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On rational equivalence in tropical geometry Allermann, Lars; Hampe, Simon; Rau, Johannes Author's Draft
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).


Chern classes of splayed intersections Aluffi, Paolo; Faber, Eleonore Published: 20150505
We generalize the Chern class relation for the transversal intersection
of two nonsingular
varieties to a relation for possibly singular varieties, under
a splayedness assumption.
We show that the relation for the ChernSchwartzMacPherson classes
holds for two splayed hypersurfaces in a nonsingular variety,
and under a `strong splayedness' assumption for more
general subschemes. Moreover, the relation is shown to hold for
the ChernFulton classes
of any two splayed subschemes.
The main tool is a formula for Segre classes of splayed
subschemes. We also discuss the Chern class relation under the
assumption that one of the
varieties is a general very ample divisor.


$p$adic and Motivic Measure on Artin $n$stacks Balwe, Chetan Published: 20140703
We define a notion of $p$adic measure on Artin $n$stacks which are
of strongly finite type over the ring of $p$adic integers. $p$adic
measure on schemes can be evaluated by counting points on the
reduction of the scheme modulo $p^n$. We show that an analogous
construction works in the case of Artin stacks as well if we count the
points using the counting measure defined by Toën. As a consequence,
we obtain the result that the Poincaré and Serre series of such
stacks are rational functions, thus extending Denef's result for
varieties. Finally, using motivic integration we show that as $p$
varies, the rationality of the Serre series of an Artin stack defined
over the integers is uniform with respect to $p$.


Lyapunov Stability and Attraction Under Equivariant Maps Barros, Carlos Braga; Rocha, Victor; Souza, Josiney Published: 20150331
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed
with
admissible families of open coverings. Assume that $\mathcal{S}$ is a
semigroup acting on both $M$ and $N$. In this paper we study the behavior of
limit sets, prolongations, prolongational limit sets, attracting sets,
attractors and Lyapunov stable sets (all concepts defined for the action of
the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations
from $M$ to $N$.


Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles Biswas, Indranil; Gómez, Tomás L.; Logares, Marina Author's Draft
We prove a Torelli theorem for the moduli space of semistable
parabolic Higgs bundles over a smooth complex projective algebraic
curve under the assumption that the parabolic weight system is
generic. When the genus is at least two, using this result
we also
prove a Torelli theorem for the moduli space of semistable
parabolic
bundles of rank at least two with generic parabolic weights.
The key
input in the proofs is a method of
J.C. Hurtubise.


The SL$(2, C)$ Casson invariant for knots and the $\hat{A}$polynomial Boden, Hans Ulysses; Curtis, Cynthia L Author's Draft
In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ Casson
invariant
to arbitrary knots $K$ in integral homology 3spheres and relate
it to the $m$degree of the $\widehat{A}$polynomial of $K$. We
prove a product formula for the $\widehat{A}$polynomial of the connected
sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity
of ${SL(2, {\mathbb C})}$ Casson knot invariant under connected sum for a large
class of knots in $S^3$. We also present an example of a nontrivial
knot $K$ in $S^3$ with trivial $\widehat{A}$polynomial and trivial
${SL(2, {\mathbb C})}$ Casson knot invariant, showing that neither of these invariants
detect the unknot.


Abelian Surfaces with an Automorphism and Quaternionic Multiplication Bonfanti, Matteo Alfonso; van Geemen Published: 20150331
We construct one dimensional families of Abelian surfaces with
quaternionic multiplication
which also have an automorphism of order three or four. Using Barth's
description of the moduli space of $(2,4)$polarized Abelian surfaces,
we find the Shimura curve parametrizing these Abelian surfaces in a
specific case.
We explicitly relate these surfaces to the Jacobians of genus two
curves studied by Hashimoto and Murabayashi.
We also describe a (Humbert) surface in Barth's moduli space which
parametrizes Abelian surfaces with real multiplication by
$\mathbf{Z}[\sqrt{2}]$.


