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1. CJM Online first

Benaych-Georges, Florent; Cébron, Guillaume; Rochet, Jean
Fluctuation of matrix entries and application to outliers of elliptic matrices
For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in K}$ which is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic (such random variables include for example the normalized matrix entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic independence of the projection of the matrices $\mathbf{A}_k$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other. These phenomena have already been observed with random matrices from the Single Ring Theorem.

Keywords:random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure
Categories:60B20, 15B52, 60F05, 46L54

2. CJM Online first

Wang, Zhenjian
On algebraic surfaces associated with line arrangements
For a line arrangement $\mathcal{A}$ in the complex projective plane $\mathbb{P}^2$, we investigate the compactification $\overline{F}$ in $\mathbb{P}^3$ of the affine Milnor fiber $F$ and its minimal resolution $\widetilde{F}$. We compute the Chern numbers of $\widetilde{F}$ in terms of the combinatorics of the line arrangement $\mathcal{A}$. As applications of the computation of the Chern numbers, we show that the minimal resolution is never a quotient of a ball; in addition, we also prove that $\widetilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities; finally, we compute all the Hodge numbers of some $\widetilde{F}$ by using some knowledge about the Milnor fiber monodromy of the arrangement.

Keywords:line arrangement, Milnor fiber, algebraic surface, Chern number
Categories:32S22, 32S25, 14J17, 14J29, 14J70

3. CJM Online first

Conway, Anthony
An explicit computation of the Blanchfield pairing for arbitrary links
Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing which is defined on the torsion submodule of the Alexander module of $L$. In some particular cases, namely if $L$ is a boundary link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$ can be computed explicitly; however no formula is known in general. In this article, we compute the Blanchfield pairing of any link, generalizing the aforementioned results. As a corollary, we obtain a new proof that the Blanchfield pairing is hermitian. Finally, we also obtain short proofs of several properties of $\operatorname{Bl}(L)$.

Keywords:link, Blanchfield pairing, C-complex, Alexander module

4. CJM Online first

Cordova Bulens, Hector; Lambrechts, Pascal; Stanley, Don
Rational models of the complement of a subpolyhedron in a manifold with boundary
Let $W$ be a compact simply connected triangulated manifold with boundary and $K\subset W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of $W\backslash K$ out of a model of the map of pairs $(K,K \cap \partial W)\hookrightarrow (W,\partial W)$ under some high codimension hypothesis. We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

Keywords:Lefschetz duality, Sullivan model, configuration space
Categories:55P62, 55R80

5. CJM Online first

Phan, Tuoc
Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts
This paper investigates regularity in Lorentz spaces of weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions which are measurable in $(x,t)$-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John Nirenberg space. The results are even new when the drifts are identically zero because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón-Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known, available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a "double-scaling parameter" technique, and the maximal function free approach introduced by Acerbi and Mingione.

Keywords:gradient estimate, quasi-linear parabolic equation, divergence-free drift
Categories:35B45, 35K57, 35K59, 35K61

6. CJM Online first

Glöckner, Helge
Completeness of infinite-dimensional Lie groups in their left uniformity
We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$ such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on~$G_n$, for each $n\in\mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each~$G_n$ whenever $\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood in~$G$ for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover, each $G_n$ is complete, then~$G$ is complete. We also show that the weak direct product $\bigoplus_{j\in J}G_j$ is complete for each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$. As a consequence, every strict direct limit $G=\bigcup_{n\in \mathbb{N}}G_n$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\operatorname{Diff}_c(M)$ of a paracompact finite-dimensional smooth manifold~$M$ and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$ and complete Lie group~$H$ modelled on a complete locally convex space.

Keywords:infinite-dimensional Lie group, left uniform structure, completeness
Categories:22E65, 22A05, 22E67, 46A13, 46M40, 58D05

7. CJM Online first

Hajir, Farshid; Maire, Christian
On the invariant factors of class groups in towers of number fields
For a finite abelian $p$-group $A$ of rank $d=\dim A/pA$, let $\mathbb{M}_A := \log_p |A|^{1/d}$ be its \emph{(logarithmic) mean exponent}. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\mathrm{L}/K$ of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-$p$ towers in which the mean exponent of $p$-class groups remains bounded. Several explicit examples are given with $p=2$. Turning to group theory, we introduce an invariant $\underline{\mathbb{M}}(G)$ attached to a finitely generated pro-$p$ group $G$; when $G=\operatorname{Gal}(\mathrm{L}/\mathrm{K})$, where $\mathrm{L}$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}(G)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\mathrm{L}/\mathrm{K}$. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Keywords:class field tower, ideal class group, pro-p group, p-adic analytic group, Brauer-Siegel Theorem
Categories:11R29, 11R37

