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1. CJM Online first

Ng, P. W.; Skoufranis, P.
Closed convex hulls of unitary orbits in certain simple real rank zero C$^*$-algebras
In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C$^*$-algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

Keywords:convex hull of unitary orbits, real rank zero C*-algebras simple, eigenvalue function, majorization
Category:46L05

2. CJM Online first

Harrison-Trainor, Matthew; Melnikov, Alexander; Miller, Russell
On computable field embeddings and difference closed fields
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of computable difference fields into computable difference closed fields.

Keywords:computable algebra, algebraic field, difference field, extension of automorphism
Categories:03D45, 03C57, 12Y05

3. CJM Online first

Ghaani Farashahi, Arash
A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups
This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated to the Weil's formula and $1\le p\lt \infty$. We then present a structured class of abstract linear representations of the Banach convolution function algebras $L^p(G/H,\mu)$.

Keywords:homogeneous space, linear representation, continuous unitary representation, convolution function algebra, compact group, convolution, involution
Categories:43A85, 47A67, 20G05

4. CJM Online first

Manon, Christopher
Toric geometry of $SL_2(\mathbb{C})$ free group character varieties from outer space
Culler and Vogtmann defined a simplicial space $O(g)$ called outer space to study the outer automorphism group of the free group $F_g$. Using representation theoretic methods, we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g, SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety of $F_g,$ reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O(g)$ defines a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g, SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a symplectomorphism on a dense, open subset of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ with respect to natural symplectic structures on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$, and we show that each $v$ defines a topological decomposition of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O(g)$ all arise as divisorial valuations built from an associated projective compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$

Keywords:character variety, outer space, analytification, compactification, integrable system
Categories:14M25, 14T05, 14D20

5. CJM Online first

Diacu, Florin
The classical $N$-body problem in the context of curved space
We provide the differential equations that generalize the Newtonian $N$-body problem of celestial mechanics to spaces of constant Gaussian curvature, $\kappa$, for all $\kappa\in\mathbb R$. In previous studies, the equations of motion made sense only for $\kappa\ne 0$. The system derived here does more than just include the Euclidean case in the limit $\kappa\to 0$: it recovers the classical equations for $\kappa=0$. This new expression of the laws of motion allows the study of the $N$-body problem in the context of constant curvature spaces and thus offers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals.

Keywords:N-body problem, spaces of constant curvature

6. CJM Online first

Lee, Jungyun; Lee, Yoonjin
Regulators of an infinite family of the simplest quartic function fields
We explicitly find regulators of an infinite family $\{L_m\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where $\mathbb{F}_q$ is the finite field of order $q$ with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields, where they have the same conductors. We obtain a lower bound on the class numbers of the family $\{L_m\}$ and some result on the divisibility of the divisor class numbers of cyclotomic function fields which contain $\{L_m\}$ as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field $\mathbb{F}_q (t)$ in $\{L_m\}$.

Keywords:regulator, function field, quartic extension, class number
Categories:11R29, 11R58

7. CJM Online first

Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia
On the asymptotic growth of Bloch-Kato--Shafarevich-Tate groups of modular forms over cyclotomic extensions
We study the asymptotic behaviour of the Bloch--Kato--Shafarevich--Tate group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.

Keywords:cyclotomic extension, Shafarevich-Tate group, Bloch-Kato Selmer group, modular form, non-ordinary prime, p-adic Hodge theory
Categories:11R18, 11F11, 11R23, 11F85

8. CJM Online first

Pan, Shu-Yen
$L$-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction
The preservation principle of local theta correspondences of reductive dual pairs over a $p$-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams/Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

Keywords:local theta correspondence, supercuspidal representation, preservation principle, Langlands functoriality
Categories:22E50, 11F27, 20C33

9. CJM Online first

Lee, Hun Hee; Youn, Sang-gyun
New deformations of convolution algebras and Fourier algebras on locally compact groups
In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information of the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

Keywords:Fourier algebra, convolution algebra, operator algebra, Beurling algebra
Categories:43A20, 43A30, 47L30, 47L25

10. CJM Online first

Hartglass, Michael
Free product C*-algebras associated to graphs, free differentials, and laws of loops
We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that arises from a weighted graph $(\Gamma, \mu)$, specific cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma, \mu)$, and study the structure of its positive cone. We then study the $*$-algebra, $\mathcal{A}$, generated by the generators of $\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show that certain ``loop" elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements $x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

Keywords:free probability, C*-algebra
Category:46L09

11. CJM Online first

Ovchinnikov, Alexey; Wibmer, Michael
Tannakian categories with semigroup actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.

Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06

12. CJM Online first

Chen, Xianghong; Seeger, Andreas
Convolution powers of Salem measures with applications
We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-\alpha}$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb{N}$.

