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3  Free Bessel Laws Banica, T.; Belinschi, S. T.; Capitaine, M.; Collins, B.
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.


38  Asymptotic Formulae for Pairs of Diagonal Cubic Equations Brüdern, Jörg; Wooley, Trevor D.
We investigate the number of integral solutions possessed by a
pair of diagonal cubic equations in a large box. Provided that the number of
variables in the system is at least fourteen, and in addition the number of
variables in any nontrivial linear combination of the underlying forms is at
least eight, we obtain an asymptotic formula for the number of integral
solutions consistent with the product of local densities associated with the
system.


55  Pseudolocality for the Ricci Flow and Applications Chau, Albert; Tam, LuenFai; Yu, Chengjie
Perelman established a differential LiYauHamilton
(LYH) type inequality for fundamental solutions of the conjugate
heat equation corresponding to the Ricci flow on compact manifolds.
As an application of the LYH inequality,
Perelman proved a pseudolocality result for the Ricci flow on
compact manifolds. In this article we provide the details for the
proofs of these results in the case of a complete noncompact
Riemannian manifold. Using these results we prove that under
certain conditions, a finite time singularity of the Ricci flow
must form within a compact set. The conditions are satisfied by
asymptotically flat manifolds. We also prove a long time existence
result for the K\"ahlerRicci flow on complete nonnegatively curved K\"ahler
manifolds.


86  On Vojta's $1+\varepsilon$ Conjecture Chen, Xi
We give another proof of Vojta's $1+\varepsilon$ conjecture.


104  Reversibility of Interacting FlemingViot Processes with Mutation, Selection, and Recombination Feng, Shui; Schmuland, Byron; Vaillancourt, Jean; Zhou, Xiaowen
Reversibility of the FlemingViot process with mutation, selection,
and recombination is well understood. In this paper, we study the
reversibility of a system of FlemingViot processes that live on a
countable number of colonies interacting with each other through
migrations between the colonies. It is shown that reversibility
fails when both migration and mutation are nontrivial.


123  Strong and Extremely Strong Ditkin sets for the Banach Algebras $A_p^r(G)=A_p\cap L^r(G)$ Granirer, Edmond E.
Let $A_p(G)$ be the FigaTalamanca,
Herz Banach Algebra on $G$; thus $A_2(G)$
is the Fourier algebra. Strong Ditkin (SD) and
Extremely Strong Ditkin (ESD) sets for the Banach algebras
$A_p^r(G)$ are investigated for abelian and nonabelian
locally compact groups $G$. It is shown that SD and ESD sets
for $A_p(G)$ remain SD and ESD sets for $A_p^r(G)$,
with strict inclusion for ESD sets. The case for the strict
inclusion of SD sets is left open.


136  Transcendental Nature of Special Values of $L$Functions Gun, Sanoli; Murty, M. Ram; Rath, Purusottam
In this paper, we study the nonvanishing and transcendence of
special values of a varying class of $L$functions and their derivatives.
This allows us to investigate the transcendence of Petersson norms
of certain weight one modular forms.


153  Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet Hambly, B. M.
We establish the asymptotic behaviour of the partition function, the
heat content, the integrated eigenvalue counting function, and, for
certain points, the ondiagonal heat kernel of generalized
Sierpinski carpets. For all these functions the leading term is of
the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$
and $\phi$ a periodic function. We also discuss similar results for
the heat content of affine nested fractals.


181  Characterizations of Continuous and Discrete $q$Ultraspherical Polynomials Ismail, Mourad E. H.; Obermaier, Josef
We characterize the continuous $q$ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the AskeyWilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$Jacobi polynomials is given.


200  An Explicit Polynomial Expression for a $q$Analogue of the 9$j$ Symbols Rahman, Mizan
Using standard transformation and summation formulas for basic
hypergeometric series we obtain an explicit polynomial form of the
$q$analogue of the 9$j$ symbols, introduced by the author in a
recent publication. We also consider a limiting case in which the
9$j$ symbol factors into two Hahn polynomials. The same
factorization occurs in another limit case of the corresponding
$q$analogue.