8. CJM Online first

Dyachenko, Mikhail; Mukanov, Askhat; Tikhonov, Sergey
Uniform convergence of trigonometric series with general monotone coefficients
We study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

Keywords:trigonometric series, general monotone sequence, uniform convergence
Categories:42A16, 42A20, 42A32

9. CJM Online first

Ha, Junsoo
Smooth Polynomial Solutions to a Ternary Additive Equation
Let $\mathbf{F}_{q}[T]$ be the ring of polynomials over the finite field of $q$ elements, and $Y$ be a large integer. We say a polynomial in $\mathbf{F}_{q}[T]$ is $Y$-smooth if all of its irreducible factors are of degree at most $Y$. We show that a ternary additive equation $a+b=c$ over $Y$-smooth polynomials has many solutions. As an application, if $S$ is the set of first $s$ primes in $\mathbf{F}_{q}[T]$ and $s$ is large, we prove that the $S$-unit equation $u+v=1$ has at least $\exp(s^{1/6-\epsilon}\log q)$ solutions.

Keywords:smooth number, polynomial over a finite field, circle method
Categories:11T55, 11D04, 11L07, 11T23

10. CJM Online first

McDiarmid, Colin; Wood, David R.
Edge-Maximal Graphs on Surfaces
We prove that for every surface $\Sigma$ of Euler genus $g$, every edge-maximal embedding of a graph in $\Sigma$ is at most $O(g)$ edges short of a triangulation of $\Sigma$. This provides the first answer to an open problem of Kainen (1974).

Keywords:graph, surface, embedding

11. CJM Online first

Hare, Kathryn; Hare, Kevin; Ng, Michael Ka Shing
Local dimensions of measures of finite type II -- Measures without full support and with non-regular probabilities
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho x+d_{j}$ and probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested in the self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval. Under some mild technical assumptions, we prove that there exists a subset of supp$\mu $ of full $\mu $ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support and we show that the dimension of the support can be computed using only information about the essential class. To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the $kth$ convolution of the associated Cantor measure has local dimension at $x\in (0,1)$ tending to 1 as $k$ tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

Keywords:multi-fractal analysis, local dimension, IFS, finite type
Categories:28A80, 28A78, 11R06

12. CJM Online first

Geroldinger, Alfred; Zhong, Qinghai
Long sets of lengths with maximal elasticity
We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $\Delta_{\rho} (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $\rho (H)$. We study $\Delta_{\rho} (H)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

Keywords:transfer Krull monoid, weakly Krull monoid, set of length, elasticity
Categories:13A05, 13F05, 16H10, 16U30, 20M13

13. CJM Online first

Elazar, Boaz; Shaviv, Ary
Schwartz functions on real algebraic varieties
We define Schwartz functions, tempered functions and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.

Keywords:real algebraic geometry, Schwartz function, tempered distribution
Categories:14P99, 14P05, 22E45, 46A11, 46F05

14. CJM Online first

Handelman, David
Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk
Categories:37A05, 06F25, 28D05, 46B40, 60G50

15. CJM Online first

Courtney, Kristin; Shulman, Tatiana
Elements of $C^*$-algebras attaining their norm in a finite-dimensional representation
We characterize the class of RFD $C^*$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^*$-algebra is finite-dimensional, which is equivalent to the $C^*$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^*$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

Keywords:AF-telescope, RFD, projective
Categories:46L05, 47A67

16. CJM Online first

Cohen, David Bruce
Lipschitz 1-connectedness for some solvable Lie groups
A space X is said to be Lipschitz 1-connected if every L-Lipschitz loop in X bounds a O(L)-Lipschitz disk. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.

