For an equivariant commutative ring spectrum $R$, $pi_0 R$ has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yie
lds the additional structure of a Tambara functor. If $R$ is an $N_infty$ ring spectrum in the category of genuine $G$-spectra, then all possible additive transfers are present and $pi_0 R$ has the structure of an incomplete Tambara functor. However, if $R$ is an $N_infty$ ring spectrum in a category of incomplete $G$-spectra, the situation is more subtle. In this paper, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.
Comparing and aligning large datasets is a pervasive problem occurring across many different knowledge domains. We introduce and study MREC, a recursive decomposition algorithm for computing matchings between data sets. The basic idea is to partition
the data, match the partitions, and then recursively match the points within each pair of identified partitions. The matching itself is done using black box matching procedures that are too expensive to run on the entire data set. Using an absolute measure of the quality of a matching, the framework supports optimization over parameters including partitioning procedures and matching algorithms. By design, MREC can be applied to extremely large data sets. We analyze the procedure to describe when we can expect it to work well and demonstrate its flexibility and power by applying it to a number of alignment problems arising in the analysis of single cell molecular data.
In this paper, we construct incomple
The development of single-cell technologies provides the opportunity to identify new cellular states and reconstruct novel cell-to-cell relationships. Applications range from understanding the transcriptional and epigenetic processes involved in meta
zoan development to characterizing distinct cells types in heterogeneous populations like cancers or immune cells. However, analysis of the data is impeded by its unknown intrinsic biological and technical variability together with its sparseness; these factors complicate the identification of true biological signals amidst artifact and noise. Here we show that, across technologies, roughly 95% of the eigenvalues derived from each single-cell data set can be described by universal distributions predicted by Random Matrix Theory. Interestingly, 5% of the spectrum shows deviations from these distributions and present a phenomenon known as eigenvector localization, where information tightly concentrates in groups of cells. Some of the localized eigenvectors reflect underlying biological signal, and some are simply a consequence of the sparsity of single cell data; roughly 3% is artifactual. Based on the universal distributions and a technique for detecting sparsity induced localization, we present a strategy to identify the residual 2% of directions that encode biological information and thereby denoise single-cell data. We demonstrate the effectiveness of this approach by comparing with standard single-cell data analysis techniques in a variety of examples with marked cell populations.
We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(-)$, and it describes the $E_2$ term of the Kunneth spectral sequence for relative $THH$.
Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholts construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.
We study the problem of distinguishing between two distributions on a metric space; i.e., given metric measure spaces $({mathbb X}, d, mu_1)$ and $({mathbb X}, d, mu_2)$, we are interested in the problem of determining from finite data whether or not
$mu_1$ is $mu_2$. The key is to use pairwise distances between observations and, employing a reconstruction theorem of Gromov, we can perform such a test using a two sample Kolmogorov--Smirnov test. A real analysis using phylogenetic trees and flu data is presented.
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 m
otivated 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.
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.
This note compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the Bokstedt smash product and the Hill-Hopkins-Ravenel norm.
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 in
finity 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.