Equivariant map queer Lie superalgebras Calixto, Lucas; Moura, Adriano; Savage, Alistair Author's Draft
An equivariant map queer Lie superalgebra is the Lie superalgebra
of regular maps from an algebraic variety (or scheme) $X$ to
a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect
to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$.
In this paper, we classify all irreducible finitedimensional
representations of the equivariant map queer Lie superalgebras
under the assumption that $\Gamma$ is abelian and acts freely
on $X$. We show that such representations are parameterized
by a certain set of $\Gamma$equivariant finitely supported maps
from $X$ to the set of isomorphism classes of irreducible finitedimensional
representations of $\mathfrak{q}$. In the special case where $X$ is the
torus, we obtain a classification of the irreducible finitedimensional
representations of the twisted loop queer superalgebra.


Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems Carcamo, Cristian; Vidal, Claudio Published: 20150128
In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.


The frequency of elliptic curve groups over prime finite fields Chandee, Vorrapan; David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan Author's Draft
Letting $p$ vary over all primes and $E$ vary over all elliptic
curves over the finite field $\mathbb{F}_p$, we study the frequency to
which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
It is wellknown that the only permissible groups are of the
form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$.
Given such a candidate group, we let $M(G_{m,k})$ be the frequency
to which the group $G_{m,k}$ arises in this way.
Previously, the second and fourth named authors determined an
asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes
in short arithmetic progressions. In this paper, we prove several
unconditional bounds for $M(G_{m,k})$, pointwise and on average. In
particular, we show that $M(G_{m,k})$ is bounded above by a constant
multiple of the expected quantity when $m\le k^A$ and that the
conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups
$G_{m,k}$ when $m\le k^{1/4\epsilon}$.
We also apply our methods to study the frequency to which a given
integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.


On twofaced families of noncommutative random variables Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul Author's Draft
We demonstrate that the notions of bifree independence and combinatorialbifree
independence of twofaced families are equivalent using a diagrammatic
view of binoncrossing partitions.
These diagrams produce an operator model on a Fock space suitable
for representing any twofaced family of noncommutative random
variables.
Furthermore, using a Kreweras complement on binoncrossing partitions
we establish the expected formulas for the multiplicative convolution
of a bifree pair of twofaced families.


The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank Cojocaru, Alina Carmen; Shulman, Andrew Michael Published: 20150728
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study
the first elementary divisor
$d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies.
In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$
and prove that, on average, it has a large norm. Our work is motivated by the study of J.P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.


On a linear refinement of the PrékopaLeindler inequality Colesanti, Andrea; Gómez, Eugenia Saorín; Nicolás, Jesus Yepes Author's Draft
If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are nonnegative measurable
functions, then the PrékopaLeindler inequality asserts that
the integral of the Asplund sum (provided that it is measurable)
is greater or equal than the $0$mean of the integrals of $f$
and $g$.
In this paper we prove that under the sole assumption that $f$
and $g$ have
a common projection onto a hyperplane, the PrékopaLeindler
inequality admits a linear refinement. Moreover, the same inequality
can be obtained when assuming that both projections (not necessarily
equal as functions) have the same integral. An analogous approach
may be also carried out for the socalled BorellBrascampLieb
inequality.


Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution da Silva, Genival Jr.; Kerr, Matt; Pearlstein, Gregory Author's Draft
We collect evidence in support of a conjecture of Griffiths,
Green
and Kerr
on the arithmetic of extension classes of
limiting
mixed Hodge structures arising from semistable degenerations
over
a number field. After briefly summarizing how a result of Iritani
implies this conjecture for a collection of hypergeometric
CalabiYau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (nonhypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the MumfordTate
group (which is $G_{2}$) of the family of 6folds, and the theory
of boundary components of MumfordTate domains.


