
The following papers are the latest research papers available from the Canadian Journal of Mathematics.
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Mixed perverse sheaves on flag varieties for Coxeter groups Achar, Pramod N.; Riche, Simon; Vay, Cristian Author's Draft
In this paper we construct an abelian category of "mixed perverse
sheaves" attached to any realization of a Coxeter group, in
terms of the associated EliasWilliamson diagrammatic category.
This construction extends previous work of the first two authors,
where we worked with parity complexes instead of diagrams, and
we extend most of the properties known in this case to the general
setting. As an application we prove that the split Grothendieck
group of the EliasWilliamson diagrammatic category is isomorphic
to the corresponding Hecke algebra, for any choice of realization.


Geometry of infinitely presented small cancellation groups and quasihomomorphisms Arzhantseva, Goulnara; Druţu, Cornelia Author's Draft
We study the geometry of infinitely presented groups satisfying
the small cancellation condition $C'(1/8)$, and introduce a standard
decomposition (called the crisscross decomposition) for the elements
of such groups. Our method yields a direct construction of a
linearly independent set of power continuum in the kernel of
the comparison map between the bounded and the usual group cohomology
in degree $2$, without the use of free subgroups and extensions.


Isomorphic structure of Cesàro and Tandori spaces Astashkin, Sergey V.; Lesnik, Karol; Maligranda, Lech Published: 20180406
We investigate the isomorphic structure of the Cesàro spaces
and their duals, the Tandori spaces.
The main result states that the Cesàro function space $Ces_{\infty}$
and its sequence counterpart
$ces_{\infty}$ are isomorphic, which answers the question posted
previously.
This is rather surprising
since $Ces_{\infty}$ (like the known Talagrand's example)
has no natural lattice predual.
We prove that $ces_{\infty}$ is not isomorphic to ${\ell}_{\infty}$
nor is $Ces_{\infty}$ isomorphic to the
Tandori space $\widetilde{L_1}$ with the norm $\f\_{\widetilde{L_1}}=
\\widetilde{f}\_{L_1},$ where
$\widetilde{f}(t):= \operatorname{esssup}_{s \geq t} f(s).$ Our investigation
involves also an examination of the
Schur and DunfordPettis properties of Cesàro and Tandori
spaces.
In particular, using results of Bourgain we show that a wide
class of CesàroMarcinkiewicz and
CesàroLorentz spaces have the latter property.


Geometry of uniform spanning forest components in high dimensions Barlow, Martin T.; Járai, Antal A. Author's Draft
We study the geometry of the component of the origin
in the uniform spanning forest of $\mathbb{Z}^d$
and give bounds on the size of balls in the intrinsic metric.


Cubic twin prime polynomials are counted by a modular form BarySoroker, Lior; Stix, Jakob M. Author's Draft
We present the geometry lying behind counting twin prime polynomials
in $\mathbb{F}_q[T]$ in general.
We compute cohomology and explicitly count points by means of
a twisted Lefschetz trace formula
applied to these parametrizing varieties for cubic twin
prime polynomials.
The elliptic curve $X^3 = Y(Y1)$ occurs in the geometry, and
thus counting cubic twin prime polynomials involves the associated
modular form. In theory, this approach can be extended to higher
degree twin primes, but the computations become harder.


A Boltzmann approach to percolation on random triangulations Bernardi, Olivier; Curien, Nicolas; Miermont, Grégory Author's Draft
We study the percolation model on Boltzmann triangulations using
a generating function approach. More precisely, we consider a
Boltzmann model on the set of finite planar triangulations, together
with a percolation configuration (either sitepercolation or
bondpercolation) on this triangulation.
By enumerating triangulations with boundaries according to both
the boundary length and the number of vertices/edges on the boundary,
we are able to identify a phase transition for the geometry of
the origin cluster.
For instance, we show that the probability that a percolation
interface has length $n$ decays exponentially with $n$ except
at a particular value $p_c$ of the percolation parameter $p$
for which the decay is polynomial (of order $n^{10/3}$). Moreover,
the probability that the origin cluster has size $n$ decays
exponentially if $p\lt p_c$ and polynomially if $p\geq p_c$.


Ramification of the Eigencurve at classical RM points Betina, Adel Author's Draft
J.Bellaïche and M.Dimitrov have shown that the $p$adic eigencurve
is smooth but not étale over the weight space at $p$regular
theta series attached to a character of a real quadratic field
$F$ in which $p$ splits. In this paper we prove the existence
of an isomorphism between the subring fixed by the AtkinLehner
involution of the completed local ring of the eigencurve at these
points and an universal ring representing a pseudodeformation
problem. Additionally, we give a precise criterion for which
the ramification index is exactly $2$.
We finish this paper by proving the smoothness of the nearly
ordinary and ordinary Hecke algebras for Hilbert modular forms
over $F$ at the overconvergent cuspidal Eisenstein points, being
the base change lift for $\operatorname{GL}(2)_{/F}$ of these theta series.
Our approach uses deformations and pseudodeformations of reducible
Galois representations.


