arXiv subjects

David Barnes

Publications and source records attributed to David Barnes.

At least 55 records · Page 3Linked to original sources

arXiv2015

Capturing Goodwillie's Derivative

Recent work of Biedermann and Röndigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with an action of the symmetric group on n objects. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.

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arXiv2015

Comparing the orthogonal and homotopy functor calculi

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor which sends a vector space V to F evaluated at the one-point compactification of V. In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

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arXiv2014

A monoidal algebraic model for rational SO(2)-spectra

The category of rational SO(2)-equivariant spectra admits an algebraic model. That is, there is an abelian category A(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)-equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra? The category A(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non-Noetherian ring. Hence the standard techniques for constructing a monoidal model structure cannot be applied. In this paper we construct a monoidal model structure on A(SO(2)) and show that the derived product on the homotopy category is compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K_p-local stable homotopy category. We also provide a monoidal Quillen equivalence to a simpler monoidal model category R_*-modules that has explicit generating sets. Having monoidal model structures on A(SO(2)) and R_*-modules removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)-equivariant spectra.

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arXiv2012

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E-localisation of this model category. We study the properties of this new construction and relate it to some well-known categories.

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arXiv2012

Fast Holographic Deconvolution: a new technique for precision radio interferometry

We introduce the Fast Holographic Deconvolution method for analyzing interferometric radio data. Our new method is an extension of A-projection/software-holography/forward modeling analysis techniques and shares their precision deconvolution and widefield polarimetry, while being significantly faster than current implementations that use full direction-dependent antenna gains. Using data from the MWA 32 antenna prototype, we demonstrate the effectiveness and precision of our new algorithm. Fast Holographic Deconvolution may be particularly important for upcoming 21 cm cosmology observations of the Epoch of Reionization and Dark Energy where foreground subtraction is intimately related to the precision of the data reduction.

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arXiv2012

Model Categories for Orthogonal Calculus

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.

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arXiv2012

Stable left and right Bousfield localisations

We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

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arXiv2012

Rational Equivariant Rigidity

We prove that if G is the circle group or a profinite group, then the all of the homotopical information of the category of rational G-spectra is captured by triangulated structure of the rational G-equivariant stable homotopy category. That is, for G profinite or S1, the rational G-equivariant stable homotopy category is rigid. For the case of profinite groups this rigidity comes from an intrinsic formality statement, so we carefully relate the notion of intrinsic formality of a differential graded algebra to rigidity.

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arXiv2011

Monoidality of Franke's Exotic Model

We discuss the monoidal structure on Franke's algebraic model for the $K_{(p)}$-local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers.

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arXiv2011

Local Framings

Framings provide a way to construct Quillen functors from simplicial sets to any given model category. A more structured set-up studies stable frames giving Quillen functors from spectra to stable model categories. We will investigate how this is compatible with Bousfield localisation to gain insight into the deeper structure of the stable homotopy category. We further show how these techniques relate to rigidity questions and how they can be used to study algebraic model categories.

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arXiv2011

First Spectroscopic Imaging Observations of the Sun at Low Radio Frequencies with the Murchison Widefield Array Prototype

We present the first spectroscopic images of solar radio transients from the prototype for the Murchison Widefield Array (MWA), observed on 2010 March 27. Our observations span the instantaneous frequency band 170.9-201.6 MHz. Though our observing period is characterized as a period of `low' to `medium' activity, one broadband emission feature and numerous short-lived, narrowband, non-thermal emission features are evident. Our data represent a significant advance in low radio frequency solar imaging, enabling us to follow the spatial, spectral, and temporal evolution of events simultaneously and in unprecedented detail. The rich variety of features seen here reaffirms the coronal diagnostic capability of low radio frequency emission and provides an early glimpse of the nature of radio observations that will become available as the next generation of low frequency radio interferometers come on-line over the next few years.

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arXiv2010

Rational Z_p-Equivariant Spectra

We find a simple algebraic model for rational G-equivariant spectra, where G is the p-adic integers, via a series of Quillen equivalences. This model, along with an Adams short exact sequence, will allow us to easily perform constructions and calculations.

