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    <subfield code="a">Using an integrated luminosity of 320 $\mathrm{pb}^{-1}$ collected by LHCb in the first part of the 2011 run, we reconstruct a large sample of the charmless charged two-body decay modes of $B$ hadrons, providing measurements of the direct $CP$ asymmetries $A_{CP}(B^0 \rightarrow K\pi)=-0.088 \pm 0.011\mathrm{(stat)} \pm 0.008\mathrm{(syst)}$ and $A_{CP}(B^0_s \rightarrow \pi K)=0.27 \pm 0.08\mathrm{(stat)} \pm 0.02\mathrm{(syst)}$. The former constitutes the best measurement in the world as well as the first observation of $CP$ violation at LHCb, while the latter is the first evidence of $CP$ violation in the $B^0_s \rightarrow \pi K$ decay. Furthermore, we measure the branching fractions $\mathcal{BR}(B^0 \rightarrow K^+K^-)=(0.13 ^{+0.06}_{-0.05} \mathrm{(stat)} \pm 0.07\mathrm{(syst)})\times10^{-6}$ and $\mathcal{BR}(B^0_s \rightarrow \pi^+\pi^-)=(0.98 ^{+0.23}_{-0.19} \mathrm{(stat)} \pm 0.11\mathrm{(syst)})\times10^{-6}$. In particular, the absence of a $B^0_s \rightarrow \pi^+\pi^- $ signal is rejected with a statistical significance of 5.3$\sigma$.</subfield>
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    <subfield code="a">For a single scalar field with unit sound speed, there are exactly three distinct cosmological solutions which produce a scale invariant spectrum of curvature perturbations in a dynamical attractor background, assuming vacuum initial conditions: slow-roll inflation; a slowly contracting adiabatic ekpyrotic phase, described by a rapidly-varying equation of state; and an adiabatic ekpyrotic phase on a slowly expanding background. Of these three, only inflation remains weakly coupled over a wide range of modes, while the other scenarios can produce at most 12 e-folds of scale invariant and gaussian modes. In this paper, we investigate how allowing the speed of sound of fluctuations to evolve in time affects this classification. While in the presence of a variable sound speed there are many more scenarios which are scale invariant at the level of the two-point function, they generically suffer from strong coupling problems similar to those in the canonical case. There is, however, an exceptional case with superluminal sound speed, which suppresses non-gaussianities and somewhat alleviates strong coupling issues. We show that even these scenarios are constrained and only able to produce at most 28 e-folds of scale invariant and gaussian perturbations. This condition results from the combined requirements of matching the observed amplitude of curvature perturbations, demanding that the Hubble parameter remain sub-Planckian, and keeping non-gaussianities under control. We therefore conclude that inflation remains the unique cosmological scenario, assuming a single degree of freedom on an attractor background, capable of producing arbitrarily many scale invariant modes while remaining weakly coupled. Alternative mechanisms must inevitably be unstable or rely on multiple degrees of freedom.</subfield>
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    <subfield code="a">On Truncation of irreducible representations of Chevalley groups</subfield>
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    <subfield code="a">We prove part of a higher rank analogue of the Mazur-Gouvea Conjecture. More precisely, let $\tilde{\bf G}$ be a connected, reductive ${\Bbb Q}$-split group and let $\Gamma$ be an arithmetic subgroup of $\tilde{\bf G}$. We show that the dimension of the slope $\alpha$ subspace of the cohomology of $\Gamma$ with values in an irreducible $\tilde{\bf G}$-module $L$ is bounded independently of $L$. The proof is elementary making only use of general principles of the representation theory of algebraic groups; it is based on consideration of certain truncations of irreducible representations of Chevalley groups.</subfield>
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