Free Multivariate w*Semicrossed Products: Reflexivity and the Bicommutant Property Bickerton, Robert T.; Kakariadis, Evgenios T.A. Published: 20180321
We study w*semicrossed products over actions of the free semigroup
and the free abelian semigroup on (possibly nonselfadjoint)
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.


Logarithmes des points rationnels des variétés abéliennes Bosser, Vincent; Gaudron, Éric Published: 20180618
Nous démontrons une généralisation
du théorème des périodes de Masser et Wüstholz
où la période est remplacée par un logarithme non
nul $u$ d'un point rationnel $p$ d'une variété abélienne
définie sur un corps de nombres. Nous en déduisons des
minorations explicites de la norme de $u$ et de la hauteur de
NéronTate de $p$ qui dépendent des invariants classiques
du problème dont la dimension et la hauteur de Faltings de
la variété abélienne. Les démonstrations reposent
sur une construction de transcendance du type Gel'fondBaker
de la théorie des formes linéaires de logarithmes dans
laquelle se greffent des formules explicites provenant de la
théorie des pentes d'Arakelov.


Linear maps preserving matrices of local spectral radius zero at a fixed vector Bourhim, Abdellatif; Costara, Constantin Published: 20180817
In this paper, we characterize linear maps on matrix spaces which
preserve
matrices of local spectral radius zero at some fixed nonzero
vector.


A basis theorem for the degenerate affine oriented BrauerClifford supercategory Brundan, Jonathan; Comes, Jonathan; Kujawa, Jonathan Robert Author's Draft
We introduce the oriented BrauerClifford and degenerate affine oriented BrauerClifford supercategories. These are diagrammatically defined monoidal supercategories which provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.


Casselman's Basis of Iwahori vectors and KazhdanLusztig polynomials Bump, Daniel; Nakasuji, Maki Published: 20180620
A problem in representation theory of $p$adic groups
is the computation of the Casselman basis of
Iwahori fixed vectors in the spherical principal series
representations, which are dual to the intertwining
integrals. We shall express the transition matrix
$(m_{u,v})$ of the Casselman basis to another natural basis in
terms of certain polynomials which are deformations
of the KazhdanLusztig Rpolynomials. As an application
we will obtain certain new functional equations
for these transition matrices under the algebraic
involution sending the residue cardinality $q$ to
$q^{1}$. We will also obtain a new proof of a
surprising result of Nakasuji and Naruse that
relates the matrix $(m_{u,v})$ to its inverse.


Powers in orbits of rational functions: cases of an arithmetic dynamical MordellLang conjecture Cahn, Jordan; Jones, Rafe; Spear, Jacob Author's Draft
Let $K$ be a finitely generated field of characteristic
zero. We study, for fixed $m \geq 2$, the rational functions
$\phi$ defined over $K$ that have a $K$orbit containing infinitely
many distinct $m$th powers. For $m \geq 5$ we show the only such
functions are those of the form $cx^j(\psi(x))^m$ with $\psi
\in K(x)$, and for $m \leq 4$ we show the only additional cases
are certain Lattès maps and four families of rational functions
whose special properties appear not to have been studied before.


CalabiYau quotients of hyperkähler fourfolds Camere, Chiara; Garbagnati, Alice; Mongardi, Giovanni Author's Draft
The aim of this paper is to construct CalabiYau 4folds as
crepant resolutions of the quotients of a hyperkähler 4fold
$X$ by a non symplectic involution $\alpha$. We first compute
the Hodge numbers of a CalabiYau constructed in this way in
a general setting and then we apply the results to several specific
examples of non symplectic involutions, producing CalabiYau
4folds with different Hodge diamonds. Then we restrict ourselves
to the case where $X$ is the Hilbert scheme of two points on
a K3 surface $S$ and the involution $\alpha$ is induced by a
non symplectic involution on the K3 surface. In this case we
compare the CalabiYau 4fold $Y_S$, which is the crepant resolution
of $X/\alpha$, with the CalabiYau 4fold $Z_S$, constructed
from $S$ through the BorceaVoisin construction. We give several
explicit geometrical examples of both these CalabiYau 4folds
describing maps related to interesting linear systems as well
as a rational $2:1$ map from $Z_S$ to $Y_S$.