222  Limit Theorems for Additive Conditionally Free Convolution Wang, JiunChau
In this paper we determine the limiting distributional behavior for
sums of infinitesimal conditionally free random variables. We show that the weak
convergence of classical convolution and that of conditionally free convolution
are equivalent for measures in an infinitesimal triangular array,
where the measures may have unbounded support. Moreover, we use these
limit theorems to study the conditionally free infinite divisibility. These results
are obtained by complex analytic methods without reference to the
combinatorics of cfree convolution.


241  Multiple ZetaFunctions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula Essouabri, Driss; Matsumoto, Kohji; Tsumura, Hirofumi
We prove the holomorphic continuation of certain multivariable multiple
zetafunctions whose coefficients satisfy a suitable recurrence condition.
In fact, we introduce more general vectorial zetafunctions and prove their
holomorphic continuation. Moreover, we show a vectorial sum formula among
those vectorial zetafunctions from which some generalizations of the
classical sum formula can be deduced.


277  Locally Indecomposable Galois Representations Ghate, Eknath; Vatsal, Vinayak
In a previous paper
the authors showed that, under some technical
conditions,
the local Galois representations attached to the members of
a nonCM family of ordinary cusp forms are indecomposable for all
except possibly finitely many
members of the family. In this paper we use deformation theoretic
methods to give examples of nonCM families for
which every classical member of weight at least two has a locally
indecomposable Galois representation.


298  A Variant of Lehmer's Conjecture, II: The CMcase Gun, Sanoli; Murty, V. Kumar
Let $f$ be a normalized Hecke eigenform with rational integer Fourier
coefficients. It is an interesting question to know how often an
integer $n$ has a factor common with the $n$th Fourier coefficient of
$f$. It has been shown in previous papers that this happens very often. In this
paper, we give an asymptotic formula for the number of integers $n$
for which $(n, a(n)) = 1$, where $a(n)$ is the $n$th Fourier coefficient of
a normalized Hecke eigenform $f$ of weight $2$ with rational integer
Fourier coefficients and having complex multiplication.


327  Discrete Series for $p$adic $SO(2n)$ and Restrictions of Representations of $O(2n)$ Jantzen, Chris
In this paper we give a classification of discrete series for
$SO(2n,F)$, $F$ $p$adic, similar to that of
MœglinTadić for
the other classical groups. This is obtained by taking the
MœglinTadić classification for $O(2n,F)$ and studying how the
representations restrict to $SO(2n,F)$. We then extend this to an
analysis of how admissible representations of $O(2n,F)$ restrict.


381  A Complete Classification of AI Algebras with the Ideal Property Ji, Kui ; Jiang, Chunlan
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$algebra inductive limit
of a sequence
$$
A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3}
\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,
$$
where
$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,
$X^{i}_n$ are $[0,1]$, $k_n$, and
$[n,i]$ are positive integers.
Suppose that $A$ has the
ideal property: each closed twosided ideal of $A$ is generated by
the projections inside the ideal, as a closed twosided ideal.
In this article, we give a complete classification of AI algebras with the ideal property.


413  Generating Functions for Hecke Algebra Characters Konvalinka, Matjaž; Skandera, Mark
Certain polynomials in $n^2$ variables that serve as generating
functions for symmetric group characters are sometimes called
($S_n$) character immanants.
We point out a close connection between the identities of
LittlewoodMerrisWatkins
and GouldenJackson, which relate $S_n$ character immanants
to the determinant, the permanent and MacMahon's Master Theorem.
From these results we obtain a generalization
of Muir's identity.
Working with the quantum polynomial ring and the Hecke algebra
$H_n(q)$, we define quantum immanants that are generating
functions for Hecke algebra characters.
We then prove quantum analogs of the LittlewoodMerrisWatkins identities
and selected GouldenJackson identities
that relate $H_n(q)$ character immanants to
the quantum determinant, quantum permanent, and quantum Master Theorem
of GaroufalidisL\^eZeilberger.
We also obtain a generalization of Zhang's quantization of Muir's
identity.


