arXiv subjects

Andrew Baker

Publications and source records attributed to Andrew Baker.

At least 37 records · Page 2Linked to original sources

arXiv2017

$\mathcal{E}_\infty$ ring spectra and elements of Hopf invariant $1$

The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the $\mathcal{E}_\infty$ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces $B\mathrm{SO},\,B\mathrm{Spin},\,B\mathrm{String}$. We show that the homology of these Thom spectra are all extended comodule algebras of the form $\mathcal{A}_*\square_{\mathcal{A}(r)_*}P_*$ over the dual Steenrod algebra $\mathcal{A}_*$ with $\mathcal{A}_*\square_{\mathcal{A}(r)_*}\mathbb{F}_2$ as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra $H\mathbb{Z}$, $k\mathrm{O}$ or $\mathrm{tmf}$, however apart from the first case, we have no concrete results on this.

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arXiv2017

Calculating with topological André-Quillen theory, I: Homotopical properties of universal derivations and free commutative $S$-algebras

We adopt the viewpoint that topological Andé-Quillen theory for commutative $S$-algebras should provide usable (co)homology theories for doing calculations in the sense traditional within Algebraic Topology. Our main emphasis is on homotopical properties of universal derivations, especially their behaviour in multiplicative homology theories. There are algebraic derivation properties, but also deeper properties arising from the homotopical structure of the free algebra functor $\mathbb{P}_R$ and its relationship with extended powers of spectra. In the connective case in ordinary $\bmod{\,p}$ homology, this leads to useful formulae involving Dyer-Lashof operations in the homology of commutative $S$-algebras. Although many of our results could no doubt be obtained using stabilisation, our approach seems more direct. We also discuss a reduced free algebra functor $\tilde{\mathbb{P}}_R$.

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arXiv2015

Frobenius Green functors

These notes provide an informal introduction to a type of Mackey functor that arises naturally in algebraic topology in connection with Morava $K$-theory of classifying spaces of finite groups. The main aim is to identify key algebraic aspects of the Green functor structure obtained by applying a Morava $K$-theory to such classifying spaces.

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arXiv2014

Power operations and coactions in highly commutative homology theories

Power operations in the homology of infinite loop spaces, and $H_\infty$ or $E_\infty$ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p, the power operations of Kudo, Araki, Dyer and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required.On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable formulae for the interaction between power operations and homology coactions in a wide class of examples; our approach makes crucial use of modern frameworks for spectra with well behaved smash products. In the case of mod $p$ homology, our formulae extend those of Bisson and Joyal to odd primes. We also show how to exploit our results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof action on the dual Steenrod algebra.

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arXiv2014

$MSp$ localized away from $2$ and odd formal group laws

We investigate the relationship between complex and symplectic cobordism localized away from the prime~$2$ and show that these theories are related much as a real Lie group is related to its complexification. This suggests that ideas from the theory of symmetric spaces might be used to illuminate these subjects. In particular, we give an explicit equivalence of ring spectra \[ MSp[1/2]\wedge Sp/U_+\simeq MU[1/2] \] and deduce that $MU[1/2]$ is a wedge of copies of $MSp[1/2]$. We discuss the implications for the structure of the stable operation algebra $MSp[1/2]^*MSp[1/2]$ and the dual cooperation algebra $MSp[1/2]_*MSp[1/2]$. Finally we describe some related Witt vector algebra and apply our results to the study of formal involutions on the category of formal group laws over a $\mathbb{Z}[1/2]$-algebra.

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arXiv2013

$BP$: Close encounters of the $E_\infty$ kind

Inspired by Stewart Priddy's cellular model for the $p$-local Brown-Peterson spectrum $BP$, we give a construction of a $p$-local $E_\infty$ ring spectrum $R$ which is a close approximation to $BP$. Indeed we can show that if $BP$ admits an $E_\infty$ structure then these are weakly equivalent as $E_\infty$ ring spectra. Our inductive cellular construction makes use of power operations on homotopy groups to define homotopy classes which are then killed by attaching $E_\infty$ cells.