Categorical aspects of quantum groups: multipliers and intrinsic groups Daws, Matthew Author's Draft
We show that the assignment of the (left) completely bounded
multiplier algebra
$M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group
$\mathbb G$, and
the assignment of the intrinsic group, form functors between
appropriate
categories. Morphisms of locally compact quantum
groups can be described by Hopf $*$homomorphisms between universal
$C^*$algebras, by bicharacters, or by special sorts of coactions.
We show that the whole
theory of completely bounded multipliers can be lifted to the
universal
$C^*$algebra level, and that then the different pictures of
both multipliers
(reduced, universal, and as centralisers)
and morphisms interact in extremely natural ways. The intrinsic
group of a
quantum group can be realised as a class of multipliers, and
so our techniques
immediately apply. We also show how to think of the intrinsic
group using
the universal $C^*$algebra picture, and then, again, show how
the differing
views on the intrinsic group interact naturally with morphisms.
We show that
the intrinsic group is the ``maximal classical'' quantum subgroup
of a locally
compact quantum group, show that it is even closed in the strong
Vaes sense,
and that the intrinsic group functor is an adjoint to the inclusion
functor
from locally compact groups to quantum groups.


Kernels in the category of formal group laws Demchenko, Oleg; Gurevich, Alexander Author's Draft
Fontaine described the category of formal groups over the ring
of Witt vectors over a finite field
of characteristic $p$ with the aid of triples consisting of the
module of logarithms,
the Dieudonné module and the morphism from the former to the
latter. We propose
an explicit construction for the kernels in this category in
term of Fontaine's triples.
The construction is applied to the formal norm homomorphism in
the case of an unramified extension
of $\mathbb{Q}_p$ and of a totally ramified extension of degree less
or equal than $p$. A similar
consideration applied to a global extension allows us to establish
the existence of a strict
isomorphism between the formal norm torus and a formal group
law coming from $L$series.


Optimization related to some nonlocal problems of Kirchhoff type Emamizadeh, Behrouz; Farjudian, Amin; ZivariRezapour, Mohsen Author's Draft
In this paper we introduce two rearrangement optimization
problems, one being a maximization and the other a minimization
problem, related to a nonlocal boundary value problem of Kirchhoff
type. Using the theory of rearrangements as developed by
G. R. Burton we are able to show that both problems are solvable,
and derive the corresponding optimality conditions. These conditions
in turn provide information concerning the locations of the
optimal
solutions. The strict convexity of the energy functional plays
a
crucial role in both problems. The popular case in which the
rearrangement class (i.e., the admissible set) is generated
by a
characteristic function is also considered. We show that in
this
case, the maximization problem gives rise to a free boundary
problem
of obstacle type, which turns out to be unstable. On the other
hand,
the minimization problem leads to another free boundary problem
of
obstacle type, which is stable. Some numerical results are
included
to confirm the theory.


Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property Fernández Bretón, David J. Author's Draft
We answer two questions of Hindman, Steprāns and Strauss,
namely we prove that every
strongly summable
ultrafilter on an abelian group is sparse and has the trivial
sums property. Moreover we
show that in most
cases the sparseness of the given ultrafilter is a
consequence of its being isomorphic to a union ultrafilter. However,
this does not happen
in all cases:
we also construct (assuming Martin's Axiom for countable partial
orders, i.e.
$\operatorname{cov}(\mathcal{M})=\mathfrak c$), on the
Boolean group, a strongly summable ultrafilter that
is not additively isomorphic to any union ultrafilter.


Frobenius distribution for quotients of Fermat curves of prime exponent Fité, Francesc; González, Josep; Lario, Joan Carles Author's Draft
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime
exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$
of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians
$\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell2$, where
$\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by
certain subgroups of automorphisms of $\mathcal{C}$. It is well known
that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely
simple abelian variety $B_k$ with complex multiplication. We call
degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM
type. For a nondegenerate pair $(\ell,k)$, we compute the SatoTate
group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized
SatoTate Conjecture for it, and give an explicit method to compute
the moments and measures of the involved distributions. Regardless of
$(\ell,k)$ being degenerate or not, we also obtain Frobenius
equidistribution results for primes of certain residue degrees in the
$\ell$th cyclotomic field. Key to our results is a detailed study of
the rank of certain generalized Demjanenko matrices.


Strongly incompressible curves GarciaArmas, Mario Author's Draft
Let $G$ be a finite group. A faithful $G$variety $X$ is called
strongly incompressible if every dominant $G$equivariant rational
map of $X$ onto another faithful $G$variety $Y$ is birational.
We settle the problem of existence of strongly incompressible
$G$curves for any finite group $G$ and any base field $k$ of
characteristic zero.