436  Simplicial Complexes and Open Subsets of NonSeparable LFSpaces Mine, Kotaro; Sakai, Katsuro
Let $F$ be a nonseparable LFspace homeomorphic to
the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$,
where $\aleph_0 < \tau_1 < \tau_2 < \cdots$.
It is proved that
every open subset $U$ of $F$ is homeomorphic to the product $K \times F$
for some locally finitedimensional simplicial complex $K$ such that
every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$
with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$
(and $\operatorname{card} K^{(0)} \le \tau$),
and, conversely, if $K$ is such a simplicial complex,
then the product $K \times F$ can be embedded in $F$ as an open set,
where $K$ is the polyhedron of $K$ with the metric topology.


460  Monotonically Controlled Mappings Pavlíček, Libor
We study classes of mappings between finite and infinite dimensional
Banach spaces that are monotone and mappings which are differences
of monotone mappings (DM). We prove a RadóReichelderfer estimate
for monotone mappings in finite dimensional spaces that remains
valid for DM mappings. This provides an alternative proof of the
Fréchet differentiability a.e. of DM mappings. We establish a
Morreytype estimate for the distributional derivative of monotone
mappings. We prove that a locally DM mapping between finite
dimensional spaces is also globally DM. We introduce and study a new
class of the socalled UDM mappings between Banach spaces, which
generalizes the concept of curves of finite variation.


481  The Ample Cone for a K3 Surface Baragar, Arthur
In this paper, we give several pictorial fractal
representations of the ample or Kähler cone for surfaces in a
certain class of $K3$ surfaces. The class includes surfaces
described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a
sufficiently large number field $K$ that have a line parallel to
one of the axes and have Picard number four. We relate the
Hausdorff dimension of this fractal to the asymptotic growth of
orbits of curves under the action of the surface's group of
automorphisms. We experimentally estimate the Hausdorff dimension
of the fractal to be $1.296 \pm .010$.


500  OneParameter Continuous Fields of Kirchberg Algebras. II Dadarlat, Marius; Elliott, George A.; Niu, Zhuang
Parallel to the first two authors' earlier classification of separable, unita
oneparameter, continuous fields of Kirchberg algebras with torsion free
$\mathrm{K}$groups supported in one dimension, oneparameter, separable, uni
continuous fields of AFalgebras are classified by their ordered
$\mathrm{K}_0$sheaves. EffrosHandelmanShen type theorems are pr separable
unital oneparameter continuous fields of AFalgebras and Kirchberg algebras.


533  On Best Proximity Points in Metric and Banach Spaces Espínola, Rafa; FernándezLeón, Aurora
In this paper we study the existence and uniqueness of
best proximity points of cyclic contractions as well as the convergence
of iterates to such proximity points. We take two different approaches,
each one leading to different results that complete, if not improve,
other similar results in the theory. Results in this paper stand for Banach
spaces, geodesic metric spaces and metric spaces. We also include an appendix
on CAT$(0)$ spaces where we study the particular behavior of these spaces
regarding the problems we are concerned with.


551  Topological Free Entropy Dimensions in Nuclear C$^*$algebras and in Full Free Products of Unital C$^*$algebras Hadwin, Don; Li, Qihui; Shen, Junhao
In the paper, we introduce a new concept,
topological orbit dimension of an $n$tuple of elements in a unital
C$^{\ast}$algebra. Using this concept, we conclude that Voiculescu's
topological free
entropy dimension of every finite family of selfadjoint generators of a
nuclear C$^{\ast}$algebra is less than or equal to $1$. We also show that the
Voiculescu's topological free entropy dimension is additive in the full free
product of some unital C$^{\ast}$algebras. We show that the unital full free
product of Blackadar and Kirchberg's unital MF
algebras is also an MF algebra. As an application, we obtain that
$\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.