A Galois correspondence for reduced crossed products of unital simple C$^*$algebras by discrete groups Cameron, Jan; Smith, Roger R. Published: 20180718
Let a discrete group $G$ act on a unital simple C$^*$algebra
$A$ by outer automorphisms. We establish a Galois correspondence
$H\mapsto A\rtimes_{\alpha,r}H$ between subgroups of $G$ and
C$^*$algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{\alpha,r}G$,
where
$A\rtimes_{\alpha,r}G$ denotes the reduced crossed product. For
a twisted dynamical system $(A,G,\alpha,\sigma)$, we also prove
the corresponding result for the reduced twisted crossed product
$A\rtimes^\sigma_{\alpha,r}G$.


A CR Analogue of Yau's Conjecture On Pseudoharmonic Functions of Polynomial Growth Chang, DerChen; Chang, ShuCheng; Han, Yingbo; Tie, Jingzhi Author's Draft
In this paper, we first derive the CR volume doubling property,
CR Sobolev
inequality, and mean value inequality. We then apply them to
prove the CR
analogue of Yau's conjecture on the space consisting of all pseudoharmonic
functions of polynomial growth of degree at most $d$ in a complete
noncompact
pseudohermitian $(2n+1)$manifold. As a byproduct, we obtain
the CR analogue
of volume growth estimate and Gromov precompactness theorem.


Annihilators of the ideal class group of a cyclic extension of an imaginary quadratic field Chapdelaine, Hugo; Kuċera, Radan Author's Draft
The aim of this paper is to study the group of elliptic units
of a
cyclic extension $L$ of an imaginary quadratic field $K$ such
that the degree $[L:K]$ is a power of an odd prime $p$.
We construct an explicit root of the usual top generator of this
group and
we use it to obtain an annihilation result of the $p$Sylow subgroup
of the ideal class group of $L$.


Unperforated pairs of operator spaces and hyperrigidity of operator systems Clouâtre, Raphaël Published: 20180322
We study restriction and extension properties for states on C$^*$algebras
with an eye towards hyperrigidity of operator systems. We use
these ideas to provide supporting evidence for Arveson's hyperrigidity
conjecture. Prompted by various characterizations of hyperrigidity
in terms of states, we examine unperforated pairs of selfadjoint
subspaces in a C$^*$algebra. The configuration of the subspaces
forming an unperforated pair is in some sense compatible with
the order structure of the ambient C$^*$algebra. We prove
that commuting pairs are unperforated, and obtain consequences
for hyperrigidity. Finally, by exploiting recent advances in
the tensor theory of operator systems, we show how the weak expectation
property can serve as a flexible relaxation of the notion of
unperforated pairs.


Lipschitz 1connectedness for some solvable Lie groups Cohen, David Bruce Published: 20180424
A space X is said to be Lipschitz 1connected if every LLipschitz loop in X bounds a O(L)Lipschitz disk. A Lipschitz 1connected 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 1connected.


Elements of $C^*$algebras attaining their norm in a finitedimensional representation Courtney, Kristin; Shulman, Tatiana Published: 20171204
We characterize the class of RFD $C^*$algebras as those containing
a dense subset of elements that attain their norm under a finitedimensional
representation. We show further that this subset is the whole
space precisely when every irreducible representation of the
$C^*$algebra is finitedimensional, which is equivalent to the
$C^*$algebra having no simple infinitedimensional AF subquotient.
We apply techniques from this proof to show the existence of
elements in more general classes of $C^*$algebras whose norms
in finitedimensional representations fit certain prescribed
properties.


BakryÉmery Curvature Functions on Graphs Cushing, David; Liu, Shiping; Peyerimhoff, Norbert Published: 20180705
We study local properties of the BakryÉmery curvature function
$\mathcal{K}_{G,x}:(0,\infty]\to \mathbb{R}$ at a vertex $x$ of a graph
$G$
systematically. Here $\mathcal{K}_{G,x}(\mathcal{N})$ is defined as the optimal
curvature lower bound $\mathcal{K}$ in the BakryÉmery curvaturedimension
inequality $CD(\mathcal{K},\mathcal{N})$ that $x$ satisfies. We provide upper
and
lower bounds for the curvature functions, introduce fundamental
concepts like curvature sharpness and $S^1$out regularity,
and
relate the curvature functions of $G$ with various spectral
properties of (weighted) graphs constructed from local structures
of
$G$. We prove that the curvature functions of the Cartesian
product
of two graphs $G_1,G_2$ are equal to an abstract product of
curvature functions of $G_1,G_2$. We explore the curvature
functions
of Cayley graphs and many particular (families of) examples.
We
present various conjectures and construct an infinite increasing
family of $6$regular graphs which satisfy $CD(0,\infty)$ but
are
not Cayley graphs.


Colouring squares of clawfree graphs de Joannis de Verclos, Rémi; Kang, Ross J.; Pastor, Lucas Published: 20171213
Is there some absolute $\varepsilon > 0$ such that for any clawfree
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.


Uniform convergence of trigonometric series with general monotone coefficients Dyachenko, Mikhail; Mukanov, Askhat; Tikhonov, Sergey Published: 20180102
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.


