Approximation of subcategories by abelian subcategories

Let $\mathcal{C}$ be an abelian category and let $\Lambda : \mathcal{C}\rightarrow\mathcal{C}$ be an idempotent functor which is not right exact, so that the zeroth left derived functor $L_0\Lambda$ does not necessarily coincide with $\Lambda$. In this paper we show that, under mild conditions on $\Lambda$, $L_0\Lambda$ is also idempotent, and the category of $L_0\Lambda$-complete objects of $\mathcal{C}$ is the smallest exact subcategory of $\mathcal{C}$ containing the $\Lambda$-complete objects. In the main application, where $\Lambda$ is the $I$-adic completion functor on a category of modules, this gives us that the category of "$L$-complete modules," studied by Greenlees-May and Hovey-Strickland, is not an ad hoc construction but is in fact characterized by a universal property. Generalizations are also given to the case of relative derived functors, in the sense of relative homological algebra

The structure of the classifying ring of formal groups with complex multiplication

If $A$ is a commutative ring, there exists a classifying ring $L^A$ of formal groups with complex multiplication by $A$, i.e., "formal $A$-modules." In this paper, the basic properties of the functor that sends $A$ to $L^A$ are developed and studied. When $A$ is a Dedekind domain, the problem of computing $L^A$ was studied by M. Lazard, by V. Drinfeld, and by M. Hazewinkel, who showed that $L^A$ is a polynomial algebra whenever $A$ is a discrete valuation ring or a (global) number ring of class number $1$, Hazewinkel observed that $L^A$ is not necessarily polynomial for more general Dedekind domains $A$, but no computations of $L^A$ have ever appeared in any case when $L^A$ is not a polynomial algebra. In the present paper, the ring $L^A$ is computed, modulo torsion, for all Dedekind domains $A$ of characteristic zero, including many cases in which $L^A$ fails to be a polynomial algebra. Qualitative features (lifting and extensions) of the moduli theory of formal modules are then derived.Comment: Rewritten completely, from top to bottom, since the last version. with many new result

From Mathematics to Medicine: A Practical Primer on Topological Data Analysis (TDA) and the Development of Related Analytic Tools for the Functional Discovery of Latent Structure in fMRI Data

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. “Structure” within fMRI data is determined by dynamic fluctuations in spatially distributed signals over time, and TDA is well positioned to help researchers better characterize mass dynamics of the signal by rigorously capturing shape within it. To accurately motivate this idea, we a) survey an established method in TDA (“persistent homology”) to reveal and describe how complex structures can be extracted from data sets generally, and b) describe how persistent homology can be applied specifically to fMRI data. We provide explanations for some of the mathematical underpinnings of TDA (with expository figures), building ideas in the following sequence: a) fMRI researchers can and should use TDA to extract structure from their data; b) this extraction serves an important role in the endeavor of functional discovery, and c) TDA approaches can complement other established approaches toward fMRI analyses (for which we provide examples). We also provide detailed applications of TDA to fMRI data collected using established paradigms, and offer our software pipeline for readers interested in emulating our methods. This working overview is both an inter-disciplinary synthesis of ideas (to draw researchers in TDA and fMRI toward each other) and a detailed description of methods that can motivate collaborative research

Moduli and cohomology

Thesis (Ph. D.)--University of Rochester. Dept. of Mathematics, 2008.We provide an algebro-geometric framework for Hopf algebroids in which they become equivalent to at algebraic stacks, modulo many technical details, which we describe and handle. We produce isomorphisms in at cohomology between objects on either side of this equivalence, and we produce resolutions and spectral sequences for performing basic cohomological tasks in the relevant categories. We apply these results to the moduli stacks and classifying Hopf algebroids of formal A-modules, with A a number ring, obtaining some splitting results about these moduli objects. We lay the groundwork for further methods, currently in preparation, generalizing chromatic theory from stable homotopy theory for general use in algebraic geometry as well as advanced applications in computational stable homotopy