591  Rank One Reducibility for Metaplectic Groups via Theta Correspondence Hanzer, Marcela; Muić, Goran
We calculate reducibility for the representations of
metaplectic groups induced from cuspidal representations of
maximal parabolic subgroups via theta correspondence, in terms of the
analogous representations of the odd orthogonal groups. We also
describe the lifts of all relevant subquotients.


616  A Modular Quintic CalabiYau Threefold of Level 55 Lee, Edward
In this note we search the parameter space of HorrocksMumford quintic
threefolds and locate a CalabiYau threefold that is modular, in the
sense that the $L$function of its middledimensional cohomology is
associated with a classical modular form of weight 4 and level 55.


634  On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms Lü, Guangshi
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even
integral weight $k$ for the full modular group.
Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$th normalized Fourier coefficients of
two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively.
In this paper we are able to show the following results about higher
moments of Fourier coefficients of holomorphic cusp forms.


648  Spectral Asymptotics of Laplacians Associated with Onedimensional Iterated Function Systems with Overlaps Ngai, SzeMan
We set up a framework for computing the spectral dimension of a class of onedimensional
selfsimilar measures that are defined by iterated function systems
with overlaps and satisfy a family of secondorder selfsimilar
identities. As applications of our result we obtain the spectral dimension
of important measures such as the infinite Bernoulli convolution
associated with the golden ratio and convolutions of Cantortype measures.
The main novelty of our result is that the iterated function systems
we consider are not postcritically finite and do not satisfy the
wellknown open set condition.


689  Higher Rank Wavelets Olphert, Sean; Power, Stephen C.
A theory of higher rank multiresolution analysis is given in the
setting of abelian multiscalings. This theory enables the
construction, from a higher rank MRA, of finite wavelet sets
whose multidilations have translates forming an orthonormal basis
in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide
simple examples we construct many nonseparable higher rank
wavelets. In particular we construct Latin square
wavelets as rank~$2$ variants of Haar wavelets. Also we construct
nonseparable scaling functions for rank $2$ variants of Meyer
wavelet scaling functions, and we construct the associated
nonseparable wavelets with compactly supported Fourier transforms.
On the other hand we show that compactly supported scaling
functions for biscaled MRAs are necessarily separable.


721  Isoresonant Complexvalued Potentials and Symmetries Autin, Aymeric
Let $X$ be a connected Riemannian manifold such that the resolvent of
the free Laplacian $(\Deltaz)^{1}$, $z\in\mathbb{C} \setminus
\mathbb{R}^+$, has a meromorphic continuation
through $\mathbb{R}^+$. The poles of this continuation are called
resonances. When $X$ has some symmetries, we construct complexvalued
potentials, $V$, such that the resolvent of $\Delta+V$, which has also
a meromorphic continuation, has the same resonances with
multiplicities as the free Laplacian.


755  On the Geometry of the Moduli Space of Real Binary Octics Chu, Kenneth C. K.
The moduli space of smooth real binary octics has five connected
components. They parametrize the real binary octics whose defining
equations have $0,\dots,4$ complexconjugate pairs of roots
respectively. We show that each of these five components has a real
hyperbolic structure in the sense that each is isomorphic as a
realanalytic manifold to the quotient of an open dense subset of
$5$dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an
arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are
commensurable to discrete hyperbolic reflection groups, and the
Vinberg diagrams of the latter are computed.


798  Representing Multipliers of the Fourier Algebra on NonCommutative $L^p$ Spaces Daws, Matthew
We show that the multiplier algebra of the Fourier algebra on a
locally compact group $G$ can be isometrically represented on a direct
sum on noncommutative $L^p$ spaces associated with the right von
Neumann algebra of $G$. The resulting image is the idealiser of the
image of the Fourier algebra. If these spaces are given their
canonical operator space structure, then we get a completely isometric
representation of the completely bounded multiplier algebra. We make
a careful study of the noncommutative $L^p$ spaces we construct and
show that they are completely isometric to those considered recently
by Forrest, Lee, and Samei. We improve a result of theirs about module
homomorphisms. We suggest a definition of a FigaTalamancaHerz
algebra built out of these noncommutative $L^p$ spaces, say
$A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to
$L^1(G)$, generalising the abelian situation.


826  Singular Moduli of Shimura Curves Errthum, Eric
The $j$function acts as a parametrization of the classical modular
curve. Its values at complex multiplication (CM) points are called
singular moduli and are algebraic integers. A Shimura curve is a
generalization of the modular curve and, if the Shimura curve has
genus~$0$, a rational parameterizing function exists and when
evaluated at a CM point is again algebraic over~$\mathbf{Q}$. This paper shows
that the coordinate maps given by N.~Elkies for the Shimura
curves associated to the quaternion algebras with discriminants $6$
and $10$ are Borcherds lifts of vectorvalued modular forms. This
property is then used to explicitly compute the rational norms of
singular moduli on these curves. This method not only verifies
conjectural values for the rational CM points, but also provides a way
of algebraically calculating the norms of CM points with arbitrarily
large negative discriminant.


862  Linear Combinations of Composition Operators on the Bloch Spaces Hosokawa, Takuya; Nieminen, Pekka J.; Ohno, Shûichi
We characterize the compactness of linear combinations of analytic
composition operators on the Bloch space. We also study
their boundedness and compactness on the little Bloch space.


878  The Toric Geometry of Triangulated Polygons in Euclidean Spac Howard, Benjamin; Manon, Christopher; Millson, John
Speyer and Sturmfels associated Gröbner toric
degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$
of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each
trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations
induce toric
degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the
space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line.
Our goal in this paper is to give a
geometric (Euclidean polygon) description of the toric fibers
and describe the action of the
compact part of the torus
as "bendings of polygons".
We prove the conjecture of Foth and Hu that
the toric fibers are homeomorphic
to the spaces defined by Kamiyama and Yoshida.


938  AVCourant Algebroids and Generalized CR Structures LiBland, David
We construct a generalization of Courant algebroids that are
classified by the third cohomology group $H^3(A,V)$, where $A$ is a
Lie Algebroid, and $V$ is an $A$module. We see that both Courant
algebroids and $\mathcal{E}^1(M)$ structures are examples of
them. Finally we introduce generalized CR structures on a manifold,
which are a generalization of generalized complex structures, and show
that every CR structure and contact structure is an example of a
generalized CR structure.


961  Low Frequency Estimates for Long Range Perturbations in Divergence Form Bouclet, JeanMarc
We prove a uniform control as $ z \rightarrow 0 $ for the resolvent $
(Pz)^{1} $ of long range perturbations $ P $ of the Euclidean
Laplacian in divergence form by combining positive commutator
estimates and properties of Riesz transforms. These estimates hold in
dimension $d \geq 3 $ when $ P $ is defined on $ \mathbb{R}^d $ and in dimension $ d \geq 2 $ when $ P $ is defined outside a compact obstacle with Dirichlet boundary conditions.


992  The Arithmetic of Genus Two Curves with (4,4)Split Jacobians Bruin, Nils; Doerksen, Kevin
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$torsion, and we derive the relation their absolute invariants
satisfy.


1025  Universal Series on a Riemann Surface Clouâtre, Raphaël
Every holomorphic function on a compact subset of a Riemann surface can
be uniformly approximated by partial sums of a given series of functions.
Those functions behave locally like the classical fundamental solutions
of the CauchyRiemann operator in the plane.


1038  Critical Points and Resonance of Hyperplane Arrangements Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement
$\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship
between the critical set of $\Phi_\lambda$, the variety defined by the vanishing
of the oneform $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$.
For arrangements satisfying certain conditions, we show that if $\lambda$ is
resonant in dimension $p$, then the critical set
of $\Phi_\lambda$ has codimension
at most $p$. These include all free arrangements and all rank $3$ arrangements.


