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Andrew J. Blumberg

Publications and source records attributed to Andrew J. Blumberg.

At least 55 records · Page 3Linked to original sources
arXiv2016

Genomic data analysis in tree spaces

Recently, an elegant approach in phylogenetics was introduced by Billera-Holmes-Vogtmann that allows a systematic comparison of different evolutionary histories using the metric geometry of tree spaces. In many problem settings one encounters heavily populated phylogenetic trees, where the large number of leaves encumbers visualization and analysis in the relevant evolutionary moduli spaces. To address this issue, we introduce tree dimensionality reduction, a structured approach to reducing large phylogenetic trees to a distribution of smaller trees. We prove a stability theorem ensuring that small perturbations of the large trees are taken to small perturbations of the resulting distributions. We then present a series of four biologically motivated applications to the analysis of genomic data, spanning cancer and infectious disease. The first quantifies how chemotherapy can disrupt the evolution of common leukemias. The second examines a link between geometric information and the histologic grade in relapsed gliomas, where longer relapse branches were specific to high grade glioma. The third concerns genetic stability of xenograft models of cancer, where heterogeneity at the single cell level increased with later mouse passages. The last studies genetic diversity in seasonal influenza A virus. We apply tree dimensionality reduction to 24 years of longitudinally collected H3N2 hemagglutinin sequences, generating distributions of smaller trees spanning between three and five seasons. A negative correlation is observed between the influenza vaccine effectiveness during a season and the variance of the distributions produced using preceding seasons' sequence data. We also show how tree distributions relate to antigenic clusters and choice of influenza vaccine. Our formalism exposes links between viral genomic data and clinical observables such as vaccine selection and efficacy.

q-bio.GN↗
arXiv2016

Defending against malicious peripherals with Cinch

Malicious peripherals designed to attack their host computers are a growing problem. Inexpensive and powerful peripherals that attach to plug-and-play buses have made such attacks easy to mount. Making matters worse, commodity operating systems lack coherent defenses, and users are often unaware of the scope of the problem. We present Cinch, a pragmatic response to this threat. Cinch uses virtualization to attach peripheral devices to a logically separate, untrusted machine, and includes an interposition layer between the untrusted machine and the protected one. This layer regulates interaction with devices according to user-configured policies. Cinch integrates with existing OSes, enforces policies that thwart real-world attacks, and has low overhead.

cs.OS↗
arXiv2016

Incomplete Tambara functors

For a "genuine" equivariant commutative ring spectrum $R$, $π_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by $N_\infty$ operads. In a precise sense, these interpolate between "naive" and "genuine" equivariant ring structures. In this paper, we describe the algebraic analogue of $N_\infty$ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as $π_0$ of $N_\infty$ algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of $N_\infty$ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.

math.AT↗
arXiv2015

Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable infinity category parametrized over objects of an infinity topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures. Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

math.AT↗
arXiv2015

Uniqueness of the multiplicative cyclotomic trace

Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible. We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spectra give rise to E_{n-1} ring algebraic K-theory spectra. Along the way, we develop a "multiplicative Morita theory", establishing a symmetric monoidal equivalence between the infinity category of small idempotent-complete stable infinity categories and the Morita localization of the infinity category of spectral categories.

math.KT↗
arXiv2015

Operadic multiplications in equivariant spectra, norms, and transfers

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N-infinity operads, equivariant generalizations of E-infinity operads. Algebras in equivariant spectra over an N-infinity operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N-infinity operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces. To explain the relationship between norms, transfers, and N-infinity operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general universe $U$ need not determine the same algebras. Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

math.AT↗
arXiv2014

Homotopical resolutions associated to deformable adjunctions

Given an adjunction connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calculus in general model categories.