Keywords:Dehn function, solvable group, lipschitz $1$-connectedness
Categories:20F65, 22E25

17. CJM Online first

Osaka, Hiroyuki; Teruya, Tamotsu
The Jiang-Su absorption for inclusions of unital C*-algebras
We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras $P \subset A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E\colon A \rightarrow A^G$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital C*-algebras which is closed under the following conditions: (1) If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$. (2) If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$. (3) If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection, then $pAp \in \mathcal{C}$. Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial $\mathcal{C}$-algebra in the sense of Fan and Fang and a conditional expectation $E\colon A \rightarrow P$ is of index-finite type with the tracial Rokhlin property, then $P$ is a unital local tracial $\mathcal{C}$-algebra. The main result is that if $A$ is simple, separable, unital nuclear, Jiang-Su absorbing and $E\colon A \rightarrow P$ has the tracial Rokhlin property, then $P$ is Jiang-Su absorbing. As an application, when an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property, then for any subgroup $H$ of $G$ the fixed point algebra $A^H$ and the crossed product algebra $A \rtimes_{\alpha_{|H}} H$ is Jiang-Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W(A)$ is hereditary to $W(P)$ if $A$ is simple, separable, exact, unital, and $E\colon A \rightarrow P$ has the tracial Rokhlin property.

Keywords:Jiang-Su absorption, inclusion of C*-algebra, strict comparison
Categories:46L55, 46L35

18. CJM Online first

Bickerton, Robert T.; Kakariadis, Evgenios T.A.
Free Multivariate w*-Semicrossed Products: Reflexivity and the Bicommutant Property
We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products of factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bicommutant property if and only if the ambient algebra of the dynamics does also.

Keywords:reflexivity, semicrossed product
Categories:47A15, 47L65, 47L75, 47L80

19. CJM Online first

Luo, Caihua
Spherical fundamental lemma for metaplectic groups
In this paper, we prove the spherical fundamental lemma for metaplectic group $Mp_{2n}$ based on the formalism of endoscopy theory by J.Adams, D.Renard and Wen-Wei Li.

Keywords:metaplectic group, endoscopic group, elliptic stable trace formula, fundamental lemma

20. CJM Online first

Georgescu, Magdalena Cecilia
Integral Formula for Spectral Flow for $p$-Summable Operators
Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace $\tau$. For a path of self-adjoint Breuer-Fredholm operators, the spectral flow measures the net amount of spectrum which moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer-Fredholm operator affiliated with $\mathcal{N}$. If the unbounded operator is p-summable (that is, its resolvents are contained in the ideal $L^p$), then it is possible to obtain an integral formula which calculates spectral flow. This integral formula was first proven by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\theta$-summable operators, and then using Laplace transforms to obtain a p-summable formula. In this paper, we present a direct proof of the p-summable formula, which is both shorter and simpler than theirs.

Keywords:spectral flow, $p$-summable Fredholm module
Categories:19k56, 46L87, , 58B34

21. CJM Online first

de Joannis de Verclos, Rémi; Kang, Ross J.; Pastor, Lucas
Colouring squares of claw-free graphs
Is there some absolute $\varepsilon > 0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\chi(G^2) \le (2-\varepsilon) \omega(G)^2$, where $\omega(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case of $G$ the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erdős and Nešetřil.

Keywords:graph colouring, Erdős--Nešetřil conjecture, claw-free graphs
Categories:05C15, 05C35, 05C70

22. CJM Online first

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.
Geometric classification of graph C*-algebras over finite graphs
We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountably many ideals. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the $C^*$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^*$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^*$-algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantum lens spaces defined by Hong and Szymański, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.

Keywords:graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalence
Categories:46L35, 46L80, 46L55, 37B10

23. CJM Online first

Zhang, Chao
Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
For a Shimura variety of Hodge type with hyperspecial level structure at a prime~$p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\gt 2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l(w)$ (i.e. the length of $w$) if it is non-empty. We also determine the closure of each stratum.

Keywords:Shimura variety, F-zip
Categories:14G35, 11G18

24. CJM Online first

Asakura, Masanori; Otsubo, Noriyuki
CM periods, CM Regulators and Hypergeometric Functions, I
We prove the Gross-Deligne conjecture on CM periods for motives associated with $H^2$ of certain surfaces fibered over the projective line. Then we prove for the same motives a formula which expresses the $K_1$-regulators in terms of hypergeometric functions ${}_3F_2$, and obtain a new example of non-trivial regulators.

Keywords:period, regulator, complex multiplication, hypergeometric function
Categories:14D07, 19F27, 33C20, 11G15, 14K22

25. CJM 2017 (vol 69 pp. 992)

Bremner, Murray; Dotsenko, Vladimir
Classification of Regular Parametrized One-relation Operads
Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\ ; \] such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_\sigma\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $((a_1a_2)a_3)=0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

Keywords:parametrized one-relation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis
Categories:18D50, 13B25, 13P10, 13P15, 15A54, 17-04, , , , , 17A30, 17A50, 20C30, 68W30
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