraically or isometrically isomorphic. This generalizes
results of Davidson, Ramsey, Shalit,
and the author.
Keywords:nonselfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, NevanlinnaPick kernels, isomorphism problem Categories:47L30, 46E22, 47A13 

5. CJM Online first
 Levin, Aaron; Wang, Julie TzuYueh

On nonArchimedean curves omitting few components and their arithmetic analogues
Let $\mathbf{k}$ be an algebraically closed field complete with respect
to a nonArchimedean absolute value of arbitrary characteristic.
Let $D_1,\dots, D_n$ be effective nef divisors intersecting
transversally in an $n$dimensional nonsingular projective variety
$X$.
We study the degeneracy of nonArchimedean analytic maps from
$\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric
conditions. When $X$ is a rational ruled surface and $D_1$ and
$D_2$ are ample, we obtain a necessary and sufficient condition
such that
there is no nonArchimedean analytic map from $\mathbf{k}$ into $X\setminus
D_1 \cup D_2$.
Using the dictionary between nonArchimedean Nevanlinna theory
and Diophantine approximation that originated in
earlier work with T. T. H. An, %
we also study arithmetic analogues of these problems, establishing
results on integral points on these varieties over $\mathbb{Z}$
or the ring of integers of an imaginary quadratic field.
Keywords:nonArchimedean Picard theorem, nonArchimedean analytic curves, integral points Categories:11J97, 32P05, 32H25 

6. CJM Online first
 Izumi, Masaki; Morrison, Scott; Penneys, David

Quotients of $A_2 * T_2$
We study unitary quotients of the free product unitary pivotal
category $A_2*T_2$.
We show that such quotients are parametrized by an integer $n\geq
1$ and an $2n$th root of unity $\omega$.
We show that for $n=1,2,3$, there is exactly one quotient and
$\omega=1$.
For $4\leq n\leq 10$, we show that there are no such quotients.
Our methods also apply to quotients of $T_2*T_2$, where we have
a similar result.
The essence of our method is a consistency check on jellyfish
relations.
While we only treat the specific cases of $A_2 * T_2$ and $T_2
* T_2$, we anticipate that our technique can be extended to a
general method for proving nonexistence of planar algebras with
a specified principal graph.
During the preparation of this manuscript, we learnt of Liu's
independent result on composites of $A_3$ and $A_4$ subfactor
planar algebras
(arxiv:1308.5691).
In 1994, BischHaagerup showed that the principal graph of a
composite of $A_3$ and $A_4$ must fit into a certain family,
and Liu has classified all such subfactor planar algebras.
We explain the connection between the quotient categories and
the corresponding composite subfactor planar algebras.
As a corollary of Liu's result, there are no such quotient categories
for $n\geq 4$.
This is an abridged version of
arxiv:1308.5723.
Keywords:pivotal category, free product, quotient, subfactor, intermediate subfactor Category:46L37 

7. CJM 2014 (vol 67 pp. 152)
 Lescop, Christine

On Homotopy Invariants of Combings of Threemanifolds
Combings of compact, oriented $3$dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$valued invariant $\theta_G$ of torsion combings
on closed $3$manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$invariant, which is the simplest configuration
space integral invariant of rational homology $3$balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the CassonWalker invariant.
The article also includes a selfcontained presentation of combings
for $3$manifolds.
Keywords:Spin$^c$structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, CassonWalker invariant, perturbative expansion of ChernSimons theory, configuration space integrals Categories:57M27, 57R20, 57N10 

8. CJM 2014 (vol 67 pp. 527)
 Brugallé, Erwan; Shaw, Kristin

Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory
We provide some new local obstructions to
approximating
tropical curves in
smooth tropical surfaces. These obstructions are based on
a
relation between tropical and complex intersection theories which is
also established here. We give
two applications of the methods developed in this paper.
First we classify all locally irreducible approximable 3valent fan tropical
curves in a
fan tropical plane.
Secondly, we prove that a generic nonsingular
tropical surface
in tropical projective 3space contains finitely
many approximable tropical lines
if
it is of degree 3, and contains no approximable tropical lines if
it is of degree 4 or more.
Keywords:tropical geometry, amoebas, approximation of tropical varieties, intersection theory Categories:14T05, 14M25 

9. CJM 2014 (vol 67 pp. 667)
 Nishinou, Takeo

Toric Degenerations, Tropical Curve, and GromovWitten Invariants of Fano Manifolds
In this paper, we give a tropical method for computing GromovWitten
type invariants
of Fano manifolds of special type.
This method applies to those Fano manifolds which admit toric
degenerations
to toric Fano varieties with singularities allowing small resolutions.
Examples include (generalized) flag manifolds of type A, and
some moduli space
of rank two bundles on a genus two curve.
Keywords:Fano varieties, GromovWitten invariants, tropical curves Category:14J45 