On the rate of convergence of empirical measures in $\infty$transportation distance Garcia Trillos, Nicolas; Slepcev, Dejan Published: 20150302
We consider random i.i.d. samples of absolutely continuous measures
on bounded connected domains.
We prove an upper bound on the $\infty$transportation distance
between the measure and the empirical measure of the sample.
The bound is optimal in terms of scaling with the number of sample
points.


Bilinear and quadratic forms on rational modules of split reductive groups Garibaldi, Skip; Nakano, Daniel K. Author's Draft
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.


Strong Logarithmic Sobolev Inequalities for LogSubharmonic Functions Graczyk, Piotr; Kemp, Todd; Loeb, JeanJacques Author's Draft
We prove an intrinsic equivalence between strong
hypercontractivity and a strong logarithmic Sobolev
inequality for the cone of logarithmically subharmonic
(LSH) functions. We introduce a new large class of measures,
Euclidean regular and exponential type, in addition to all compactlysupported
measures, for which this equivalence holds. We prove a Sobolev
density theorem through LSH functions and use it to prove
the equivalence of strong
hypercontractivity and the strong logarithmic Sobolev
inequality for such logsubharmonic
functions.


Les $\theta$régulateurs locaux d'un nombre algébrique  Conjectures $p$adiques Gras, Georges Author's Draft
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that
$\operatorname{Reg}_\infty (\eta) \ne 0$.
We define the local $\theta$regulators $\Delta_p^\theta(\eta)
\in \mathbb{F}_p$
for the $\mathbb{Q}_p\,$irreducible characters $\theta$ of
$G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \,
V_\theta$ is associated with
$\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta
\geq 1$.
Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$
modulo $p$ in the factorization
$\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of
$\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]}
}$ (normalized $p$adic regulator).
From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal
L}^\theta \simeq \delta \, V_\theta\big )
\leq p^{ f \delta^2}$ ($f \geq 1$ is a residue degree) and the
BorelCantelli heuristic,
we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$
is a $p$adic unit or that
$p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$
with $f=\delta=1$); this obstruction may be lifted assuming the
existence of a binomial probability law
confirmed through numerical studies
(groups $C_3$, $C_5$, $D_6$).
This conjecture would imply that, for all $p$ large enough,
Fermat quotients, normalized $p$adic
regulators are $p$adic units and that
number fields are $p$rational.
We recall some deep cohomological results that
may strengthen such conjectures.


Characterizing the absolute continuity of the convolution of orbital measures in a classical Lie algebra Gupta, Sanjiv Kumar; Hare, Kathryn Author's Draft
Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension
$d$. It is
a classical result that the convolution of any $d$ nontrivial,
$G$invariant,
orbital measures is absolutely continuous with respect to
Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ nontrivial
orbits
has nonempty interior. The number $d$ was later reduced to the
rank of the
Lie algebra (or rank $+1$ in the case of type $A_{n}$). More
recently, the
minimal integer $k=k(X)$ such that the $k$fold convolution of
the orbital
measure supported on the orbit generated by $X$ is an absolutely
continuous
measure was calculated for each $X\in \mathfrak{g}$.


The lower bound on the EulerPoincaré characteristic of certain surfaces of general type with a linear pencil of hyperelliptic curves Ishida, Hirotaka Author's Draft
Let $S$ be a surface of general type.
In this article, when there exists a relatively minimal hyperelliptic
fibration $f \colon S
\rightarrow \mathbb{P}^1$ whose slope is less than or equal to four,
we show the lower bound
on the EulerPoincaré characteristic of $S$.
Furthermore, we prove that our bound is the best possible by
giving required
hyperelliptic fibrations.


Quotients of $A_2 * T_2$ Izumi, Masaki; Morrison, Scott; Penneys, David Author's Draft
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.


