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1. CJM Online first

Levinson, Jake
 One-dimensional 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 zero-dimensional 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 one-dimensional 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 "first-order" K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof. Keywords:Schubert calculus, stable curves, Shapiro-Shapiro Conjecture, jeu de taquin, growth diagram, promotionCategories:14N15, 05E99

2. CJM 2016 (vol 68 pp. 999)

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, Bisch-Haagerup 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 subfactorCategory:46L37

3. CJM Online first

Hartz, Michael
 On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick 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 Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick 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 Nevanlinna-Pick 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:non-selfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, Nevanlinna-Pick kernels, isomorphism problemCategories:47L30, 46E22, 47A13

4. 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 frontsCategories:57R45, 53A05, 53A55

5. 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 equivalenceCategory:14T05

6. CJM Online first

Levin, Aaron; Wang, Julie Tzu-Yueh
 On non-Archimedean curves omitting few components and their arithmetic analogues Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean 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 non-Archimedean 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 non-Archimedean analytic map from $\mathbf{k}$ into $X\setminus D_1 \cup D_2$. Using the dictionary between non-Archimedean 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:non-Archimedean Picard theorem, non-Archimedean analytic curves, integral pointsCategories:11J97, 32P05, 32H25

7. CJM 2014 (vol 67 pp. 152)

Lescop, Christine
 On Homotopy Invariants of Combings of Three-manifolds 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 Casson-Walker invariant. The article also includes a self-contained 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, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integralsCategories: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 3-valent fan tropical curves in a fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space 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 theoryCategories:14T05, 14M25

9. CJM 2014 (vol 67 pp. 667)

Nishinou, Takeo
 Toric Degenerations, Tropical Curve, and Gromov-Witten Invariants of Fano Manifolds In this paper, we give a tropical method for computing Gromov-Witten 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, Gromov-Witten invariants, tropical curvesCategory: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 Jacquet-Langlands 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 Jacquet-Langlands 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, Jacquet-Langlands correspondenceCategories: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, CalderÃ³n-Zygmund operatorsCategories: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 one-to-one 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 spaceCategories:14T05, 14D22

13. CJM 2012 (vol 64 pp. 254)

Bell, Jason P.; Hare, Kevin G.
 Corrigendum to On $\mathbb{Z}$-modules of Algebraic Integers'' We fix a mistake in the proof of Theorem 1.6 in the paper in the title. Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theoremCategories:11R04, 11R06

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 numbersCategory: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 functionCategories: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 blow-ups 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 variationCategories: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 RadÃ³-Reichelderfer 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 Morrey-type 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 so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation. Keywords: monotone mapping, DM mapping, RadÃ³-Reichelderfer property, UDM mapping, differentiabilityCategories:26B05, 46G05

18. CJM 2009 (vol 61 pp. 641)

 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 vectorCategories: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, subordinationCategories: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 $d-1$ 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 Pisot-cyclotomic 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 Pisot-cyclotomic numbers of any fixed order. \end{compactenum} Keywords:algebraic integers, Pisot numbers, full rank, discriminantCategories:11R04, 11R06

21. CJM 2008 (vol 60 pp. 1067)

Kariyama, Kazutoshi
 On Types for Unramified $p$-Adic Unitary Groups Let $F$ be a non-archimedean 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 Bushnell--Kutzko 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 algebraCategories: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 setCategories:13J07, 32B20, 13J10, 32S10

23. CJM 2007 (vol 59 pp. 696)

Bangoura, Momo
 AlgÃ¨bres de Lie d'homotopie associÃ©es Ã  une proto-bigÃ¨bre de Lie On associe \a toute structure de proto-big\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 proto-big\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 proto-Lie algebra structure on a finite-dimensional 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 proto-Lie 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 skew-symmetric 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, quasi-bigÃ¨bre de Lie, proto-bigÃ¨bre de Lie, crochet dÃ©rivÃ©, jacobiateurCategories: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, cozero-subspaces, 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 \s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger condition. We get some results in the non-Lindel\"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© plankCategories: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 productCategories:53C26, 53C29
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