On the weak order of Coxeter groups Dyer, Matthew Published: 20180521
This paper provides some evidence for conjectural
relations between extensions of (right) weak order on Coxeter
groups, closure operators on root systems, and Bruhat order.
The conjecture
focused upon here refines an earlier question as to whether the
set of initial sections of reflection orders, ordered by inclusion,
forms a complete lattice.
Meet and join in weak order are described in terms of a suitable
closure operator. Galois connections are defined from the
power set of $W$ to itself, under which maximal subgroups of
certain groupoids correspond
to certain complete meet subsemilattices of weak order. An analogue
of weak order for standard parabolic subsets of any rank
of the root system
is defined, reducing to the usual weak order in rank zero, and
having some analogous properties in rank one (and conjecturally
in general).


Range spaces of coanalytic Toeplitz operators Fricain, Emmanuel; Hartmann, Andreas; Ross, William T. Published: 20180313
In this paper we discuss the range of a coanalytic Toeplitz
operator. These range spaces are closely related to de BrangesRovnyak
spaces (in some cases they are equal as sets). In order to understand
its structure, we explore when
the range space decomposes into the range of an associated analytic
Toeplitz operator and an identifiable orthogonal complement.
For certain cases, we compute this orthogonal complement in terms
of the kernel of a certain Toeplitz operator on the Hardy space
where we focus on when this kernel is a model space (backward
shift invariant subspace).
In the spirit of AhernClark, we also discuss the nontangential
boundary behavior in these range spaces. These results give us
further insight into the description of the range of a coanalytic
Toeplitz operator as well as its orthogonal decomposition. Our
AhernClark type results, which are stated in a general abstract
setting, will also have applications to related subHardy Hilbert
spaces of analytic functions such as the de BrangesRovnyak spaces
and the harmonically weighted Dirichlet spaces.


Titchmarsh's method for the approximate functional equations for $\zeta^{\prime}(s)^{2}$, $\zeta(s)\zeta^{\prime\prime}(s)$ and $\zeta^{\prime}(s)\zeta^{\prime\prime}(s)$ Furuya, Jun; Minamide, Makoto; Tanigawa, Yoshio Published: 20180523
Let $\zeta (s)$ be the Riemann zeta function. In 1929, Hardy
and Littlewood proved the approximate
functional equation for $\zeta^2(s)$ with error term $O(x^{1/2\sigma}((x+y)/t)^{1/4}\log
t)$
where $1/2\lt \sigma\lt 3/2$, $x,y \geq 1$, $xy=(t/2\pi)^2$. Later,
in 1938, Titchmarsh improved the error term by removing
the factor $((x+y)/t)^{1/4}$. In 1999, Hall showed the approximate
functional equations for $\zeta'(s)^2, \zeta(s)\zeta''(s) $
and $\zeta'(s)\zeta''(s)$ (in the range $0\lt \sigma\lt 1$) whose error
terms contain the factor $((x+y)/t)^{1/4}$.
In this paper we remove this factor from these three error terms
by using the method of Titchmarsh.


Degrees of regular sequences with a symmetric group action Galetto, Federico; Geramita, Anthony Vito; Wehlau, David Louis Published: 20180321
We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.


Integral Formula for Spectral Flow for $p$Summable Operators Georgescu, Magdalena Cecilia Published: 20180605
Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace
$\tau$. For a path of selfadjoint BreuerFredholm operators, the
spectral flow measures the net amount of spectrum which moves from
negative to nonnegative. We consider specifically the case of paths
of bounded perturbations of a fixed unbounded selfadjoint
BreuerFredholm operator affiliated with $\mathcal{N}$. If the unbounded
operator is psummable (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 psummable formula. In this paper, we present a direct proof
of the psummable formula, which is both shorter and simpler than
theirs.


Long sets of lengths with maximal elasticity Geroldinger, Alfred; Zhong, Qinghai Published: 20180118
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.


Completeness of infinitedimensional Lie groups in their left uniformity Glöckner, Helge Published: 20180427
We prove completeness for the main examples
of infinitedimensional 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 finitedimensional
Lie groups is complete, as well as the diffeomorphism group
$\operatorname{Diff}_c(M)$
of a paracompact finitedimensional 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.


Monochromatic solutions to $x + y = z^2$ Green, Ben Joseph; Lindqvist, Sofia Published: 20180117
Suppose that $\mathbb{N}$ is $2$coloured. Then there are infinitely
many monochromatic solutions to $x + y = z^2$. On the other hand,
there is a $3$colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this equation.


