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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:
