Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.Comment: 9 page

Splittings of groups and intersection numbers

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.Comment: 40 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper6.abs.htm

Representations and KK-theory of Discrete Groups

Let $\Gamma$ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for $\Gamma$, determined on its elements of finite order, which is of finite type. Then we determine the contribution of this ring to the topological $K$-theory $K^*(B\Gamma)$, obtaining an exact formula for the difference in terms of the cohomology of the centralizers of elements of finite order in $\Gamma$.Comment: 4 page

Affine algebraic groups with periodic components

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.Comment: 20 page

One Relator Quotients of Graph Products

In this paper, we generalise Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups

X-ray sources and their optical counterparts in the globular cluster M 22

Using XMM-Newton EPIC imaging data, we have detected 50 low-luminosity X-ray sources in the field of view of M 22, where 5 +/- 3 of these sources are likely to be related to the cluster. Using differential optical photometry, we have identified probable counterparts to those sources belonging to the cluster. Using X-ray spectroscopic and timing studies, supported by the optical colours, we propose that the most central X-ray sources in the cluster are cataclysmic variables, millisecond pulsars, active binaries and a blue straggler. We also identify a cluster of galaxies behind this globular cluster.Comment: 11 pages, 7 figures, accepted for publication in A&

Towards a Unified Computational Model of Contextual Interactions across Visual Modalities

The perception of a stimulus is largely determined by its surrounding. Examples abound from color (Land and McCann, 1971), disparity (Westheimer, 1986) and motion induction (Anstis and Casco, 2006) to orientation tilt effects (O’Toole and Wenderoth, 1976). Some of these phenomena have been studied individually using monkey neurophysiology techniques. In these experiments, a center stimulus is typically used to probe a cell’s classical “center” receptive field (cRF), whose activity is then modulated by an annular “surround” (extra-cRF) stimulus. While this center-surround integration (CSI) has been well characterized, a theoretical framework which unifies these different phenomena across visual modalities is lacking. Here, we present an extension of a popular cortical inhibition circuit, divisive normalization (Carandini and Heeger, 2011), which yields a computational model that is consistent with experimental data across visual modalities. We have found that a common characteristic of CSI across modalities is a shift in neural population responses induced by surround activity. Typical implementations of the divisive normalization model rely on gain control mechanisms from an ‘untuned’ suppressive pool of cells; that is, the identity of that pool is the same for every cell being suppressed. As such, the circuit cannot account for the selective shift in population response curves observed in contextual effects. In contrast, we show that the addition of an extra-classical suppressive ‘tuned’ pool of cells which selectively inhibits different parts of a population response curve is sufficient to explain complex shifts in population tuning responses. Overall, our results suggest that a normalization circuit based on two forms of inhibition, gain control and selective suppression, captures shifts in population responses associated with CSI and yields a model that seems consistent with contextual phenomena across visual modalities

Towards a Unified Model of Classical and Extra-Classical Receptive Fields

One of the major goals in neuroscience is to understand how the cortex processes information. A substantial effort has thus gone into mapping classical receptive fields (cRF) across areas of the visual cortex and characterizing input-output relationships through linear-nonlinear response functions. Recently, there has been a lot of interest in mapping the extra-classical receptive field (extra-cRF) as well, by using contextual stimuli. The extra-cRF is a region outside the cRF that modulates a cell’s response but that is incapable of driving it on its own. However, existing models typically focus on one particular visual modality (form, motion, disparity or color), and do not offer a coherent computational role for the extra-cRF. Meanwhile, because of the sheer diversity of contextual effects, we still lack a single model that is consistent with the known anatomy and physiology of the visual cortex. Here, we present an integrated computational model of early vision that comprehensively describes neural responses in the primary visual cortex across modalities (form, motion, disparity and color). The basic circuit combines “untuned” recurrent connections within the cRF with “tuned” recurrent interactions in the extra-cRF. The circuit offers a characterization of disparate contextual phenomena across visual modalities as general induction phenomena. We show that the resulting circuit seems sufficient to capture the extent of psychophysical data on color constancy – offering a possible computational-level justification for the observed center-surround interactions

Extended diffeomorphism algebras in (quantum) gravitational physics

We construct an explicit representation of the algebra of local diffeomorphisms of a manifold with realistic dimensions. This is achieved in the setting of a general approach to the (quantum) dynamics of a physical system which is characterized by the fundamental role assigned to a basic underlying symmetry. The developed mathematical formalism makes contact with the relevant gravitational notions by means of the addition of some extra structure. The specific manners in which this is accomplished, together with their corresponding physical interpretation, lead to different gravitational models. Distinct strategies are in fact briefly outlined, showing the versatility of the present conceptual framework.Comment: 20 pages, LATEX, no figure