Pathological phenomena in DenjoyCarleman classes Jaffe, Ethan Y. Published: 20150521
Let $\mathcal{C}^M$ denote a DenjoyCarleman class of $\mathcal{C}^\infty$
functions (for a given logarithmicallyconvex sequence $M = (M_n)$).
We construct: (1) a function in $\mathcal{C}^M((1,1))$ which
is nowhere in any smaller class; (2) a function on $\mathbb{R}$ which
is formally $\mathcal{C}^M$ at every point, but not in
$\mathcal{C}^M(\mathbb{R})$;
(3) (under the assumption of quasianalyticity) a smooth function
on $\mathbb{R}^p$ ($p \geq 2$) which is $\mathcal{C}^M$ on every $\mathcal{C}^M$
curve, but not in $\mathcal{C}^M(\mathbb{R}^p)$.


Functiontheoretic Properties for the Gauss Maps of Various Classes of Surfaces Kawakami, Yu Published: 20150415
We elucidate the geometric background of functiontheoretic properties
for the Gauss maps of
several classes of immersed surfaces in threedimensional space
forms, for example, minimal surfaces in Euclidean threespace, improper affine spheres in the affine threespace, and constant
mean curvature one surfaces and flat surfaces in hyperbolic threespace. To achieve this purpose, we prove an optimal curvature bound
for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for
the Gauss maps of these classes of surfaces.


Constrained approximation with Jacobi weights Kopotun, Kirill; Leviatan, Dany; Shevchuk, Igor Author's Draft
In this paper, we prove that, for $\ell=1$ or $2$, the rate of
best $\ell$monotone polynomial approximation in the $L_p$
norm ($1\leq p \leq \infty$) weighted by the Jacobi weight
$w_{\alpha,\beta}(x)
:=(1+x)^\alpha(1x)^\beta$ with $\alpha,\beta\gt 1/p$
if $p\lt \infty$, or $\alpha,\beta\geq
0$ if $p=\infty$,
is bounded by an appropriate $(\ell+1)$st modulus of smoothness
with the same weight, and that this rate cannot be bounded by
the $(\ell+2)$nd modulus. Related results on constrained weighted
spline approximation and applications of our estimates are also
given.


On nonArchimedean curves omitting few components and their arithmetic analogues Levin, Aaron; Wang, Julie TzuYueh Author's Draft
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.


Monotone Classes of Dendrites MartínezdelaVega, Veronica; Mouron, Christopher Author's Draft
Continua $X$ and $Y$ are monotone equivalent
if there exist monotone onto maps $f:X\longrightarrow Y$ and
$g:Y\longrightarrow X$. A continuum $X$ is isolated with respect
to monotone maps if every continuum that is monotone equivalent
to $X$ must also be homeomorphic to
$X$. In this paper we show that a dendrite $X$ is isolated with
respect to
monotone maps if and only if the set of ramification points of
$X$ is
finite. In this way we fully characterize the classes of dendrites
that are
monotone isolated.


Geometric invariants of cuspidal edges Martins, Luciana de Fátima; Saji, Kentaro Author's Draft
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.


Metric spaces admitting lowdistortion embeddings into all $n$dimensional Banach spaces Ostrovskii, Mikhail; Randrianantoanina, Beata Author's Draft
For a fixed $K\gg 1$ and
$n\in\mathbb{N}$, $n\gg 1$, we study metric
spaces which admit embeddings with distortion $\le K$ into each
$n$dimensional Banach space. Classical examples include spaces
embeddable
into $\log n$dimensional Euclidean spaces, and equilateral spaces.


Centrevalued Index for Toeplitz Operators with Noncommuting Symbols Phillips, John; Raeburn, Iain Author's Draft
We formulate and prove a ``winding number'' index
theorem for certain ``Toeplitz'' operators in the same spirit
as GohbergKrein, Lesch and others. The ``number'' is replaced
by a selfadjoint operator in a subalgebra $Z\subseteq Z(A)$
of a unital $C^*$algebra, $A$. We assume a faithful $Z$valued
trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf
R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the
infinitesimal generator of $\alpha$ and $u$ is invertible in
$\operatorname{dom}(\delta)$ then the
``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{1})\in
Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$
to a von Neumann setting
$(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$
and $\mathfrak{Z}=Z^{\prime\prime}.$
Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von
Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$trace
on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf
R}$
corresponding to the nonnegative spectrum of the generator of
$\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and
$\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$
is the embedding then we define for $u\in A^{1}$, $T_u=P\tilde\pi(u)
P$
and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$valued
index of $T_u$ is the negative of the winding operator.
In outline the proof follows the proof of the scalar case done
previously by the authors. The main difficulty is making sense
of the constructions with the scalars replaced by $\mathfrak{Z}$ in
the von Neumann setting. The construction of the dual $\mathfrak{Z}$trace
on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development
of a $\mathfrak{Z}$Hilbert Algebra theory. We show that certain of
these Fredholm operators fiber as a ``section'' of Fredholm operators
with scalarvalued index and the centrevalued index fibers as
a section of the scalarvalued indices.


