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721  SubRiemannian Geometry on the Sphere $\mathbb{S}^3$ Calin, Ovidiu; Chang, DerChen; Markina, Irina
We discuss the subRiemannian
geometry induced by two noncommutative
vector fields which are left invariant
on the Lie group $\mathbb{S}^3$.


740  On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings Caprace, PierreEmmanuel; Haglund, Frédéric
Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if
$\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We
prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0)
realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup
of a Coxeter group is Gromovhyperbolic if and only if it does not contain a free abelian group of rank 2. Our
result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits
buildings.


762  The Hilbert Coefficients of the Fiber Cone and the $a$Invariant of the Associated Graded Ring D'Cruz, Clare; Puthenpurakal, Tony J.
Let $(A,\m)$ be a Noetherian local ring with infinite residue
field and let $I$ be an ideal in $A$ and let $F(I) =
\bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$.
We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$
when the $a$invariant of the associated graded ring $G(I)$ is negative.


779  Residual Spectra of Split Classical Groups and their Inner Forms Grbac, Neven
This paper is concerned with the residual spectrum of the
hermitian quaternionic classical groups $G_n'$ and $H_n'$ defined
as algebraic groups for a quaternion algebra over an algebraic
number field. Groups $G_n'$ and
$H_n'$ are not quasisplit. They are inner forms of the split
groups $\SO_{4n}$ and $\Sp_{4n}$. Hence, the parts of the residual
spectrum of $G_n'$ and $H_n'$ obtained in this paper are compared
to the corresponding parts for the split groups $\SO_{4n}$ and
$\Sp_{4n}$.


807  Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients Hong, Sunggeum; Kim, Joonil; Yang, Chan Woo
We prove the $L^p(\mathbb{R}^d)$ ($1<p\le \infty$) boundedness of
the maximal operators associated with a family of vector polynomials
given by the form
$\{(2^{k_1}\mathfrak{p}_1(t),\dots,2^{k_d}\mathfrak{p}_d(t)):
t\in\mathbb{R} \}.$ Furthermore, by using the lifting argument, we
extend this result to a general class of vector polynomials whose
coefficients are of the form constant times $2^k$.


828  Twisted GrossZagier Theorems Howard, Benjamin
The theorems of GrossZagier and Zhang relate the N\'eronTate
heights of complex multiplication points on the modular curve $X_0(N)$
(and on Shimura curve analogues) with the central derivatives of
automorphic $L$function. We extend these results to include certain
CM points on modular curves of the form
$X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues).
These results are motivated by applications to Hida theory
that can be found in the companion article
"Central derivatives of $L$functions in Hida families", Math.\ Ann.\
\textbf{399}(2007), 803818.


888  Face Ring Multiplicity via CMConnectivity Sequences Novik, Isabella; Swartz, Ed
The multiplicity conjecture of Herzog, Huneke, and Srinivasan
is verified for the face rings of the following classes of
simplicial complexes: matroid complexes, complexes of dimension
one and two,
and Gorenstein complexes of dimension at most four.
The lower bound part of this conjecture is also established for the
face rings of all doubly CohenMacaulay complexes whose 1skeleton's
connectivity does not exceed the codimension plus one as well as for
all $(d1)$dimensional $d$CohenMacaulay complexes.
The main ingredient of the proofs is a new interpretation
of the minimal shifts in the resolution of the face ring
$\field[\Delta]$ via the CohenMacaulay connectivity of the
skeletons of $\Delta$.


904  The Face Semigroup Algebra of a Hyperplane Arrangement Saliola, Franco V.
This article presents a study of an algebra spanned by the faces of a
hyperplane arrangement. The quiver with relations of the algebra is
computed and the algebra is shown to be a Koszul algebra.
It is shown that the algebra depends only on the intersection lattice of
the hyperplane arrangement. A complete system of primitive orthogonal
idempotents for the algebra is constructed and other algebraic structure
is determined including: a description of the projective indecomposable
modules, the Cartan invariants, projective resolutions of the simple
modules, the Hochschild homology and cohomology, and the Koszul dual
algebra. A new cohomology construction on posets is introduced, and it is
shown that the face semigroup algebra is isomorphic to the cohomology
algebra when this construction is applied to the intersection lattice of
the hyperplane arrangement.


930  Prolongations and Computational Algebra Sidman, Jessica; Sullivant, Seth
We explore the geometric notion of prolongations in the setting of
computational algebra, extending results of Landsberg and Manivel
which relate prolongations to equations for secant varieties. We also
develop methods for computing prolongations that are combinatorial in
nature. As an application, we use prolongations to derive a new
family of secant equations for the binary symmetric model in
phylogenetics.


950  Infinitesimal Invariants in a Function Algebra Tange, Rudolf
Let $G$ be a reductive connected linear algebraic group
over an algebraically closed field of positive
characteristic and let $\g$ be its Lie algebra.
First we extend a wellknown result about the Picard group of a
semisimple group to reductive groups.
Then we prove that if the derived group is simply connected
and $\g$ satisfies a
mild condition, the algebra $K[G]^\g$ of regular functions
on $G$ that are invariant under the action of $\g$ derived
from the conjugation action is a unique factorisation domain.