10. CJM 2013 (vol 66 pp. 566)
 Choiy, Kwangho

Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$adic Inner Forms
Let $F$ be a $p$adic field of characteristic $0$, and let $M$ be an $F$Levi subgroup of a connected reductive $F$split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local JacquetLanglands type correspondence between $M$ and its $F$inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of
MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local JacquetLanglands correspondence. It can be applied to a simply connected simple $F$group of type $E_6$ or $E_7$, and a connected reductive $F$group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.
Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, JacquetLanglands correspondence Categories:22E50, 11F70, 22E55, 22E35 

11. CJM 2013 (vol 65 pp. 1217)
 Cruz, Victor; Mateu, Joan; Orobitg, Joan

Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the
complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of
the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f
(z)$, have first derivatives locally in $X(\mathbb C)$, provided that the
Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.
Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³nZygmund operators Categories:30C62, 35J99, 42B20 

12. CJM 2012 (vol 65 pp. 120)
 Francois, Georges; Hampe, Simon

Universal Families of Rational Tropical Curves
We introduce the notion of families of $n$marked
smooth rational tropical curves over smooth tropical varieties and
establish a onetoone correspondence between (equivalence classes of)
these families and morphisms
from smooth tropical varieties into the moduli space of $n$marked
abstract rational tropical curves $\mathcal{M}_{n}$.
Keywords:tropical geometry, universal family, rational curves, moduli space Categories:14T05, 14D22 

13. CJM 2012 (vol 64 pp. 254)
14. CJM 2011 (vol 64 pp. 345)
 McKee, James; Smyth, Chris

Salem Numbers and Pisot Numbers via Interlacing
We present a general construction of Salem numbers via rational
functions whose zeros and poles mostly lie on the unit circle and
satisfy an interlacing condition. This extends and unifies earlier
work. We then consider the ``obvious'' limit points of the set of Salem
numbers produced by our theorems and show that these are all Pisot
numbers, in support of a conjecture of Boyd. We then show that all
Pisot numbers arise in this way. Combining this with a theorem of
Boyd, we produce all Salem numbers via an interlacing construction.
Keywords:Salem numbers, Pisot numbers Category:11R06 

15. CJM 2011 (vol 64 pp. 573)
 Nawata, Norio

Fundamental Group of Simple $C^*$algebras with Unique Trace III
We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$unital $C^*$algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$algebras.
We show that there exist separable stably projectionless $C^*$algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.
Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function Categories:46L05, 46L08, 46L35 

16. CJM 2011 (vol 64 pp. 409)
 Rainer, Armin

Lifting Quasianalytic Mappings over Invariants
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blowups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 

17. CJM 2011 (vol 63 pp. 460)
 Pavlíček, Libor

Monotonically Controlled Mappings
We study classes of mappings between finite and infinite dimensional
Banach spaces that are monotone and mappings which are differences
of monotone mappings (DM). We prove a RadÃ³Reichelderfer estimate
for monotone mappings in finite dimensional spaces that remains
valid for DM mappings. This provides an alternative proof of the
FrÃ©chet differentiability a.e. of DM mappings. We establish a
Morreytype estimate for the distributional derivative of monotone
mappings. We prove that a locally DM mapping between finite
dimensional spaces is also globally DM. We introduce and study a new
class of the socalled UDM mappings between Banach spaces, which
generalizes the concept of curves of finite variation.
Keywords: monotone mapping, DM mapping, RadÃ³Reichelderfer property, UDM mapping, differentiability Categories:26B05, 46G05 

18. CJM 2009 (vol 61 pp. 641)
 Maeda, Sadahiro; Udagawa, Seiichi

Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions
For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.
Keywords:space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector Categories:53C40, 53C42 

19. CJM 2009 (vol 61 pp. 566)
 Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.

Convex Subordination Chains in Several Complex Variables
In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\z\,\z\)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.
Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordination Categories:32H02, 30C45 

20. CJM 2009 (vol 61 pp. 264)
 Bell, J. P.; Hare, K. G.

On $\BbZ$Modules of Algebraic Integers
Let $q$ be an algebraic integer of degree $d \geq 2$.
Consider the rank of the multiplicative subgroup of $\BbC^*$ generated
by the conjugates of $q$.
We say $q$ is of {\em full rank} if either the rank is $d1$ and $q$
has norm $\pm 1$, or the rank is $d$.
In this paper we study some properties of $\BbZ[q]$ where $q$ is an
algebraic integer of full rank.
The special cases of when $q$ is a Pisot number and when $q$ is a Pisotcyclotomic number
are also studied.
There are four main results.
\begin{compactenum}[\rm(1)]
\item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive
integer,
then there are only finitely many $m$ such that
$\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$.
\item If $q$ and $r$ are algebraic integers of degree $d$ of full rank
and $\BbZ[q^n] = \BbZ[r^n]$ for
infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$,
where
$r'$ is some conjugate of $r$ and $\omega$ is some root of unity.
\item Let $r$ be an algebraic integer of degree at most $3$.
Then there are at most $40$ Pisot numbers $q$ such that
$\BbZ[q] = \BbZ[r]$.
\item There are only finitely many Pisotcyclotomic numbers of any fixed
order.
\end{compactenum}
Keywords:algebraic integers, Pisot numbers, full rank, discriminant Categories:11R04, 11R06 