math.AT↗
arXiv2014

Moduli Spaces of Phylogenetic Trees Describing Tumor Evolutionary Patterns

Cancers follow a clonal Darwinian evolution, with fitter subclones replacing more quiescent cells, ultimately giving rise to macroscopic disease. High-throughput genomics provides the opportunity to investigate these processes and determine specific genetic alterations driving disease progression. Genomic sampling of a patient's cancer provides a molecular history, represented by a phylogenetic tree. Cohorts of patients represent a forest of related phylogenetic structures. To extract clinically relevant information, one must represent and statistically compare these collections of trees. We propose a framework based on an application of the work by Billera, Holmes and Vogtmann on phylogenetic tree spaces to the case of unrooted trees of intra-individual cancer tissue samples. We observe that these tree spaces are globally nonpositively curved, allowing for statistical inference on populations of patient histories. A projective tree space is introduced, permitting visualizations of aggregate evolutionary behavior. Published data from three types of human malignancies are explored within our framework.

q-bio.QM↗
arXiv2014

An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology

We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum $R$, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space $X$ to the infinity category of free rank one $R$-modules, which we show is a model for $BGL_1 R$; we show that $BGL_1 R$ classifies homotopy sheaves of rank one $R$-modules, which we call $R$-line bundles. We use our $R$-module Thom spectrum to define the twisted $R$-homology and cohomology of an $R$-line bundle over a space $X$, classified by a map from $X$ to $BGL_1 R$, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. An earlier version of this paper was part of arXiv:0810.4535.

math.AT↗
arXiv2014

Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom spectra and orientations for associative ring spectra. Our work is based on a new model of the Thom spectrum as a derived smash product. An earlier version of this paper was part of arXiv:0810.4535.

math.AT↗
arXiv2014

Localization for THH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories

We prove a conjecture of Hesselholt and Ausoni-Rognes, establishing localization cofiber sequences of spectra for THH(ku) and TC(ku). These sequences support Hesselholt's view of the map l to ku as a "tamely ramified" extension of ring spectra, and validate the hypotheses necessary for Ausoni's simplified computation of V(1)_* K(KU). In order to make sense of the relative term THH(ku|KU) in the cofiber sequence and prove these results, we develop a theory of THH and TC of Waldhausen categories and prove the analogues of Waldhausen's theorems for K-theory. We resolve the longstanding confusion about localization sequences in THH and TC, and establish a specialized devissage theorem.

math.KT↗
arXiv2013

Quantitative Homotopy Theory in Topological Data Analysis

This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in terms of the combinatorial notion of contiguity. We generalize the Simplicial Approximation Theorem to show that the contiguity complex approximates the homotopy type of the mapping space as we subdivide the domain. We describe algorithms for approximating the rate of growth of the components of the contiguity complex under subdivision of the domain; this procedure allows us to computationally distinguish spaces with isomorphic homology but different homotopy types.

cs.CG↗
arXiv2013

K-theory of endomorphisms via noncommutative motives

In this article we study the K-theory of endomorphisms using noncommutative motives. We start by extending the K-theory of endomorphisms functor from ordinary rings to (stable) infinity categories. We then prove that this extended functor KEnd(-) not only descends to the category of noncommutative motives but moreover becomes co-represented by the noncommutative motive associated to the tensor algebra S[t] of the sphere spectrum S. Using this co-representability result, we then classify all the natural transformations of KEnd(-) in terms of an integer plus a fraction between polynomials with constant term 1; this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative co-algebra structure of S[t], we explain how the (rational) Witt vectors can also be recovered from the symmetric monoidal category of noncommutative motives. Along the way we show that the K_0-theory of endomorphisms of a connective ring spectrum R equals the K_0-theory of endomorphisms of the underlying ordinary ring π_0(R).

math.AT↗
arXiv2013

A universal characterization of higher algebraic K-theory

In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that non-connective algebraic K-theory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the "Thomason-Trobaugh-Neeman" localization theorem. To prove these results, we construct and study two stable infinity categories of "noncommutative motives"; one associated to additivity and another to localization. In these stable infinity categories, Waldhausen's S. construction corresponds to the suspension functor and connective and non-connective algebraic K-theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K-theory of every scheme, stack, and ring spectrum can be recovered from these categories of noncommutative motives. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable infinity categories. We also explain in detail the comparison between the infinity categorical version of Waldhausen K-theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.

math.KT↗
arXiv2012

Localization theorems in topological Hochschild homology and topological cyclic homology

We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory. We also deduce versions of Thomason's blow-up formula and the projective bundle formula for THH and TC.

math.KT↗