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Uniqueness of the von Neumann continuous factor Ara, Pere; Claramunt, Joan Author's Draft
For a division ring $D$, denote by $\mathcal M_D$ the $D$ring
obtained as the completion of the direct limit $\varinjlim_n
M_{2^n}(D)$
with respect to the metric induced by its unique rank function.
We prove that, for any ultramatricial $D$ring $\mathcal B$ and
any
nondiscrete extremal pseudorank function $N$ on $\mathcal B$,
there is an isomorphism of $D$rings $\overline{\mathcal B} \cong \mathcal
M_D$, where $\overline{\mathcal B}$ stands
for the completion of $\mathcal B$ with respect to the pseudometric
induced by $N$.
This generalizes a result of von Neumann. We also show a corresponding
uniqueness result for $*$algebras over fields $F$ with positive
definite involution, where the
algebra $\mathcal M_F$ is endowed with its natural involution
coming from the $*$transpose involution on each of the factors
$M_{2^n}(F)$.


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.


On Dirichlet spaces with a class of superharmonic weights Bao, Guanlong; Göğüş, Nihat Gökhan; Pouliasis, Stamatis Published: 20170519
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the AlexanderTaylorUllman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and innerouter factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.


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.


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 $p\geq p_c$.


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.


Partial Hasse invariants, partial degrees, and the canonical subgroup Bijakowski, Stephane Published: 20170512
If the Hasse invariant of a $p$divisible group is small enough,
then one can construct a canonical subgroup inside its $p$torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.


Logarithmes des points rationnels des variétés abéliennes Bosser, Vincent; Gaudron, Éric Author's Draft
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.


Casselman's Basis of Iwahori vectors and KazhdanLusztig polynomials Bump, Daniel; Nakasuji, Maki Author's Draft
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.


A Galois correspondence for reduced crossed products of unital simple C$^*$algebras by discrete groups Cameron, Jan; Smith, Roger R. Author's Draft
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
$A\rtimes^\sigma_{\alpha,r}G$.


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 Author's Draft
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.


An explicit computation of the Blanchfield pairing for arbitrary links Conway, Anthony Published: 20180226
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 $\operatorname{Bl}(L)$.


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 Author's Draft
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.


Multiplication formulas and canonical bases for quantum affine gln Du, Jie; Zhao, Zhonghua Published: 20170816
We will give a representationtheoretic proof for the multiplication
formula
in the RingelHall algebra
$\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first
application, we see immediately the existence of Hall polynomials
for cyclic quivers, a fact established
by J. Y. Guo and C. M. Ringel,
and derive a recursive formula
to compute them.
We will further use the formula and the construction of a certain
monomial base for $\mathfrak{H}_\Delta(n)$ given
by Deng, Du, and Xiao
together with the double RingelHall algebra realisation of
the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$
given by
Deng, Du, and Fu
to develop some algorithms and to compute the canonical basis
for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples,
we will show explicitly the part of the canonical basis
associated with modules of Lowey length at most $2$ for the quantum
group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.


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 Author's Draft
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).


Schwartz functions on real algebraic varieties Elazar, Boaz; Shaviv, Ary Published: 20180123
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 nonaffine case.


Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras Elduque, Alberto Published: 20171219
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.


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 Author's Draft
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 Author's Draft
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.


Inequalities for the surface area of projections of convex bodies Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros Published: 20170327
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.


Completeness of infinitedimensional Lie groups in their left uniformity Glöckner, Helge Author's Draft
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.


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.


Local dimensions of measures of finite type II  Measures without full support and with nonregular probabilities Hare, Kathryn; Hare, Kevin; Ng, Michael Ka Shing Published: 20170926
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
selfsimilar measure $\mu =\sum p_{j}\mu \circ S_{j}^{1}$, of
finite type.
In this paper we study the multifractal analysis of such measures,
extending the theory to measures arising from nonregular probabilities
and
whose support is not necessarily an interval.