21. CJM 2008 (vol 60 pp. 1067)
 Kariyama, Kazutoshi

On Types for Unramified $p$Adic Unitary Groups
Let $F$ be a nonarchimedean local field of residue characteristic
neither 2 nor 3 equipped with a galois involution with fixed field
$F_0$, and let $G$ be a symplectic group over $F$ or an unramified
unitary group over $F_0$. Following the methods of BushnellKutzko for
$\GL(N,F)$, we define an analogue of a simple type attached to a
certain skew simple stratum, and realize a type in $G$. In
particular, we obtain an irreducible supercuspidal representation of
$G$ like $\GL(N,F)$.
Keywords:$p$adic unitary group, type, supercuspidal representation, Hecke algebra Categories:22E50, 22D99 

22. CJM 2008 (vol 60 pp. 721)
 Adamus, J.; Bierstone, E.; Milman, P. D.

Uniform Linear Bound in Chevalley's Lemma
We obtain a uniform linear bound for the Chevalley function at a point in
the source of an analytic mapping that is regular in the sense of
Gabrielov. There is a version of
Chevalley's lemma also along a fibre, or at a point of the image of a proper
analytic mapping. We get a uniform linear bound for the Chevalley
function of a closed Nash (or formally Nash) subanalytic set.
Keywords:Chevalley function, regular mapping, Nash subanalytic set Categories:13J07, 32B20, 13J10, 32S10 

23. CJM 2007 (vol 59 pp. 696)
 Bangoura, Momo

AlgÃ¨bres de Lie d'homotopie associÃ©es Ã une protobigÃ¨bre de Lie
On associe \`a toute structure de protobig\`ebre de Lie sur un espace
vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie
d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre
ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres,
tous les crochets $n$aires sont nuls pour $n \geq 4$ du fait qu'ils
proviennent d'une structure de protobig\`ebre de Lie. Plus
g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de
l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une
collection d'applications multilin\'eaires antisym\'etriques sur
l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les
identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres
de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour
le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de
$F$ et de~$F^*$.
To any protoLie algebra structure on a finitedimensional vector
space~$F$, we associate homotopy Lie algebra structures defined on
the suspension of the exterior algebra of $F$ and that of its dual
$F^*$, respectively. In these algebras, all $n$ary brackets for $n
\geq 4$ vanish because the brackets are defined by the protoLie
algebra structure. More generally, to any element of odd degree in
the exterior algebra of the direct sum of $F$ and $F^*$, we associate
a set of multilinear skewsymmetric mappings on the suspension of the
exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized
Jacobi identities, defining the homotopy Lie algebras, if the given
element is of square zero with respect to the big bracket of the
exterior algebra of the direct sum of $F$ and~$F^*$.
Keywords:algÃ¨bre de Lie d'homotopie, bigÃ¨bre de Lie, quasibigÃ¨bre de Lie, protobigÃ¨bre de Lie, crochet dÃ©rivÃ©, jacobiateur Categories:17B70, 17A30 

24. CJM 2007 (vol 59 pp. 465)
 Barr, Michael; Kennison, John F.; Raphael, R.

Searching for Absolute $\mathcal{CR}$Epic Spaces
In previous papers, Barr and Raphael investigated the situation of a
topological space $Y$ and a subspace $X$ such that the induced map
$C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative
rings (with units). We call such an embedding a $\CR$epic embedding
and we say that $X$ is absolute $\CR$epic if every embedding of $X$
is $\CR$epic. We continue this investigation. Our most notable
result shows that a Lindel\"of space $X$ is absolute $\CR$epic if a
countable intersection of $\beta X$neighbourhoods of $X$ is a $\beta
X$neighbourhood of $X$. This condition is stable under countable
sums, the formation of closed subspaces, cozerosubspaces, and being
the domain or codomain of a perfect map. A strengthening of the
Lindel\"of property leads to a new class with the same closure
properties that is also closed under finite products. Moreover, all
\scompact spaces and all Lindel\"of $P$spaces satisfy this stronger
condition. We get some results in the nonLindel\"of case that are
sufficient to show that the Dieudonn\'e plank and some closely related
spaces are absolute $\CR$epic.
Keywords:absolute $\mathcal{CR}$epics, countable neighbourhoo9d property, amply LindelÃ¶f, DiuedonnÃ© plank Categories:18A20, 54C45, 54B30 

25. CJM 2005 (vol 57 pp. 1012)
 Karigiannis, Spiro

Deformations of $G_2$ and $\Spin(7)$ Structures
We consider some deformations of $G_2$structures on $7$manifolds. We
discover a canonical way to deform a $G_2$structure by a vector field in
which the associated metric gets ``twisted'' in some way by the
vector cross product. We present a system of partial differential
equations for an unknown vector field $w$ whose solution would
yield a manifold with holonomy $G_2$. Similarly we consider analogous
constructions for $\Spin(7)$structures on $8$manifolds. Some of
the results carry over directly, while others do not because of the
increased complexity of the $\Spin(7)$ case.
Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross product Categories:53C26, 53C29 