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arXiv2013

On the homology of regular quotients

We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. For $s>1$, we easily deduce that the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction map $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras.

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arXiv2013

Some properties of the Thom spectrum over loop suspension of complex projective space

This note provides a reference for some properties of the Thom spectrum $Mξ$ over $ΩΣ\CPi$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $Mξ$-cohomology of $\CPi$ and show that $Mξ^*(\CPi)$ injects into power series over the algebra of non-symmetric functions. We show that $Mξ$ gives rise to a commutative formal group law over the non-commutative ring $π_*Mξ$. We also discuss how $Mξ$ and some real and quaternionic analogues behave with respect to spectra that are related to these Thom spectra by splittings and by maps.

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arXiv2012

On the cohomology of loop spaces for some Thom spaces

In this paper we identify conditions under which the cohomology $H^*(ΩMξ;\k)$ for the loop space $ΩMξ$ of the Thom space $Mξ$ of a spherical fibration $ξ\downarrow B$ can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class $e(ξ)\in H^n(B;\k)$ vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum $Σ^\inftyΩMξ$ has a local splitting replacing the James splitting of $ΣΩMξ$ when $Mξ$ is a suspension.

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arXiv2012

Brauer groups for commutative $S$-algebras

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.

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arXiv2011

Some properties of Lubin-Tate cohomology for classifying spaces of finite groups

We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory $K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a $G$-Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it ramifies. As a consequence, it follows that the $E_n$-theory Eilenberg-Moore spectral sequence for $G$ and $BG$ does not always converge to its expected target.

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arXiv2010

The Herschel-SPIRE Legacy Survey (HSLS): the scientific goals of a shallow and wide submillimeter imaging survey with SPIRE

A large sub-mm survey with Herschel will enable many exciting science opportunities, especially in an era of wide-field optical and radio surveys and high resolution cosmic microwave background experiments. The Herschel-SPIRE Legacy Survey (HSLS), will lead to imaging data over 4000 sq. degrees at 250, 350, and 500 micron. Major Goals of HSLS are: (a) produce a catalog of 2.5 to 3 million galaxies down to 26, 27 and 33 mJy (50% completeness; 5 sigma confusion noise) at 250, 350 and 500 micron, respectively, in the southern hemisphere (3000 sq. degrees) and in an equatorial strip (1000 sq. degrees), areas which have extensive multi-wavelength coverage and are easily accessible from ALMA. Two thirds of the of the sources are expected to be at z > 1, one third at z > 2 and about a 1000 at z > 5. (b) Remove point source confusion in secondary anisotropy studies with Planck and ground-based CMB data. (c) Find at least 1200 strongly lensed bright sub-mm sources leading to a 2% test of general relativity. (d) Identify 200 proto-cluster regions at z of 2 and perform an unbiased study of the environmental dependence of star formation. (e) Perform an unbiased survey for star formation and dust at high Galactic latitude and make a census of debris disks and dust around AGB stars and white dwarfs.

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arXiv2010

Intense Star Formation within Resolved Compact Regions in a Galaxy at z=2.3

Massive galaxies in the early Universe have been shown to be forming stars at surprisingly high rates. Prominent examples are dust-obscured galaxies which are luminous when observed at sub-millimeter (sub-mm) wavelengths and which may be forming stars at rates upto 1,000Mo/yr. These intense bursts of star formation are believed to be driven by mergers between gas rich galaxies. However, probing the properties of individual star-forming regions within these galaxies is beyond the spatial resolution and sensitivity of even the largest telescopes at present. Here, we report observations of the sub-mm galaxy SMMJ2135-0102 at redshift z=2.3259 which has been gravitationally magnified by a factor of 32 by a massive foreground galaxy cluster lens. This cosmic magnification, when combined with high-resolution sub-mm imaging, resolves the star-forming regions at a linear scale of just ~100 parsecs. We find that the luminosity densities of these star-forming regions are comparable to the dense cores of giant molecular clouds in the local Universe, but they are ~100x larger and 10^7 times more luminous. Although vigorously star-forming, the underlying physics of the star formation processes at z~2 appears to be similar to that seen in local galaxies even though the energetics are unlike anything found in the present-day Universe.

