Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [Seg04] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW02]. Contents 1. Introduction and results 2. The cobordism category and its sheaves 2.1. The cobordism category 2.2. Recollection from [MW02] on sheaves 2.3. A sheaf model for the cobordism category 2.4. Cocycle sheaves 3. The Thom spectra and their sheaves 3.1. The spectrum MT (d) and its infinite loop space 3.2. Using Phillips' submersion theorem 4. Proof of the main theorem 5. Tangential structures 6. Connectedness issues 6.1. Discussion 6.2. Surgery 6.3. Connectivity 6.4. Parametrized surgery 7. Harer type stability and C 2 References

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In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction Z × BΓ + ∞ has an infinite loop space structure. This result and the tools developed in [BM] to analyse transfer maps, are used here to show the following splitting theorem. Letwhere Ω ∞ E i denotes the infinite loop space associated to the spectrum E i . The homology of Ω ∞ E i is known, and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group. This splitting also detects all the Miller-Morita-Mumford classes. Our results suggest a homotopy theoretic refinement of the Mumford conjecture. The above p-adic splitting uses a certain infinite loop mapthat induces an isomorphims in rational cohomology precisely if the Mumford conjecture is true. We suggest that α ∞ might be a homotopy equivalence.

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Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU , BU , BSO, BO, BSp, BGL ∞ (R) + and Q 0 (S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital E ∞ -ring spectra. We introduce the notion of q-nilpotent K-theory of a CW-complex X for any q ≥ 2, which extends the notion of commutative K-theory defined by Adem-Gómez, and show that it is represented by Z × B(q, U ), were B(q, U ) is the q-th term of the aforementioned filtration of BU .For the proof we introduce an alternative way of associating an infinite loop space to a commutative I-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative I-rig and show that they give rise to non-unital E ∞ -ring spectra.

Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data—often with outliers, artifacts, and mislabeled points—such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

The operad studied in conformal field theory and introduced ten years ago by G. Segal [S] is built out of moduli spaces of Riemann surfaces. We show here that this operad which at first sight is a double loop space operad is indeed an infinite loop space operad. This leads to a new proof of the fact that the classifying space of the stable mapping class group Z × BΓ + ∞ , is an infinite loop space after plus construction [T2]. This new approach has various advantages. In particular, the infinite loop space structure is more explicid. (1991): 55P47, 32G15, 81T40, 55R35 Mathematics Subject Classification