Hyperdescent and \'etale K-theory

We study the \'etale sheafification of algebraic K-theory, called \'etale K-theory. Our main results show that \'etale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we show that \'etale K-theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the distinction, which we investigate in detail, between sheaves and hypersheaves of spectra on \'etale sites.Comment: 89 pages, v3: various corrections and edit

Arithmetic duality in algebraic K-theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 37-38).Let X be a regular arithmetic curve or point (meaning a regular separated scheme of finite type over Z which is connected and of Krull dimension </= 1). We define a compactly-supported variant Kc(X) of the algebraic K-theory spectrum K(X), and establish the basic functoriality of Kc. Briefly, K, behaves as if it were dual to K. Then we give this duality some grounding: for every prime t invertible on X, we define a natural l-adic pairing between Kc(X) and K(X). This pairing is of an explicit homotopy-theoretic nature, and reflects a simple relation between spheres, tori, and real vector spaces. Surprisingly, it has the following two properties: first (a consequence of work of Rezk), when one tries to compute it the e-adic logarithm inevitably appears; and second, it can be used to give a new description of the global Artin map, one which makes the Artin reciprocity law manifest.by Dustin Clausen.Ph.D

K-theory and topological cyclic homology of henselian pairs

Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic rings. In addition, we show that $K$-theory with finite coefficients satisfies continuity for complete noetherian rings which are $F$-finite modulo $p$. Our main new ingredient is a basic finiteness property of $\mathrm{TC}$ with finite coefficients.Comment: 59 pages, revised and final versio

Descent and vanishing in chromatic algebraic KK-theory via group actions

We prove some $K$-theoretic descent results for finite group actions on stable $\infty$-categories, including the $p$-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if $R$ is an $\mathbb{E}_\infty$-ring spectrum with $L_{T(n)}R=0$, then $L_{T(n+1)}K(R)=0$. Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.Comment: 47 pages, comments welcom

The RESET project: constructing a European tephra lattice for refined synchronisation of environmental and archaeological events during the last c. 100 ka

This paper introduces the aims and scope of the RESET project (. RESponse of humans to abrupt Environmental Transitions), a programme of research funded by the Natural Environment Research Council (UK) between 2008 and 2013; it also provides the context and rationale for papers included in a special volume of Quaternary Science Reviews that report some of the project's findings. RESET examined the chronological and correlation methods employed to establish causal links between the timing of abrupt environmental transitions (AETs) on the one hand, and of human dispersal and development on the other, with a focus on the Middle and Upper Palaeolithic periods. The period of interest is the Last Glacial cycle and the early Holocene (c. 100-8 ka), during which time a number of pronounced AETs occurred. A long-running topic of debate is the degree to which human history in Europe and the Mediterranean region during the Palaeolithic was shaped by these AETs, but this has proved difficult to assess because of poor dating control. In an attempt to move the science forward, RESET examined the potential that tephra isochrons, and in particular non-visible ash layers (cryptotephras), might offer for synchronising palaeo-records with a greater degree of finesse. New tephrostratigraphical data generated by the project augment previously-established tephra frameworks for the region, and underpin a more evolved tephra 'lattice' that links palaeo-records between Greenland, the European mainland, sub-marine sequences in the Mediterranean and North Africa. The paper also outlines the significance of other contributions to this special volume: collectively, these illustrate how the lattice was constructed, how it links with cognate tephra research in Europe and elsewhere, and how the evidence of tephra isochrons is beginning to challenge long-held views about the impacts of environmental change on humans during the Palaeolithic. © 2015 Elsevier Ltd.RESET was funded through Consortium Grants awarded by the Natural Environment Research Council, UK, to a collaborating team drawn from four institutions: Royal Holloway University of London (grant reference NE/E015905/1), the Natural History Museum, London (NE/E015913/1), Oxford University (NE/E015670/1) and the University of Southampton, including the National Oceanography Centre (NE/01531X/1). The authors also wish to record their deep gratitude to four members of the scientific community who formed a consultative advisory panel during the lifetime of the RESET project: Professor Barbara Wohlfarth (Stockholm University), Professor Jørgen Peder Steffensen (Niels Bohr Institute, Copenhagen), Dr. Martin Street (Romisch-Germanisches Zentralmuseum, Neuwied) and Professor Clive Oppenheimer (Cambridge University). They provided excellent advice at key stages of the work, which we greatly valued. We also thank Jenny Kynaston (Geography Department, Royal Holloway) for construction of several of the figures in this paper, and Debbie Barrett (Elsevier) and Colin Murray Wallace (Editor-in-Chief, QSR) for their considerable assistance in the production of this special volume.Peer Reviewe

Robust estimation of bacterial cell count from optical density

Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals &lt;1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data