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arXiv2009

$L$-complete Hopf algebroids and their comodules

We investigate Hopf algebroids in the category of $L$-complete modules over a commutative Noetherian regular complete local ring. The main examples are provided by the Hopf algebroids associated to Lubin-Tate spectra in the K(n)-local stable homotopy category and we show that these have Landweber filtrations for all finitely generated discrete modules. Along the way we investigate the canonical Hopf algebras associated to Hopf algebroids over fields and introduce a notion of unipotent Hopf algebroid generalising that for Hopf algebras. In two appendices we continue the discussion of the connections with twisted group rings, and expand on a result of Hovey on the non-exactness of coproducts of L-complete modules.

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arXiv2009

Increasing the Number of Underrepresented Minorities in Astronomy: Executive Summary

Promoting racial and ethnic diversity is critically important to the future success and growth of the field of astronomy. The raw ability, drive and interest required to excel in the field is distributed without regard to race, gender, or socioeconomic background. By not actively promoting diversity in our field we risk losing talented people to other professions (or losing them entirely), which means that there will be astronomical discoveries that simply won't get made. There is demonstrated evidence that STEM fields benefit from diverse perspectives on problems that require more complex thought processes. This is especially relevant to a field like astronomy where more and more work is being done collaboratively. The lack of notable growth in African American, Hispanic, and Native American representation in astronomy indicates that the 'pipeline' for these individuals is systemically leaky at critical junctures. Substantially more effort must be directed toward improving the educational and career development of minorities to insure that these potential colleagues are supported through the process. However, simply recognizing that the pipeline is faulty is woefully inadequate. There must be very specific, targeted solutions to help improve the situation. With this in mind, we offer two position papers addressing specific areas of improvement that we identify as (a) essential for any foreseeable progress in the field, and (b) attainable in the 2010-2020 decade. These position papers focus primarily on African Americans, Hispanics, and Native Americans. Although we do not directly address issues of Asian Americans, Pacific Islanders, and other groups, many of the recommendations made here can be adapted to address issues faced by these groups as well.

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arXiv2009

Increasing the Number of Underrepresented Minorities in Astronomy at the Undergraduate, Graduate, and Postdoctoral Levels (Paper I)

If the ethnic makeup of the astronomy profession is to achieve parity with the general population within one generation (~30 years), the number of underrepresented minorities earning graduate degrees in astronomy and astrophysics must increase in the coming decade by a factor of 5 to 10. To accomplish this, the profession must develop and invest in mechanisms to more effectively move individuals across critical educational junctures to the PhD and beyond. Early and continuous research engagement starting in the undergraduate years is critical to this vision, in which the federally funded research internship programs (e.g. NSF REU, NASA GSRP) and national centers/observatories play a vital role. Regionally based partnerships with minority-serving institutions (MSIs) are crucial for tapping extant pools of minority talent, as are post-baccalaurate and/or masters degree "bridging" programs that provide critical stepping stones to the PhD. Because of the strong undergraduate physics, engineering, and computer science backgrounds of many students from MSIs, we suggest that instrument development and large scale computing/data-mining are particularly promising avenues for engagement in the coming decade.

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arXiv2009

Increasing the Number of Underrepresented Minorities in Astronomy Through K-12 Education and Public Outreach (Paper II)

In order to attract, recruit and retain underrepresented minority students to pursue Astronomy and related fields, we must ensure that there continues to be a well qualified pool of graduate and undergraduate students from which to recruit. This required pool of people are today's elementary, middle and high school students. The Astronomy community must be proactive in demonstrating the importance of pursing scientific study and careers to these students and their parents. Only by actively engaging these communities can U.S Astronomy hope to increase the numbers of minority PhDs and continue to be a leader in Astronomical discovery and knowledge.

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arXiv2008

Galois extensions of Lubin-Tate spectra

Let E_n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E^{nr}_n whose coefficients are built from the coefficients of E_n and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E^{nr}_n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E^{nr}_n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.

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