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961  Moduli Spaces of Vector Bundles over a Real Curve: $\mathbb Z/2$Betti Numbers Baird, Thomas
Moduli spaces of real bundles over a real curve arise naturally
as Lagrangian submanifolds of the moduli space of semistable
bundles over a complex curve. In this paper, we adapt the methods
of AtiyahBott's ``YangMills over a Riemann Surface'' to compute
$\mathbb Z/2$Betti numbers of these spaces.


993  Expression d'un facteur epsilon de paire par une formule intégrale BeuzartPlessis, Raphaël
Let $E/F$ be a quadratic extension of $p$adic fields and
let $d$, $m$ be nonnegative integers of distinct parities. Fix
admissible irreducible tempered representations $\pi$ and $\sigma$ of
$GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and
$\sigma$ are conjugatedual. That is to say $\pi\simeq \pi^{\vee,c}$
and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial
$F$automorphism of $E$. This implies, we can extend $\pi$ to an
unitary representation $\tilde{\pi}$ of a nonconnected group
$GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same
way. We state and prove an integral formula for
$\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of
$\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the
local GanGrossPrasad conjecture for unitary groups.


1050  Random Walks in Degenerate Random Environments Holmes, Mark; Salisbury, Thomas S.
We study the asymptotic behaviour of random walks in i.i.d. random
environments on $\mathbb{Z}^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2valued environment, and show that this does not hold for 3valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but nontrivial conditions on the distribution of the environment.
Our results include generalisations (to the nonelliptic setting) of 01 laws for directional transience, and in 2dimensions the existence of a deterministic limiting velocity.


1078  Values of Twisted Tensor $L$functions of Automorphic Forms Over Imaginary Quadratic Fields Lanphier, Dominic; Skogman, Howard
Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$
denote the adeles of $K$.
We find special values at all of the critical points of twisted
tensor $L$functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$,
and establish Galois equivariance of the values.
To investigate the values, we determine the archimedean factors
of a class of integral representations of these $L$functions,
thus proving a conjecture due to Ghate. We also investigate
analytic properties of these $L$functions, such as their functional
equations.


1110  On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
We consider the obstacle problem for the Schrödinger evolution
in the exterior of the unit ball with Dirichlet boundary condition. Under
the radial symmetry we compute explicitly the fundamental solution
for the linear Dirichlet Schrödinger
propagator $e^{it\Delta_D}$
and give a robust algorithm to prove sharp $L^1 \rightarrow
L^{\infty}$ dispersive estimates. We showcase the analysis in
dimensions $n=5,7$. As an application, we obtain global
wellposedness and scattering for defocusing energycritical NLS on
$\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary
condition and radial data in these dimensions.


1143  Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space Plevnik, Lucijan; Šemrl, Peter
Let $\mathcal{H}$ and $\mathcal{K}$ be infinitedimensional separable
Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$.
We describe the general form of pairs of bijective maps $\phi , \psi :
{\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair
$U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description
of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known
structural results for maps on idempotents are easy consequences.


1167  Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves Rotger, Victor; de VeraPiquero, Carlos
The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an AtkinLehner quotient over $\mathbb{Q}$.