Nori motives of curves with modulus and Laumon $1$motives Ivorra, Florian; Yamazaki, Takao Published: 20180118
Let $k$ be a number field. We describe the category of Laumon
$1$isomotives over $k$ as the universal category in the sense
of Nori associated with a quiver representation built out of
smooth proper $k$curves with two disjoint effective divisors
and a notion of $H^1_\mathrm{dR}$ for such "curves with modulus".
This result extends and relies on the theorem of J. Ayoub
and L. BarbieriViale that describes Deligne's category
of $1$isomotives in terms of Nori's Abelian category of motives.


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 $\operatorname{GL}(2)$.


A forcing axiom deciding the generalized Souslin Hypothesis LambieHanson, Chris; Rinot, Assaf Author's Draft
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.


Spherical fundamental lemma for metaplectic groups Luo, Caihua Published: 20170901
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 WenWei Li.


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.


Congruences for modular forms mod 2 and quaternionic $S$ideal classes Martin, Kimball Published: 20170720
We prove many simultaneous congruences mod 2 for elliptic and
Hilbert modular forms
among forms with different AtkinLehner eigenvalues. The proofs
involve the notion of quaternionic $S$ideal classes and the
distribution of AtkinLehner signs among
newforms.


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.


EdgeMaximal Graphs on Surfaces McDiarmid, Colin; Wood, David R. Published: 20171002
We prove that for every surface $\Sigma$ of Euler genus $g$,
every edgemaximal 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).


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.


The RudinShapiro sequence and similar sequences are normal along squares Müllner, Clemens Author's Draft
We prove that digital sequences modulo $m$ along squares are
normal,
which covers some prominent sequences like the sum of digits
in base $q$ modulo $m$, the RudinShapiro sequence and some generalizations.
This gives, for any base, a class of explicit normal numbers
that can be efficiently generated.


Lorentz estimates for weak solutions of quasilinear parabolic equations with singular divergencefree drifts Phan, Tuoc Author's Draft
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.


Doubled Khovanov homology Rushworth, William Published: 20180214
We define a homology theory of virtual links built out of the
direct sum of the standard Khovanov complex with itself, motivating
the name doubled Khovanov homology. We demonstrate that
it can be used to show that some virtual links are nonclassical,
and that it yields a condition on a virtual knot being the connect
sum of two unknots. Further, we show that doubled Khovanov homology
possesses a perturbation analogous to that defined by Lee in
the classical case and define a doubled Rasmussen invariant.
This invariant is used to obtain various cobordism obstructions;
in particular it is an obstruction to sliceness. Finally, we
show that the doubled Rasmussen invariant contains the odd writhe
of a virtual knot, and use this to show that knots with nonzero
odd writhe are not slice.


$P$adic $L$functions for GL$_2$ Salazar, Daniel Barrera; Williams, Chris Author's Draft
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.


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.


A new proof of the HansenMullen irreducibility conjecture Tuxanidy, Aleksandr; Wang, Qiang Author's Draft
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.


An explicit ManinDem'janenko theorem in elliptic curves Viada, Evelina Published: 20180124
Let $\mathcal{C}$ be a curve of genus at least $2$ embedded in $E_1
\times \cdots \times E_N$ where the $E_i$ are elliptic curves
for $i=1,\dots, N$. In this article we give an explicit sharp
bound for the NéronTate height of the points of $\mathcal{C}$ contained
in the union of all algebraic subgroups of dimension
$\lt \max(r_\mathcal{C}t_\mathcal{C},t_\mathcal{C})$
where $t_\mathcal{C}$, respectively $r_\mathcal{C}$, is the minimal dimension
of a translate, respectively of a torsion variety, containing
$\mathcal{C}$.


Pointwise convergence of solutions to the Schrödinger equation on manifolds Wang, Xing; Zhang, Chunjie Author's Draft
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 Author's Draft
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.


On a class of fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds Yuan, Rirong Published: 20180123
In this paper we study a class of second order fully nonlinear
elliptic equations
containing gradient terms on compact Hermitian manifolds and
obtain a priori estimates under
proper assumptions close to optimal.
The analysis developed here should
be useful to deal with other Hessian equations containing gradient
terms in other contexts.