On positive definiteness over locally compact quantum groups Runde, Volker; Viselter, Ami Author's Draft
The notion of positivedefinite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various wellknown results about
positivedefinite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positivedefinite functions, comparison
of
various topologies, positivedefinite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.


The Weak bprinciple: Mumford Conjecture Sadykov, Rustam Author's Draft
In this note we introduce and study a new class of maps called
oriented colored broken submersions. This is the simplest class
of maps that satisfies a version of the bprinciple and in dimension
$2$ approximates the class of oriented submersions well in the
sense that
every oriented colored broken submersion of dimension $2$ to
a closed simply connected manifold is bordant to a submersion.
We show that the MadsenWeiss theorem (the standard Mumford Conjecture)
fits a general setting of the bprinciple. Namely, a version
of the bprinciple for
oriented colored broken submersions together with the Harer
stability theorem and MillerMorita theorem implies the MadsenWeiss
theorem.


Lower Escape Rate of Symmetric Jumpdiffusion Processes Shiozawa, Yuichi Author's Draft
We establish an integral test on the lower escape rate
of symmetric jumpdiffusion processes generated by regular Dirichlet
forms.
Using this test, we can find the speed of particles escaping
to infinity.
We apply this test to symmetric jump processes of variable order. We also derive the upper and lower escape rates of time changed
processes
by using those of underlying processes.


Quantum families of invertible maps and related problems Skalski, Adam; Sołtan, Piotr Author's Draft
The notion of families of quantum invertible maps (C$^*$algebra
homomorphisms satisfying Podleś' condition) is employed to strengthen
and reinterpret several results concerning universal quantum
groups acting on finite quantum spaces. In particular Wang's
quantum automorphism groups are shown to be universal with respect
to quantum families of invertible maps. Further the construction
of the Hopf image of Banica and Bichon is phrased in the purely
analytic language and employed to define the quantum subgroup
generated by a family of quantum subgroups or more generally
a family of quantum invertible maps.


Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials Stange, Katherine E. Author's Draft
Assuming Lang's conjectured lower bound on the heights of nontorsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and nontorsion
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 nonsingular 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 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.


Nonstable $K_1$functors of multiloop groups Stavrova, Anastasia Author's Draft
Let $k$ be a field of characteristic 0. Let $G$ be a reductive
group over the ring of Laurent polynomials
$R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains
a maximal $R$torus, and
that every semisimple normal subgroup of $G$ contains a twodimensional
split torus $\mathbf{G}_m^2$.
We show that the natural map of nonstable $K_1$functors, also
called Whitehead groups,
$K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective,
and an isomorphism if $G$ is semisimple.
As an application, we provide a way to compute the difference
between the
full automorphism group of a Lie torus (in the sense of YoshiiNeher)
and the subgroup generated by
exponential automorphisms.


Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$ Takeda, Shuichiro Author's Draft
Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi
subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage
in the $n$fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$,
we construct (under a certain
technical assumption, which is always satisfied when $n=2$) an
automorphic representation $\pi$
of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the
representations $\pi_1,\dots,\pi_k$. This is
the global analogue of the metaplectic tensor product
defined by P. Mezo in the sense that locally at each place $v$,
$\pi_v$ is equivalent to the local metaplectic tensor product of
$\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all
of $\pi_i$ are cuspidal (resp. squareintegrable modulo center), then
the metaplectic tensor product is cuspidal (resp. squareintegrable
modulo center). We also show that (both
locally and globally) the metaplectic tensor product behaves in the
expected way under the action of a Weyl group element, and show the
compatibility with parabolic inductions.

