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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.