1058  $S_3$covers of Schemes Easton, Robert W.
We analyze flat $S_3$covers of schemes, attempting to create
structures parallel to those found in the abelian and triple cover
theories. We use an initial local analysis as a guide in finding a
global description.


1083  Decomposition of Splitting Invariants in Split Real Groups Kaletha, Tasho
For a maximal torus in a quasisplit semisimple simplyconnected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.


1107  Genericity of Representations of pAdic $Sp_{2n}$ and Local Langlands Parameters Liu, Baiying
Let $G$ be the $F$rational points of the symplectic group $Sp_{2n}$,
where $F$ is a nonArchimedean local field
of characteristic
$0$. Cogdell, Kim, PiatetskiShapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$case of a
conjecture of
GrossPrasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.


1137  Distribution Algebras on padic Groups and Lie Algebras Moy, Allen
When $F$ is a $p$adic field, and $G={\mathbb
G}(F)$ is the group of $F$rational points of a connected algebraic
$F$group, the complex vector space ${\mathcal H}(G)$ of compactly
supported locally constant distributions on $G$ has a natural
convolution product that makes it into a ${\mathbb C}$algebra
(without an identity) called the Hecke algebra. The Hecke algebra is a
partial analogue for $p$adic groups of the enveloping algebra of a
Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of
an identity element, and the process $G \mapsto \mathcal{H}(G)$
is not a functor. Bernstein introduced an enlargement
$\mathcal{H}\,\hat{\,}(G)$
of $\mathcal{H}(G)$. The algebra
$\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left
essentially compact. We show that the process $G \mapsto
\mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow
H$ is a morphism of $p$adic groups, let $F(\tau) \colon
\mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be
the morphism of $\mathbb{C}$algebras. We identify the kernel of
$F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$adic
Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra,
$\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the
natural projection, we show that $F(\tau)$ maps $G$invariant distributions
on $\mathcal{G}$ to $N_G (\mathfrak{m})$invariant distributions on
$\mathfrak{m}$. Finally, we exhibit a natural family of $G$invariant
essentially compact distributions on $\mathfrak{g}$ associated with a
$G$invariant nondegenerate symmetric bilinear form on ${\mathfrak g}$
and in the case of $SL(2)$ show how certain members of the family can
be moved to the group.


1161  Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group Neuwirth, Stefan; Ricard, Éric
We inspect the relationship between relative Fourier
multipliers on noncommutative LebesgueOrlicz spaces of a discrete
group $\varGamma$ and relative ToeplitzSchur multipliers on
SchattenvonNeumannOrlicz classes. Four applications are given:
lacunary sets, unconditional Schauder bases for the subspace of a
Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the
norm of the Hilbert transform and the Riesz projection on
SchattenvonNeumann classes with exponent a power of 2, and the norm of
Toeplitz Schur multipliers on SchattenvonNeumann classes with
exponent less than 1.


1188  On Complemented Subspaces of NonArchimedean Power Series Spaces Śliwa, Wiesław; Ziemkowska, Agnieszka
The nonarchimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the
best known and most important examples of nonarchimedean nuclear Fréchet spaces.
We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$
has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all
bounded limit points of the double sequence
$(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It
follows that every complemented subspace of a power series space $A_p(a)$ has a
Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded.


1201  Resonant Tunneling of Fast Solitons through Large Potential Barriers Abou Salem, Walid K. ; Sulem, Catherine
We rigorously study the resonant tunneling of fast solitons through large
potential barriers for the nonlinear Schrödinger equation in
one dimension. Our approach covers the case of general nonlinearities,
both local and Hartree (nonlocal).


1220  Similar Sublattices of Planar Lattices Baake, Michael; Scharlau, Rudolf; Zeiner, Peter
The similar sublattices of a planar lattice can be classified via
its multiplier ring. The latter is the ring of rational integers in
the generic case, and an order in an imaginary quadratic field
otherwise. Several classes of examples are discussed, with special
emphasis on concrete results. In particular, we derive Dirichlet
series generating functions for the number of distinct similar
sublattices of a given index, and relate them to
zeta functions of orders in imaginary quadratic fields.


1238  Casselman's Basis of Iwahori Vectors and the Bruhat Order Bump, Daniel; Nakasuji, Maki
W. Casselman defined a basis $f_u$ of Iwahori fixed vectors of a spherical
representation $(\pi, V)$ of a split semisimple $p$adic group $G$ over a
nonarchimedean local field $F$ by the condition that it be dual to the
intertwining operators, indexed by elements $u$ of the Weyl group $W$. On
the other hand, there is a natural basis $\psi_u$, and one seeks to find the
transition matrices between the two bases. Thus, let $f_u = \sum_v \tilde{m}
(u, v) \psi_v$ and $\psi_u = \sum_v m (u, v) f_v$. Using the IwahoriHecke
algebra we prove that if a combinatorial condition is satisfied, then $m (u,
v) = \prod_{\alpha} \frac{1  q^{ 1} \mathbf{z}^{\alpha}}{1
\mathbf{z}^{\alpha}}$, where $\mathbf z$ are the Langlands parameters
for the representation and $\alpha$ runs through the set $S (u, v)$ of
positive coroots $\alpha \in \hat{\Phi}$ (the dual root system of $G$) such
that $u \leqslant v r_{\alpha} < v$ with $r_{\alpha}$ the reflection
corresponding to $\alpha$. The condition is conjecturally always satisfied
if $G$ is simplylaced and the KazhdanLusztig polynomial $P_{w_0 v, w_0 u}
= 1$ with $w_0$ the long Weyl group element. There is a similar formula for
$\tilde{m}$ conjecturally satisfied if $P_{u, v} = 1$.
This leads to various combinatorial conjectures.


1254  Constructions of Chiral Polytopes of Small Rank D'Azevedo, Antonio Breda; Jones, Gareth A.; Schulte, Egon
An abstract polytope of rank $n$ is said to be chiral if its
automorphism group has precisely two orbits on the flags, such that
adjacent flags belong to distinct orbits. This paper describes
a general method for deriving new finite chiral polytopes from old
finite chiral polytopes of the same rank. In particular, the technique
is used to construct many new examples in ranks $3$, $4$, and $5$.


1284  NonExistence of Ramanujan Congruences in Modular Forms of Level Four Dewar, Michael
Ramanujan famously found congruences like $p(5n+4)\equiv 0
\operatorname{mod} 5$ for the partition
function. We provide a method to find all simple
congruences of this type in the coefficients of the inverse of a
modular form on $\Gamma_{1}(4)$ that is nonvanishing on the upper
half plane. This is applied to answer open questions about the
(non)existence of congruences in the generating functions for
overpartitions, crank differences, and 2colored $F$partitions.


1307  A BottBorelWeil Theorem for Diagonal Indgroups Dimitrov, Ivan; Penkov, Ivan
A diagonal indgroup is a direct limit of classical affine algebraic
groups of growing rank under a class of
inclusions that contains the inclusion
$$
SL(n)\to SL(2n), \quad
M\mapsto \begin{pmatrix}M & 0 \\ 0 & M \end{pmatrix}
$$
as a typical special case. If $G$ is a diagonal indgroup and
$B\subset G$ is a Borel indsubgroup,
we consider the indvariety $G/B$ and compute the cohomology
$H^\ell(G/B,\mathcal{O}_{\lambda})$
of any $G$equivariant line bundle $\mathcal{O}_{\lambda}$ on
$G/B$. It has been known that, for a generic $\lambda$,
all cohomology groups of $\mathcal{O}_{\lambda}$ vanish, and that a
nongeneric equivariant
line bundle $\mathcal{O}_{\lambda}$ has at most one
nonzero cohomology group. The new result of this paper is a
precise description of when
$H^j(G/B,\mathcal{O}_{\lambda})$ is nonzero and the proof of the fact
that, whenever nonzero,
$H^j(G/B, \mathcal{O}_{\lambda})$ is a $G$module dual to a highest
weight module.
The main difficulty is in defining an appropriate analog $W_B$ of the
Weyl group, so that the action of $W_B$
on weights of $G$ is compatible with the analog of the Demazure
``action" of the Weyl group on the cohomology
of line bundles. The highest weight corresponding to $H^j(G/B,
\mathcal{O}_{\lambda})$ is then computed
by a procedure similar to that in the classical BottBorelWeil theorem.