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arXiv2008

Classifying Rational G-Spectra for Finite G

We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model.

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arXiv2008

Splitting Monoidal Stable Model Categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to the product of C localised at the object eS and C localised at the object (1-e)S. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.

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arXiv2008

Classifying Dihedral O(2)-Equivariant Spectra

The category of rational O(2)-equivariant spectra splits as a product of cyclic and dihedral parts. Using the classification of rational G-equivariant spectra for finite groups G, we classify the dihedral part of rational O(2)-equivariant spectra in terms of an algebraic model.

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arXiv2008

Rational Equivariant Spectra

We develop model categories of rational equivariant spectra whose homotopy categories are equivalent to the category of rational equivariant cohomology theories. We prove that given an orthogonal decomposition of the unit in the rational Burnside ring, the model category of rational equivariant spectra decomposes into a product of localisations. We use this result to reprove the classification of rational equivariant cohomology theories for finite groups and to study such cohomology theories for the group O(2). We then concentrate on a split piece of the O(2) case and relate it to rational SO(2) equivariant spectra.

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arXiv2005

Prospects for Redshifted 21-cm observations of quasar HII regions

The introduction of low-frequency radio arrays over the coming decade is expected to revolutionize the study of the reionization epoch. Observation of the contrast in redshifted 21cm emission between a large HII region and the surrounding neutral IGM will be the simplest and most easily interpreted signature. We find that an instrument like the planned Mileura Widefield Array Low-Frequency Demonstrator (LFD) will be able to obtain good signal to noise on HII regions around the most luminous quasars, and determine some gross geometric properties, e.g. whether the HII region is spherical or conical. A hypothetical follow-up instrument with 10 times the collecting area of the LFD (MWA-5000) will be capable of mapping the detailed geometry of HII regions, while SKA will be capable of detecting very narrow spectral features as well as the sharpness of the HII region boundary. The MWA-5000 will discover serendipitous HII regions in widefield observations. We estimate the number of HII regions which are expected to be generated by quasars. Assuming a late reionization at z~6 we find that there should be several tens of quasar HII regions larger than 4Mpc at z~6-8 per field of view. Identification of HII regions in forthcoming 21cm surveys can guide a search for bright galaxies in the middle of these regions. Most of the discovered galaxies would be the massive hosts of dormant quasars that left behind fossil HII cavities that persisted long after the quasar emission ended, owing to the long recombination time of intergalactic hydrogen. A snap-shot survey of candidate HII regions selected in redshifted 21cm image cubes may prove to be the most efficient method for finding very high redshift quasars and galaxies.

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arXiv2004

What do loose groups tell us about galaxy formation?

We present the results of a Parkes Multibeam HI survey of six loose groups of galaxies analogous to the Local Group. This survey was sensitive to HI-rich objects in these groups to below 10^7 M(sun) of HI, and was designed to search for low mass, gas-rich satellite galaxies and potential analogs to the high-velocity clouds seen around the Milky Way. This survey detected a total of 79 HI-rich objects associated with the six groups, half of which were new detections. While the survey identified a small number of dwarf galaxies, no star-free HI clouds were discovered. The HI mass function of the six groups appears to be roughly flat as is that of the Local Group. The cumulative velocity distribution function (CVDF) of the HI-rich halos in the six groups is identical to that of the Local Group. Both of these facts imply that these groups are true analogs to the Local Group and that the Local Group is not unique in its lack of low-mass dwarf galaxies as compared to the predictions of cold dark matter models of galaxy formation. This survey also constrains the distance to and HI masses of the compact high-velocity clouds (CHVCs) around the Milky Way. The lack of CHVC analog detections implies that they are distributed within <160 kpc of the Milky Way and have average HI masses of <4x10^5 M(sun). The spatial distribution of CHVCs is consistent with the predictions of simulations for dark matter halos. Furthermore the CVDF of Local Group galaxies plus CHVCs matches the predicted CVDF of cold dark matter simulations of galaxy formation. This provides circumstantial evidence that CHVCs may be associated with low-mass dark matter halos.

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