Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute $\mathbb{Z}/2$–Betti numbers of these spaces.

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 $\text{G}{{\text{L}}_{d}}\left( E \right)$ and $\text{G}{{\text{L}}_{m}}\left( E \right)$ respectively. We assume that $\pi $ and $\sigma $ are conjugate-dual. That is to say $\pi \,\simeq \,{{\pi }^{\vee \text{,}\,c}}$ and $\sigma \,\simeq \,{{\sigma }^{\vee ,c}}$ where $c$ is the nontrivial $F$-automorphism of $E$. This implies that we can extend $\pi $ to an unitary representation $\tilde{\pi }$ of a nonconnected group $\text{G}{{\text{L}}_{d}}\left( E \right)\,\rtimes \,\left\{ 1,\,0 \right\}$. Define $\tilde{\sigma }$ the same way. We state and prove an integral formula for $\in \left( 1/2,\,\pi \,\times \,\sigma ,\,{{\psi }_{E}} \right)$ involving the characters of $\tilde{\pi }$ and $\tilde{\sigma }$. This formula is related to the local Gan–Gross–Prasad conjecture for unitary groups.

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 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience and, in 2-dimensions, the existence of a deterministic limiting velocity.

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 $G{{L}_{2}}\left( {{\mathbb{A}}_{K}} \right)$ 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.

We consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under 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 ${{L}^{1}}\,\to \,{{L}^{\infty }}$ dispersive estimates. We showcase the analysis in dimensions $n\,=\,5,\,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical $\text{NLS}$ on $\Omega \,=\,{{\mathbb{R}}^{n}}\backslash \overline{B\left( 0,\,1 \right)}$ with Dirichlet boundary condition and radial data in these dimensions.

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and $\text{Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi ,\,\psi :\,\text{Lat}\,\mathcal{H}\,\to \,\text{Lat}\,\mathcal{K}$ having the property that for every pair $U,\,V\,\in \,\text{Lat}\,\mathcal{H}$ we have $\mathcal{H}\,=\,U\,\oplus \,V\,\Leftrightarrow \,\mathcal{K}\,=\,\phi \left( U \right)\,\oplus \,\psi \,\left( V \right)$. 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.

The purpose of this note is to introduce a method for proving the non-existence of 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 $X\left( K \right)$ 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 by the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that one may still attach a Galois representation of $\text{Gal}\left( \bar{K},\,K \right)$ with values in the quotient group $\text{GL}\left( {{T}_{\ell }}\left( A \right) \right)/\,\text{Aut}\left( A \right)$ to a point $P\,=\,\left[ A \right]\,\in \,X\left( K \right)$ represented by an abelian variety $A/\bar{K}$, provided $\text{Aut}\left( A \right)$ lies in the centre of $\text{GL}\left( {{T}_{\ell }}\left( A \right) \right)$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over $\mathbb{Q}$.