1328  On a Conjecture of Chowla and Milnor Gun, Sanoli; Murty, M. Ram; Rath, Purusottam
In this paper, we investigate a conjecture due to S. and P. Chowla and
its generalization by Milnor. These are related to the delicate
question of nonvanishing of $L$functions associated to periodic
functions at integers greater than $1$. We report on some progress in
relation to these conjectures. In a different vein, we link them to a
conjecture of Zagier on multiple zeta values and also to linear
independence of polylogarithms.


1345  Pointed Torsors Jardine, J. F.
This paper gives a characterization of homotopy fibres of inverse
image maps on groupoids of torsors that are induced by geometric
morphisms, in terms of both pointed torsors and pointed cocycles,
suitably defined. Cocycle techniques are used to give a complete
description of such fibres, when the underlying geometric morphism is
the canonical stalk on the classifying topos of a profinite group
$G$. If the torsors in question are defined with respect to a constant
group $H$, then the path components of the fibre can be identified with
the set of continuous maps from the profinite group $G$ to the group
$H$. More generally, when $H$ is not constant, this set of path components
is the set of continuous maps from a proobject in sheaves of
groupoids to $H$, which proobject can be viewed as a ``Grothendieck
fundamental groupoid".


1364  The Cubic Dirac Operator for InfiniteDimensonal Lie Algebras Meinrenken, Eckhard
Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}} \mathfrak{g}_i$ be an infinitedimensional graded
Lie algebra, with $\dim\mathfrak{g}_i<\infty$, equipped with a nondegenerate
symmetric bilinear form $B$ of degree $0$. The quantum Weil algebra
$\widehat{\mathcal{W}}\mathfrak{g}$ is a completion of the tensor product of the
enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the
KacPeterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac
operator $\mathcal{D}\in\widehat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir
element. We show that this condition holds for symmetrizable
KacMoody algebras. Extending Kostant's arguments, one obtains
generalized WeylKac character formulas for suitable ``equal rank''
Lie subalgebras of KacMoody algebras. These extend the formulas of
G. Landweber for affine Lie algebras.


1388  Nonabelian $H^1$ and the Étale Van Kampen Theorem Misamore, Michael D.
Generalized étale homotopy progroups $\pi_1^{\operatorname{ét}}(ċ{C}, x)$
associated with pointed, connected, small Grothendieck
sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois
theory and the theory of pointed torsors for discrete
groups is explained.
<br>
Applications include new rigorous proofs of some folklore results
around $\pi_1^{\operatorname{ét}}(ét(X), x)$, a description of
Grothendieck's short exact sequence for Galois descent in terms of
pointed torsor trivializations, and a new étale
van Kampen theorem that gives a simple statement about a pushout
square of progroups that works for covering
families that do not necessarily consist exclusively of
monomorphisms. A corresponding van Kampen result for
Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows.


1416  MAD Saturated Families and SANE Player Shelah, Saharon
We throw some light on the question: is there a MAD family
(a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any
two is finite) that is saturated (completely separable i.e., any
$X \subseteq \mathbb{N}$ is
included in a finite union of members of the family or includes a
member (and even continuum many members) of the family).
We prove that it is hard to prove the consistency of the negation:

