On the geometry underlying a real Lie algebra representation

Let $G$ be a real Lie group with Lie algebra $\mathfrak g$. Given a unitary representation $\pi$ of $G$, one obtains by differentiation a representation $d\pi$ of $\mathfrak g$ by unbounded, skew-adjoint operators. Representations of $\mathfrak g$ admitting such a description are called \emph{integrable,} and they can be geometrically seen as the action of $\mathfrak g$ by derivations on the algebra of representative functions $g\mapsto$, which are naturally defined on the homogeneous space $M=G/\ker\pi$. In other words, integrable representations of a real Lie algebra can always be seen as realizations of that algebra by vector fields on a homogeneous manifold. Here we show how to use the coproduct of the universal enveloping algebra of $\mathfrak g$ to generalize this to representations which are not necessarily integrable. The geometry now playing the role of $M$ is a locally homogeneous space. This provides the basis for a geometric approach to integrability questions regarding Lie algebra representations.Comment: 12 pages. Author supported by Fondecyt Postdoctoral Grant N{\deg} 311004

On Complex Manifolds and Observable Schemes

We work out the construction of a Stein manifold from a commutative Arens-Michael algebra, under assumptions that are mild enough for the process to be useful in practice. Then, we do the passage to arbitrary complex manifolds by proposing a suitable notion of scheme. We do this in the abstract language of spectral functors, in view of its potential usefulness in non-commutative geometry.Comment: 15 pages. Supported by Fondecyt Postdoctoral Grant No. 311004

Learner Corpus Research and Second Language Acquisition: an attempt at bridging the gap

Learner corpora are traditionally defined as ‘systematic collections of authentic, continuous and contextualized language use (spoken or written) by L2 learners stored in electronic format’ (Callies & Paquot 2015). With this characterization, it appears very clearly that learner corpus researchers have always been interested in exploring the output of the more open-ended types of contextualized production tasks assigned to L2 learners (e.g. Granger 2008; Tracy-Ventura & Myles 2015). The term ‘learners’ here refers to Foreign and/or Second Language learners rather than to learners acquiring their native language (L1)

Dendritic cell mediated modulation of immune responses by Mycobacterium vaccae

The contemporary hygiene hypothesis suggests that certain microorganisms that were present throughout human evolution modulate the host immune system to reduce allergy associated T helper 2 (Th2) responses and inflammatory diseases by augmenting regulatory T cells. The prototypic environmental mycobacterium, M. vaccae has been used in mouse models of asthma to support this hypothesis, but data from human models and possible mechanisms are very limited. In view of the role of dendritic cells (DCs) in shaping adaptive T cell responses, the effect of innate immune interactions between human DCs and M. vaccae on allogeneic and antigen specific DC-dependent polarisation of T cells was tested. M. vaccae can stimulate cellular activation via Toll-like receptor 2 (TLR2) and therefore was compared to a specific TLR2 ligand (Pam3CSK4) and alternative stimulation with a TLR4 ligand (LPS). M. vaccae alone induced DC-dependent inhibition of Th2 responses, in contrast to Pam3CSK4, which had the opposite effect and LPS, which had no polarising effect. Comparison of DC maturation, genome-wide transcriptional response, and cytokine production in response to each stimulus did not correlate with the specific functional effects. In particular, directly comparable DC transcriptional responses to M. vaccae and Pam3CSK4 suggested that TLR2-mediated transcriptional regulation was not sufficient for inhibition of Th2 responses. Exclusive transcriptional responses to M. vaccae implicated a role for CREB1-dependent gene expression and analysis of signalling events confirmed selective early activation of the CREB pathway by M. vaccae. Collectively, this work has established that M. vaccae interaction with DCs does inhibit human Th2 responses and that further study of the CREB pathway in this model may provide novel insight into the molecular mechanisms of DC-dependent T cell polarisation. The final chapter of results presents development and validation of a novel approach for using short interspersed elements (SINEs) as a tool for normalisation of RT-qPCR data