Keywords:convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz type
Categories:42A85, 42B99, 42B15, 42A61

13. CJM 2016 (vol 68 pp. 1382)

Zydor, Michał
La Variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires
We establish an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups. Our formula is obtained by integrating a truncated kernel à la Arthur. It has a geometric side which is a sum of distributions $J_{\mathfrak{o}}$ indexed by classes of elements of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation as well as the "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $J_{\mathfrak{o}}$ are invariant and depend only on the choice of the Haar measure on $U(n)(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of zêta functions.

Keywords:formule des traces relative
Categories:11F70, 11F72

14. CJM Online first

Nikolidakis, Eleftherios Nikolaos
Extremal sequences for the Bellman function of the dyadic maximal operator and applications to the Hardy operator
We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.

Keywords:Bellman function, dyadic, Hardy operator, maximal
Category:42B25

15. CJM Online first

Fischer, Vera; Mejia, Diego Alejandro
Splitting, Bounding, and Almost Disjointness can be quite Different
We prove the consistency of $$ \operatorname{add}(\mathcal{N})\lt \operatorname{cov}(\mathcal{N}) \lt \mathfrak{p}=\mathfrak{s} =\mathfrak{g}\lt \operatorname{add}(\mathcal{M}) = \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a} =\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c} $$ with $\mathrm{ZFC}$, where each of these cardinal invariants assume arbitrary uncountable regular values.

Keywords:cardinal characteristics of the continuum, splitting, bounding number, maximal almost-disjoint families, template forcing iterations, isomorphism-of-names
Categories:03E17, 03E35, 03E40

16. CJM 2016 (vol 68 pp. 1285)

Ehrig, Michael; Stroppel, Catharina
2-row Springer Fibres and Khovanov Diagram Algebras for Type D
We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated $\mathbb{P}^1$-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type $\operatorname D$ diagram calculus labelling the irreducible components in a convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type $\operatorname D$ setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type $\operatorname A$ to other types.

Keywords:Springer fibers, Khovanov homology, Weyl group type D
Category:17-11

17. CJM 2016 (vol 68 pp. 1120)

Stange, Katherine E.
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P \in E(\mathbb{Q})$, there is at most one integral multiple $[n]P$ such that $n \gt C$. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence $v(\Psi_n)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called \emph{elliptic troublemaker sequences}, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}(P)/h(E)$ for integer points having two large integral multiples.

Keywords:elliptic divisibility sequence, Lang's conjecture, height functions
Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52

18. CJM Online first

Grinberg, Darij
Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
The dual immaculate functions are a basis of the ring $\operatorname*{QSym}$ of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an " immaculate tableau" is defined similarly to be a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary; but each row has to weakly increase). Dual immaculate functions have been introduced by Berg, Bergeron, Saliola, Serrano and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties. In this note, we prove a conjecture of Mike Zabrocki which provides an alternative construction for the dual immaculate functions in terms of certain "vertex operators". The proof uses a dendriform structure on the ring $\operatorname*{QSym}$; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras $\operatorname*{FQSym}$ and $\operatorname*{WQSym}$.

Keywords:combinatorial Hopf algebras, quasisymmetric functions, dendriform algebras, immaculate functions, Young tableaux
Category:05E05

19. CJM Online first

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang
Strict comparison of positive elements in multiplier algebras
Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if ``finitely many extremal traces" is replaced by ``quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Keywords:strict comparison, bi-diagonal form, positive combinations
Categories:46L05, 46L35, 46L45, 47C15

20. CJM Online first

Günther, Christian; Schmidt, Kai-Uwe
$L^q$ norms of Fekete and related polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
Categories:11B83, 42A05, 30C10

21. 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 subfactor
Category:46L37

22. CJM 2016 (vol 68 pp. 1201)

Banks, Jessica; Rathbun, Matt
Monodromy Action on Unknotting Tunnels in Fiber Surfaces
In \cite{RatTOFL}, the second author showed that a tunnel of a tunnel number one, fibered link in $S^3$ can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we observe that this is true for fibered links in any 3-manifold, we analyze how the arc behaves under the monodromy action, and we show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.

Keywords:fibered, monodromy, tunnel, clean
Category:57M25

23. CJM 2016 (vol 68 pp. 961)

Greenberg, Matthew; Seveso, Marco
$p$-adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet-Langlands Correspondence
We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet-Langlands correspondence relating modular forms on $\textbf{GL}_2/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

Keywords:modular forms, p-adic families, Jacquet-Langlands correspondence, Shimura curves, eigencurves
Categories:11F11, 11F67, 11F85

24. CJM Online first

Brandes, Julia; Parsell, Scott T.
Simultaneous additive equations: Repeated and differing degrees
We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and $r$ quadratic equations.

Keywords:equations in many variables, counting solutions of Diophantine equations, applications of the Hardy-Littlewood method
Categories:11D72, 11D45, 11P55

25. CJM Online first

Saanouni, Tarek
Global and non global solutions for some fractional heat equations with pure power nonlinearity
The initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

Keywords:nonlinear fractional heat equation, global Existence, decay, blow-up
Category:35Q55
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