Stabilization phenomena in Kac-Moody algebras and quiver varieties

Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X_0 a subgraph with all vertices of degree 1 or 2. Using the crystal structure on the components of quiver varieties for X, we show that if we expand X by extending X_0, the branching multiplicities and tensor product multiplicities stabilize, provided the weights involved satisfy a condition which we call ``depth'' and are supported outside $X_0$. This extends a theorem of Kleber and Viswanath. Furthermore, we show that the weight multiplicities of such representations are polynomial in the length of X_0, generalizing the same result for A_\ell by Benkart, et al.Comment: final version, to appear in International Math Research Notices. 17 pages, 4 figure

The path space of a higher-rank graph

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutative $C^*$-subalgebra $D_\Lambda$ of $C^*(\Lambda)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph \wt\Lambda with no sources in which $\Lambda$ is embedded, and show that $\partial\Lambda$ is homeomorphic to a subset of \partial\wt\Lambda . We show that when $\Lambda$ is row-finite, we can identify $C^*(\Lambda)$ with a full corner of C^*(\wt\Lambda), and deduce that $D_\Lambda$ is isomorphic to a corner of D_{\wt\Lambda}. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.Comment: 30 pages, all figures drawn with TikZ/PGF. Updated numbering and minor corrections to coincide with published version. Updated 29-Feb-2012 to fix a compiling error which resulted in the arXiv PDF output containing two copies of the articl

A direct approach to co-universal algebras associated to directed graphs

We prove directly that if E is a directed graph in which every cycle has an entrance, then there exists a C*-algebra which is co-universal for Toeplitz-Cuntz-Krieger E-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.Comment: 9 pages; V2: the definition of a Toeplitz-Cuntz-Krieger $E$-family has been correcte

Planting on Rural School Grounds

PDF pages: 2

The changing role of cell culture in the generation of transgenic livestock

Transgenesis may allow the generation of farm animals with altered phenotype, animal models for research and animal bioreactors. Although such animals have been produced, the time and expense involved in generating transgenic livestock and then evaluating the transgene expression pattern is very restrictive. If questions about the ability and efficiency of expression could be asked solely in vitro rapid progress could be achieved. Unfortunately, experiments addressing transcriptional control in vitro have proved unreliable in their ability to indicate whether a transgene will be transcribed or not. However, initial studies suggest that cell culture may be able to predict in vivo post-transcriptional events. We review these issues and propose that strategies which engineer the transgene integration site could enhance the probability for efficient expression. This approach has now become feasible with the development of techniques allowing animals to be generated from somatic cells by nuclear transfer. The important step in this procedure is the use of cells grown in culture as the source of genetic information, allowing the selection of specific transgene integration events. This technology which has dramatically increased the potential use of transgenic livestock for both agricultural and biotechnological applications, is based on standard cell culture methodology. We are now at the start of a new era in large animal transgenics

Application of the Contouring Method to Extended Microlensed Sources

The method devised by Lewis et al. (1993) for calculating the light curve of a microlensed point source is expanded to two dimensions to enable the calculation of light curves of extended sources. This method is significantly faster than the ray shooting method that has been used in the past. The increased efficiency is used to obtain much higher resolution light curves over increased timescales. We investigate the signatures arising from different source geometries in a realistic microlensing model. We show that a large fraction of high magnification events (HMEs) in image A of Q2237+0305 involve only one caustic, and could therefore yield information on the structure of the quasar continuum through the recognition of a characteristic event shape. In addition, the cataloguing of HMEs into morphological type will, in theory, enable the direction of the transverse motion, as well as the source size to be obtained from long term monitoring.Comment: 10 pages including 4 figures. Accepted for publication in M.N.R.A.

Falling Incapacity Benefit claims in a former industrial city: policy impacts or labour market improvement?

This article provides an in-depth study of Incapacity Benefit (IB) claims in a major city and of the factors behind their changing level. It relates to the regime prior to the introduction of the Employment and Support Allowance (ESA) in 2008. Glasgow has had one of the highest levels of IB in Britain with a peak of almost one fifth of the working age population on IB or Severe Disablement Allowance (SDA). However, over the past decade the number of IB claimants in Glasgow, as in other high claiming areas, has fallen at a faster rate than elsewhere, and Glasgow now has twice the national proportion of working-age people on IB/SDA rather than its peak of three times. The rise in IB in Glasgow can be attributed primarily to deindustrialisation; between 1971 and 1991, over 100,000 manufacturing jobs were lost in the city. Policy response was belated. Lack of local statistics on IB led to a lengthy delay in official recognition of the scale of the issue, and targeted programmes to divert or return IB claimants to work did not begin on any scale until around 2004. Evidence presented in the article suggests that the reduction in claims, which has mainly occurred since about 2003, has been due more to a strengthening labour market than to national policy changes or local programmes. This gives strong support to the view that excess IB claims are a form of disguised unemployment. Further detailed evaluation of ongoing programmes is required to develop the evidence base for this complex area. However, the study casts some doubt on the need for the post-2006 round of IB reforms in high-claim areas, since rapid decline in the number of claimants was already occurring in these areas. The article also indicates the importance of close joint working between national and local agencies, and further development of local level statistics on IB claimants

Categorification of quantum symmetric pairs I

We categorify a coideal subalgebra of the quantum group of $\mathfrak{sl}_{2r+1}$ by introducing a $2$-category \`a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the BGG category $\mathcal{O}$ of type B/C.Comment: final version, to appear in Quantum Topolog