342 research outputs found
The Baum-Connes Conjecture via Localisation of Categories
We redefine the Baum-Connes assembly map using simplicial approximation in
the equivariant Kasparov category. This new interpretation is ideal for
studying functorial properties and gives analogues of the assembly maps for all
equivariant homology theories, not just for the K-theory of the crossed
product. We extend many of the known techniques for proving the Baum-Connes
conjecture to this more general setting
Cohomological descent theory for a morphism of stacks and for equivariant derived categories
In the paper we answer the following question: for a morphism of varieties
(or, more generally, stacks), when the derived category of the base can be
recovered from the derived category of the covering variety by means of descent
theory? As a corollary, we show that for an action of a reductive group on a
scheme, the derived category of equivariant sheaves is equivalent to the
category of objects, equipped with an action of the group, in the ordinary
derived category.Comment: 28 page
Chern character, loop spaces and derived algebraic geometry.
International audienceIn this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere
Majority Dynamics and Aggregation of Information in Social Networks
Consider n individuals who, by popular vote, choose among q >= 2
alternatives, one of which is "better" than the others. Assume that each
individual votes independently at random, and that the probability of voting
for the better alternative is larger than the probability of voting for any
other. It follows from the law of large numbers that a plurality vote among the
n individuals would result in the correct outcome, with probability approaching
one exponentially quickly as n tends to infinity. Our interest in this paper is
in a variant of the process above where, after forming their initial opinions,
the voters update their decisions based on some interaction with their
neighbors in a social network. Our main example is "majority dynamics", in
which each voter adopts the most popular opinion among its friends. The
interaction repeats for some number of rounds and is then followed by a
population-wide plurality vote.
The question we tackle is that of "efficient aggregation of information": in
which cases is the better alternative chosen with probability approaching one
as n tends to infinity? Conversely, for which sequences of growing graphs does
aggregation fail, so that the wrong alternative gets chosen with probability
bounded away from zero? We construct a family of examples in which interaction
prevents efficient aggregation of information, and give a condition on the
social network which ensures that aggregation occurs. For the case of majority
dynamics we also investigate the question of unanimity in the limit. In
particular, if the voters' social network is an expander graph, we show that if
the initial population is sufficiently biased towards a particular alternative
then that alternative will eventually become the unanimous preference of the
entire population.Comment: 22 page
The Universality of Einstein Equations
It is shown that for a wide class of analytic Lagrangians which depend only
on the scalar curvature of a metric and a connection, the application of the
so--called ``Palatini formalism'', i.e., treating the metric and the connection
as independent variables, leads to ``universal'' equations. If the dimension
of space--time is greater than two these universal equations are Einstein
equations for a generic Lagrangian and are suitably replaced by other universal
equations at bifurcation points. We show that bifurcations take place in
particular for conformally invariant Lagrangians and prove
that their solutions are conformally equivalent to solutions of Einstein
equations. For 2--dimensional space--time we find instead that the universal
equation is always the equation of constant scalar curvature; the connection in
this case is a Weyl connection, containing the Levi--Civita connection of the
metric and an additional vectorfield ensuing from conformal invariance. As an
example, we investigate in detail some polynomial Lagrangians and discuss their
bifurcations.Comment: 15 pages, LaTeX, (Extended Version), TO-JLL-P1/9
Triangle-generation in topological D-brane categories
Tachyon condensation in topological Landau-Ginzburg models can generally be
studied using methods of commutative algebra and properties of triangulated
categories. The efficiency of this approach is demonstrated by explicitly
proving that every D-brane system in all minimal models of type ADE can be
generated from only one or two fundamental branes.Comment: 34 page
Superbrane Actions and Geometrical Approach
We review a generic structure of conventional (Nambu-Goto and
Dirac-Born-Infeld-like) worldvolume actions for the superbranes and show how it
is connected through a generalized action construction with a doubly
supersymmetric geometrical approach to the description of super-p-brane
dynamics as embedding world supersurfaces into target superspaces.Comment: Based on talks given by the authors at the Volkov Memorial Seminar
"Supersymmetry and Quantum field Theory" (Kharkov, January 5-7, 1997), LaTeX
file, 11 pages Misprints corrected, references adde
Mechanisms of Cognitive Impairment in Cerebral Small Vessel Disease: Multimodal MRI Results from the St George's Cognition and Neuroimaging in Stroke (SCANS) Study.
Cerebral small vessel disease (SVD) is a common cause of vascular cognitive impairment. A number of disease features can be assessed on MRI including lacunar infarcts, T2 lesion volume, brain atrophy, and cerebral microbleeds. In addition, diffusion tensor imaging (DTI) is sensitive to disruption of white matter ultrastructure, and recently it has been suggested that additional information on the pattern of damage may be obtained from axial diffusivity, a proposed marker of axonal damage, and radial diffusivity, an indicator of demyelination. We determined the contribution of these whole brain MRI markers to cognitive impairment in SVD. Consecutive patients with lacunar stroke and confluent leukoaraiosis were recruited into the ongoing SCANS study of cognitive impairment in SVD (n = 115), and underwent neuropsychological assessment and multimodal MRI. SVD subjects displayed poor performance on tests of executive function and processing speed. In the SVD group brain volume was lower, white matter hyperintensity volume higher and all diffusion characteristics differed significantly from control subjects (n = 50). On multi-predictor analysis independent predictors of executive function in SVD were lacunar infarct count and diffusivity of normal appearing white matter on DTI. Independent predictors of processing speed were lacunar infarct count and brain atrophy. Radial diffusivity was a stronger DTI predictor than axial diffusivity, suggesting ischaemic demyelination, seen neuropathologically in SVD, may be an important predictor of cognitive impairment in SVD. Our study provides information on the mechanism of cognitive impairment in SVD
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