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961  Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature Andrade, Jaime; Dávila, Nestor; PérezChavela, Ernesto; Vidal, Claudio
We classify and analyze the orbits of the Kepler problem on surfaces
of constant curvature (both positive and negative, $\mathbb S^2$ and
$\mathbb H^2$, respectively) as function of the angular momentum and
the energy. Hill's region are characterized and the problem of
timecollision is studied. We also regularize the problem in
Cartesian and intrinsic coordinates, depending on the constant
angular momentum and we describe the orbits of the regularized
vector field. The phase portrait both for $\mathbb S^2$ and $\mathbb H^2$
are pointed out.


992  Classification of Regular Parametrized Onerelation Operads Bremner, Murray; Dotsenko, Vladimir
JeanLouis Loday introduced a class of symmetric operads generated
by one bilinear operation subject to one
relation making each leftnormed product of three elements equal
to a linear combination
of rightnormed products:
\[
(a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\
;
\]
such an operad is called a parametrized onerelation operad.
For a particular choice of parameters $\{x_\sigma\}$,
this operad is said to be regular if each of its components is
the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a
graded vector space, isomorphic to the tensor
algebra of $V$. We classify, over an algebraically closed field
of characteristic zero, all regular parametrized onerelation
operads.
In fact, we prove that each such operad is isomorphic to one
of the following five operads: the leftnilpotent operad
defined by the relation $((a_1a_2)a_3)=0$, the associative operad,
the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the
Poisson operad.
Our computational methods combine linear algebra over polynomial
rings, representation theory of the symmetric group, and
Gröbner bases for determinantal ideals and their radicals.


1036  Stability for the BrunnMinkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Nonlocal Isoperimetry Carlen, Eric; Maggi, Francesco
We provide a simple, general argument to obtain improvements
of concentrationtype inequalities starting from improvements
of their corresponding isoperimetrictype inequalities. We apply
this argument to obtain robust improvements of the BrunnMinkowski
inequality (for Minkowski sums between generic sets and convex
sets) and of the Gaussian concentration inequality. The former
inequality is then used to obtain a robust improvement of the
Riesz rearrangement inequality under certain natural conditions.
These conditions are compatible with the applications to a finiterange
nonlocal isoperimetric problem arising in statistical mechanics.


1064  Amenability and Covariant Injectivity of Locally Compact Quantum Groups II Crann, Jason
Building on our previous work, we study the nonrelative homology
of quantum group convolution algebras. Our main result establishes
the equivalence of amenability of a locally compact quantum group
$\mathbb{G}$ and 1injectivity of
$L^{\infty}(\widehat{\mathbb{G}})$
as an operator
$L^1(\widehat{\mathbb{G}})$module.
In particular, a locally compact group $G$ is amenable if and
only if its group von Neumann algebra
$VN(G)$
is 1injective as
an operator module over the Fourier algebra $A(G)$. As an application,
we provide a decomposability result for completely bounded
$L^1(\widehat{\mathbb{G}})$module
maps on
$L^{\infty}(\widehat{\mathbb{G}})$,
and give a simplified proof that amenable discrete
quantum groups have coamenable compact duals which avoids the
use of modular theory and the PowersStørmer inequality, suggesting
that our homological techniques may yield a new approach to the
open problem of duality between amenability and coamenability.


1087  Absolute Continuity of Wasserstein Barycenters Over Alexandrov Spaces Jiang, Yin
In this paper, we prove that, on a compact, $n$dimensional Alexandrov
space with curvature $\geqslant 1$, the Wasserstein barycenter of
Borel probability measures $\mu_1,...,\mu_m$ is absolutely continuous
with respect to the $n$dimensional Hausdorff measure if one
of them is.


1109  Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C$^*$algebras Ng, P. W.; Skoufranis, P.
In this paper, we characterize the closures of convex hulls of
unitary orbits of selfadjoint operators in unital, separable,
simple C$^*$algebras with nontrivial tracial simplex, real
rank zero, stable rank one, and strict comparison of projections
with respect to tracial states. In addition, an upper bound
for the number of unitary conjugates in a convex combination
needed to approximate a selfadjoint are obtained.


1143  The seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices Sikirić, Mathieu Dutour
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$
and radius $r$
is called empty if for any $v\in L$ we have $\Vert v 
c\Vert \geq r$.
Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay
polytope}
$P=\operatorname{conv}(S(c,r)\cap L)$.
A Delaunay polytope is called {\em perfect} if any affine transformation
$\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily
an isometry
of the space composed with an homothety.


1169  On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical Varma, Sandeep
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic
subgroup of a connected
reductive group $\operatorname{G}$ over a $p$adic field $F$. Assume that there
exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$
to an opposite parabolic subgroup.
When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$orbit,
F. Shahidi and X. Yu describe a certain distribution $D$ on
$\operatorname{M}(F)$
such that,
for irreducible unitary supercuspidal representations $\pi$ of
$\operatorname{M}(F)$ with
$\pi \cong \pi \circ \operatorname{Int} w_0$,
$\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is
irreducible
if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of
$\pi$. Since
this irreducibility is conjecturally related to $\pi$ arising
via
transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is
of interest
to realize $D$ as endoscopic transfer from a simpler distribution
on a twisted
endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations
where $\operatorname{N}$ is abelian. Here, we handle the `standard examples'
in cases
where $\operatorname{N}$ is nonabelian but admits a Zariski dense
$\operatorname{Int} \operatorname{M}$orbit.