Poles of the Standard $\mathcal{L}$function of $G_2$ and the RallisSchiffmann lift Gurevich, Nadya; Segal, Avner Author's Draft
We characterize the cuspidal representations of $G_2$ whose standard
$\mathcal{L}$function admits a pole at $s=2$ as the image of the RallisSchiffmann
lift for the commuting pair $(\widetilde{SL_2}, G_2)$ in $\widetilde{Sp_{14}}$.
The image consists of nontempered representations.
The main tool is the recent construction, by the second author,
of a family of RankinSelberg integrals representing the standard
$\mathcal{L}$function.


Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups Han, Yanchang; Han, Yongsheng; Li, Ji; Tan, Chaoqiang Published: 20180404
The Marcinkiewicz multipliers are $L^{p}$ bounded for $1\lt p\lt \infty
$ on the
Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$
(Müller, Ricci and Stein). This is
surprising in the sense that these multipliers are invariant
under a two parameter
group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while
there is
no two parameter group of automorphic dilations
on $\mathbb{H}
^{n}$. The purpose of this paper is to establish a theory of
the flag Lipschitz space on the Heisenberg group
$\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the
sense
`intermediate' between the classical Lipschitz space on the Heisenberg
group
$\mathbb{H}^{n}$ and the product Lipschitz space on
$\mathbb{C}^{n}\times \mathbb{R}$. We characterize this flag
Lipschitz space
via the LittlewoodPaley theory and prove
that flag singular integral operators, which include the
Marcinkiewicz multipliers, are bounded on these flag Lipschitz
spaces.


Nearly approximate transitivity (AT) for circulant matrices Handelman, David Author's Draft
By previous work of Giordano and the author, ergodic
actions of $\mathbf Z$ (and other discrete groups) are completely classified
measuretheoretically 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 nonAT 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.


Eisenstein series arising from Jordan algebras Hanzer, Marcela; Savin, Gordan Author's Draft
We describe poles and the corresponding residual
automorphic representations of Eisenstein series attached to
maximal
parabolic subgroups whose unipotent radicals admit Jordan algebra
structure.


Local Shtukas and Divisible Local Anderson Modules Hartl, Urs; Singh, Rajneesh Kumar Author's Draft
We develop the analog of crystalline Dieudonné theory for $p$divisible
groups in the arithmetic of function fields. In our theory $p$divisible
groups are replaced by divisible local Anderson modules, and
Dieudonné modules are replaced by local shtukas. We show that
the categories of divisible local Anderson modules and of effective
local shtukas are antiequivalent over arbitrary base schemes.
We also clarify their relation with formal Lie groups and with
global objects like Drinfeld modules, Anderson's abelian $t$modules
and $t$motives, and Drinfeld shtukas. Moreover, we discuss the
existence of a Verschiebung map and apply it to deformations
of local shtukas and divisible local Anderson modules. As a tool
we use Faltings's and Abrashkin's theory of strict modules, which
we review to some extent.


Local Shtukas and Divisible Local Anderson Modules Hartl, Urs; Singh, Rajneesh Kumar Author's Draft
We develop the analog of crystalline Dieudonné theory for $p$divisible
groups in the arithmetic of function fields. In our theory $p$divisible
groups are replaced by divisible local Anderson modules, and
Dieudonné modules are replaced by local shtukas. We show that
the categories of divisible local Anderson modules and of effective
local shtukas are antiequivalent over arbitrary base schemes.
We also clarify their relation with formal Lie groups and with
global objects like Drinfeld modules, Anderson's abelian $t$modules
and $t$motives, and Drinfeld shtukas. Moreover, we discuss the
existence of a Verschiebung map and apply it to deformations
of local shtukas and divisible local Anderson modules. As a tool
we use Faltings's and Abrashkin's theory of strict modules, which
we review to some extent.


On deformations of pairs (manifold, coherent sheaf) Iacono, Donatella; Manetti, Marco Author's Draft
We analyse infinitesimal deformations of pairs $(X,\mathcal F)$ with
$\mathcal F$ a coherent sheaf on a smooth projective variety $X$
over an algebraically closed field of characteristic $0$. We
describe a differential graded Lie algebra controlling the deformation
problem, and we prove an analog of a MukaiArtamkin Theorem about
the trace map.


Steklov problem on differential forms Karpukhin, Mikhail A. Author's Draft
In this paper we study spectral properties of the DirichlettoNeumann
map on differential forms obtained by a slight modification of
the definition due to Belishev and Sharafutdinov.
The
resulting operator $\Lambda$ is shown to be selfadjoint on the
subspace of coclosed forms and to have purely discrete spectrum
there.
We investigate properties of eigenvalues of $\Lambda$ and prove
a HerschPayneSchiffer type inequality relating products of
those eigenvalues to eigenvalues of the Hodge Laplacian on the
boundary. Moreover, nontrivial eigenvalues of $\Lambda$ are
always at least as large as eigenvalues of the DirichlettoNeumann
map defined by Raulot and Savo
Finally, we remark
that a particular case of $p$forms on the boundary of a $2p+2$dimensional
manifold shares a lot of important properties with the classical
Steklov eigenvalue problem on surfaces.


Weighted distribution of lowlying zeros of $\operatorname{GL}(2)$ $L$functions Knightly, Andrew; Reno, Caroline Author's Draft
We show that if the zeros of an automorphic $L$function are
weighted by the central value of the $L$function
or a quadratic imaginary base change,
then for certain families of holomorphic $\operatorname{GL}(2)$ newforms,
it has the effect of changing the distribution type of lowlying
zeros from orthogonal to symplectic, for test functions whose
Fourier
transforms have sufficiently restricted support.
However, if the $L$value is twisted by a nontrivial quadratic
character, the distribution type remains orthogonal.
The proofs involve two vertical equidistribution results for
Hecke
eigenvalues weighted by central twisted $L$values. One of
these
is due to Feigon
and Whitehouse, and the other
is new and involves
an asymmetric probability
measure that has not appeared in previous equidistribution
results for $\operatorname{GL}(2)$.


Growth of homology of centrebymetabelian pro$p$ groups Kochloukova, Dessislava H.; Pinto, Aline G. S. Author's Draft
For a centrebymetabelian pro$p$ group $G$
of type $FP_{2m}$, for some $m \geq 1$, we show that
$$
\sup_{M\in \mathcal{A}} rk~H_i(M,\mathbb{Z}_p) \lt \infty, \quad \mbox{for
all~} 0\leq i \leq m,
$$
where $\mathcal{A}$ is the set of all subgroups of $p$power
index in $G$ and, for a finitely generated abelian pro$p$ group
$V$, $rk~V = \dim V \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$.


The BV algebra in String Topology of classifying spaces Kuribayashi, Katsuhiko; Menichi, Luc Author's Draft
For almost any compact connected Lie group $G$ and any field
$\mathbb{F}_p$, we compute the BatalinVilkovisky
algebra $H^{*+\operatorname{dim }G}(LBG;\mathbb{F}_p)$ on the loop cohomology
of the classifying space introduced by
Chataur and the second author.
In particular, if $p$ is odd or $p=0$, this BatalinVilkovisky
algebra is isomorphic
to the Hochschild cohomology $HH^*(H_*(G),H_*(G))$. Over $\mathbb{F}_2$,
such isomorphism of BatalinVilkovisky algebras
does not hold when $G=SO(3)$ or $G=G_2$.
Our elaborate considerations on
the signs in string topology of the classifying spaces give rise
to a general theorem on graded homological conformal field theory.


A forcing axiom deciding the generalized Souslin Hypothesis LambieHanson, Chris; Rinot, Assaf Published: 20180426
We derive a forcing axiom from the conjunction
of square and diamond, and present a few applications,
primary among them being the existence of superSouslin trees.
It follows that for every uncountable cardinal $\lambda$, if
$\lambda^{++}$ is not a Mahlo cardinal in Gödel's constructible
universe,
then $2^\lambda = \lambda^+$ entails the existence of a $\lambda^+$complete
$\lambda^{++}$Souslin tree.


Boundary quotient C*algebras of products of odometers Li, Hui; Yang, Dilian Published: 20171203
In this paper, we study the boundary quotient C*algebras associated
to products of odometers. One of our main results
shows that the boundary quotient C*algebra of the standard product
of $k$ odometers
over $n_i$letter alphabets ($1\le i\le k$) is always nuclear,
and that
it is a UCT Kirchberg algebra
if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent,
if and only if the associated singlevertex $k$graph C*algebra
is simple.
To achieve this, one of our main steps is to construct a topological
$k$graph such that
its associated CuntzPimsner C*algebra is isomorphic to the
boundary quotient C*algebra.
Some relations between the boundary quotient C*algebra and the
C*algebra $\mathrm{Q}_\mathbb{N}$ introduced by Cuntz are also
investigated.


Flow polytopes and the space of diagonal harmonics Liu, Ricky Ini; Morales, Alejandro H.; Mészáros, Karola Published: 20180411
A result of Haglund implies that the $(q,t)$bigraded Hilbert
series of the space of diagonal harmonics is a $(q,t)$Ehrhart
function of the flow polytope of a complete graph with netflow
vector $(n, 1, \dots, 1).$ We study the $(q,t)$Ehrhart functions
of flow polytopes of threshold graphs with arbitrary netflow
vectors. Our results generalize previously known specializations
of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{1}$.
As a corollary to our results, we obtain a proof of a conjecture
of Armstrong, Garsia, Haglund, Rhoades and Sagan about the $(q,
q^{1})$Ehrhart function of the flow polytope of a complete
graph with an arbitrary netflow vector.


Twocolor Soergel calculus and simple transitive 2representations Mackaaij, Marco; Tubbenhauer, Daniel Author's Draft
In this paper we complete the ADElike
classification
of simple transitive $2$representations
of Soergel bimodules
in finite dihedral type, under the assumption of gradeability.
In particular, we use bipartite
graphs and zigzag algebras of ADE type to give an explicit
construction of a graded (nonstrict)
version of all these $2$representations.


Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields Macourt, Simon; Shkredov, Ilya D.; Shparlinski, Igor E. Published: 20171220
We give a new bound on collinear triples in
subgroups of prime finite
fields and use it to give some new bounds on exponential sums
with trinomials.


Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras Matsumoto, Kengo Author's Draft
In the first part of the paper, we introduce notions of
asymptotic continuous orbit equivalence
and asymptotic conjugacy in Smale spaces
and characterize them in terms of their asymptotic Ruelle algebras
with their dual actions.
In the second part, we introduce a groupoid $C^*$algebra which
is an extended version
of the asymptotic Ruelle algebra from a Smale space
and study the extended Ruelle algebras from the view points of
CuntzKrieger algebras.
As a result, the asymptotic Ruelle algebra is realized as a fixed
point algebra
of the extended Ruelle algebra under certain circle action.


Cohomological Approach to Class Field Theory in Arithmetic Topology Mihara, Tomoki Author's Draft
We establish class field theory for $3$dimensional manifolds
and knots. For this purpose, we formulate analogues of the multiplicative
group, the idèle class group, and ray class groups in a cocycletheoretic
way. Following the arguments in abstract class field theory,
we construct reciprocity maps and verify the existence theorems.


Freeness and The Partial Transposes of Wishart Random Matrices Mingo, James A.; Popa, Mihai Published: 20180412
We show that the partial transposes of complex Wishart
random matrices are asymptotically free. We also investigate
regimes where the number of blocks is fixed but the size of
the blocks increases. This gives a example where the partial
transpose produces freeness at the operator level. Finally
we investigate the case of real Wishart matrices.


On special fiber rings of modules MirandaNeto, Cleto B. Author's Draft
We prove results concerning the multiplicity as well as the CohenMacaulay
and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$
of a finitely generated $R$module $E\subsetneq R^e$ over a Noetherian
local ring $R$ with infinite residue field. Assuming that $R$
is CohenMacaulay of dimension $1$ and that $E$ has finite colength
in $R^e$, our main result establishes an asymptotic length formula
for the multiplicity of $\mathscr{F}(E)$, which, in addition to being
of independent interest, allows us to derive a CohenMacaulayness
criterion and to detect a curious relation to the BuchsbaumRim
multiplicity of $E$ in this setting. Further, we provide a Gorensteinness
characterization for $\mathscr{F}(E)$ in the more general situation where
$R$ is CohenMacaulay of arbitrary dimension and $E$ is not necessarily
of finite colength, and we notice a constraint in terms of the
second analytic deviation of the module $E$ if its reduction
number is at least three.


Lorentz estimates for weak solutions of quasilinear parabolic equations with singular divergencefree drifts Phan, Tuoc Published: 20180523
This paper investigates regularity in Lorentz
spaces of weak solutions of a class of divergence form quasilinear
parabolic equations with singular divergencefree 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 CalderónZygmund 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 "doublescaling parameter"
technique, and the maximal function free approach introduced
by Acerbi and Mingione.


$P$adic $L$functions for GL$_2$ Salazar, Daniel Barrera; Williams, Chris Published: 20180430
Since Rob Pollack and Glenn Stevens used overconvergent
modular symbols to construct $p$adic $L$functions for noncritical
slope rational modular forms, the theory has been extended to
construct $p$adic $L$functions for noncritical slope automorphic
forms over totally real and imaginary quadratic fields by the
first and second authors respectively. In this paper, we give
an analogous construction over a general number field. In particular,
we start by proving a control theorem stating that the specialisation
map from overconvergent to classical modular symbols is an isomorphism
on the small slope subspace. We then show that if one takes the
modular symbol attached to a small slope cuspidal eigenform,
then one can construct a ray class distribution from the corresponding
overconvergent symbol, that moreover interpolates critical values
of the $L$function of the eigenform. We prove that this distribution
is independent of the choices made in its construction. We define
the $p$adic $L$function of the eigenform to be this distribution.


The mod two cohomology of the moduli space of rank two stable bundles on a surface and skew Schur polynomials Scaduto, Christopher W.; Stoffregen, Matthew Published: 20180417
We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.


On an Enriques surface associated with a quartic Hessian surface Shimada, Ichiro Author's Draft
Let $Y$ be a complex Enriques surface
whose universal cover $X$ is birational to a general quartic
Hessian surface.
Using the result on the automorphism group of $X$
due to Dolgachev and Keum,
we obtain
a finite presentation of the automorphism group of $Y$.
The list of elliptic fibrations on $Y$
and the list of combinations of rational double points that can
appear on a surface birational to $Y$
are presented.
As an application,
a set of generators of
the automorphism group of the generic Enriques surface is calculated
explicitly.


Relative discrete series representations for two quotients of $p$adic $\mathbf{GL}_n$ Smith, Jerrod Manford Published: 20180313
We provide an explicit construction of representations in the
discrete spectrum of two $p$adic symmetric spaces. We consider
$\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$
and $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, where $E$
is a quadratic Galois extension of a nonarchimedean local field
$F$ of characteristic zero and odd residual characteristic. The
proof of the main result involves an application of a symmetric
space version of Casselman's Criterion for square integrability
due to Kato and Takano.


Fourier spaces and completely isometric representations of Arens product algebras Stokke, Ross Thomas Author's Draft
Motivated by the definition of a semigroup
compactification of a locally compact group and a large collection
of examples, we introduce the notion of an (operator) ``homogeneous
left dual Banach algebra" (HLDBA) over a (completely contractive)
Banach algebra $A$. We prove a Gelfandtype representation
theorem showing that every HLDBA over $A$ has a concrete realization
as an (operator) homogeneous left Arens product algebra: the
dual of a subspace of $A^*$ with a compatible (matrix) norm and
a type of left Arens product ${\scriptstyle\square}$. Examples include all left
Arens product algebras over $A$, but also  when $A$ is the
group algebra of a locally compact group  the dual of its Fourier
algebra. Beginning with any (completely) contractive (operator)
$A$module action $Q$ on a space $X$, we introduce the (operator)
Fourier space $(\mathcal F_Q(A^*), \ \cdot \_Q)$ and
prove that
$(\mathcal F_Q(A^*)^*, {\scriptstyle\square})$ is the unique (operator) HLDBA over
$A$ for which there is a weak$^*$continuous completely isometric
representation as completely bounded operators on $X^*$ extending
the dual module representation.
Applying our theory to several examples of (completely contractive)
Banach algebras $A$ and module operations, we provide new characterizations
of
familiar HLDBAs over $A$ and we recover  and often extend
 some
(completely) isometric representation theorems concerning
these HLDBAs.


A new proof of the HansenMullen irreducibility conjecture Tuxanidy, Aleksandr; Wang, Qiang Published: 20180614
We give a new proof of the HansenMullen irreducibility conjecture.
The proof relies on an application of a (seemingly new) sufficient
condition for the existence of
elements of degree $n$ in the support of functions on finite
fields.
This connection to irreducible polynomials is made via the least
period of the discrete Fourier transform (DFT) of functions with
values in finite fields.
We exploit this relation and prove, in an elementary fashion,
that a relevant function related to the DFT of characteristic
elementary symmetric functions (which produce the coefficients
of characteristic polynomials)
satisfies a simple requirement on the least period.
This bears a sharp contrast to previous techniques in literature
employed to tackle existence
of irreducible polynomials with prescribed coefficients.


Pointwise convergence of solutions to the Schrödinger equation on manifolds Wang, Xing; Zhang, Chunjie Published: 20180424
Let $(M^n,g)$ be a Riemannian manifold without
boundary. We study the amount of initial regularity is required
so that the solution to free Schrödinger equation converges
pointwise to its initial data. Assume the initial data is in
$H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the
2 dimensional Torus, we prove that $\alpha\gt \frac{1}{2}$ is enough.
For general compact manifolds, due to lacking of local smoothing
effect, it is hard to beat the bound $\alpha\gt 1$ from interpolation.
We managed to go below 1 for dimension $\leq 3$. The more interesting
thing is that, for 1 dimensional compact manifold, $\alpha\gt \frac{1}{3}$
is sufficient.


On algebraic surfaces associated with line arrangements Wang, Zhenjian Published: 20180404
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.


Squarefree values of decomposable forms Xiao, Stanley Yao Published: 20180214
In this paper we prove that decomposable forms,
or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer
coefficients which split completely into linear factors over
$\mathbb{C}$, take on infinitely many squarefree values subject
to simple necessary conditions and $\deg f \leq 2n + 2$ for all
irreducible factors $f$ of $F$. This work generalizes a theorem
of Greaves.


A special case of completion invariance for the $c_2$ invariant of a graph Yeats, Karen Published: 20180427
The $c_2$ invariant is an arithmetic graph invariant defined
by Schnetz. It is useful for understanding Feynman periods.
Brown and Schnetz conjectured that the $c_2$ invariant has a
particular symmetry known as completion invariance.
This paper will prove completion invariance of the $c_2$ invariant
in the case that we are over the field with 2 elements and the
completed graph has an odd number of vertices.
The methods involve enumerating certain edge bipartitions of
graphs; two different constructions are needed.

