arXiv subjects

Kathryn Hess

Publications and source records attributed to Kathryn Hess.

At least 19 recordsLinked to original sources

arXiv2025

Interval-sphere model structures

The bedrock of persistence theory over a single parameter is decomposition of persistence modules into intervals. In [HLM24], the authors leveraged interval decomposition to produce a cell decomposition of the minimal model of a simply connected copersistent space. The key tool was a technique called interval surgery, which involves the gluing of intervals to a persistent CDGA by means of algebraic cell attachments. In this article, we define a compact, combinatorial model categorical structure that contextualizes interval surgery as a genuine model-categorical cell attachment. We show that our new model structure is neither the injective nor the projective one and that cofibrancy is closely linked to the notion of tameness in persistence theory and algebraic notions of compactness.

math.AT↗

arXiv2025

Cell decompositions of persistent minimal models

In this article we generalize the main structure theorems of rational homotopy theory to the persistent setting. Our main motivation is the computation of an explicit finite, cellular presentation of the persistent minimal model that completely characterizes the rational homotopy type of copersistent simply-connected spaces. We achieve this via an explicit construction of the minimal model of a tame persistent CDGA as an iterated sequence of cell attachments. As an application of our results, we construct an explicit decomposition of the rational Postnikov tower of simply-connected copersistent spaces in terms of a tower of persistent Eilenberg-Maclane intervals

math.AT↗

arXiv2025

Shadows are Bicategorical Traces

Hochschild homology has proved to be an important invariant in algebra and homotopy theory, in particular due to its relevance in algebraic $K$-theory and fixed point theory, leading to the development of numerous variants of the original construction. Ponto introduced a bicategorical axiomatization of Hochschild homology-type invariants, called a shadow, which captures the essential common properties of all known variants of Hochschild homology, such as Morita invariance. In this paper we clarify the relationship between shadows and Hochschild homology. After extending the notion of Hochschild homology to bicategories in a natural manner, we prove the existence of a universal shadow on any bicategory $\mathscr{B}$, taking values in the Hochschild homology of $\mathscr{B}$, through which all other shadows on $\mathscr{B}$ factor. Shadows are thus co-represented by a bicategorical version of Hochschild homology. Using the universal shadow on the free adjunction bicategory, we can then establish a universal Morita invariance theorem, of which all known cases are immediate corollaries. Building on this understanding of shadows on bicategories, we propose an $\infty$-categorical generalization of shadows as functors out of Hochschild homology of an $(\infty,2)$-category in the sense of Berman. As a first step towards constructing relevant examples of $\infty$-categorical shadows, we define the Hochschild homology of enriched $\infty$-categorical bimodules and prove that they assemble into a shadow. As part of this work we compute the Hochschild homology of several important $2$-categories (such as the free adjunction), which can be of independent interest.

math.CT↗

arXiv2024

Constructing monads from cubical diagrams and homotopy colimits

This paper is the first step in a general program for defining cocalculus towers of functors via sequences of compatible monads. Goodwillie's calculus of homotopy functors inspired many new functor calculi in a wide range of contexts in algebra, homotopy theory and geometric topology. Recently, the third and fourth authors have developed a general program for constructing generalized calculi from sequences of compatible comonads. In this paper, we dualize the first step of the Hess-Johnson program, focusing on monads rather than comonads. We consider categories equipped with an action of the poset category $\mathcal{P}(n)$, called $\mathcal{P}(n)$-modules. We exhibit a functor from $\mathcal{P}(n)$-modules to the category of monads. The resulting monads act on categories of functors whose codomain is equipped with a suitable notion of homotopy colimits. In the final section of the paper, we demonstrate the monads used to construct McCarthy's dual calculus as an example of a monad arising from a $\mathcal{P}(n)$-module. This confirms that our dualization of the Hess-Johnson program generalizes McCarthy's dual calculus, and serves as a proof of concept for further development of this program.

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arXiv2022

Persistent Homology with Improved Locality Information for more Effective Delineation

Persistent Homology (PH) has been successfully used to train networks to detect curvilinear structures and to improve the topological quality of their results. However, existing methods are very global and ignore the location of topological features. In this paper, we remedy this by introducing a new filtration function that fuses two earlier approaches: thresholding-based filtration, previously used to train deep networks to segment medical images, and filtration with height functions, typically used to compare 2D and 3D shapes. We experimentally demonstrate that deep networks trained using our PH-based loss function yield reconstructions of road networks and neuronal processes that reflect ground-truth connectivity better than networks trained with existing loss functions based on PH. Code is available at https://github.com/doruk-oner/PH-TopoLoss.

cs.CV↗

arXiv2022

A shadow perspective on equivariant Hochschild homologies

Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.

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arXiv2021

Computational tools for twisted topological Hochschild homology of equivariant spectra

Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson.

math.AT↗

arXiv2021

A Kuenneth theorem for configuration spaces

We construct a spectral sequence converging to the homology of the ordered configuration spaces of a product of parallelizable manifolds. To identify the second page of this spectral sequence, we introduce a version of the Boardman--Vogt tensor product for linear operadic modules, a purely algebraic operation. Using the rational formality of the little cubes operads, we show that our spectral sequence collapses in characteristic zero.

math.AT↗

arXiv2021

From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on $n+1$ leaf nodes fall into $(2n-1)!!$ distinct strata, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., $(n+1)!n!2^{-n}$. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data and opens the door to doing more precise science.

math.AT↗

arXiv2021

Graph Pseudometrics from a Topological Point of View

We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has often been observed that phenomena exhibited by real-world networks reflect the topology of their flag complexes, as measured, for example, by Betti numbers or simplex counts. As it is often computationally expensive (or even unfeasible) to determine such topological features exactly, it would be extremely valuable to have pseudometrics on the set of directed graphs that can both detect the topological differences and be computed efficiently. To facilitate work in this direction, we introduce methods to measure how well a graph pseudometric captures the topology of a directed graph. We then use these methods to evaluate some well-established pseudometrics, using test data drawn from several families of random graphs.

math.AT↗

arXiv2021

giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration

We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ implementations. The library's ability to handle various types of data is rooted in a wide range of preprocessing techniques, and its strong focus on data exploration and interpretability is aided by an intuitive plotting API. Source code, binaries, examples, and documentation can be found at https://github.com/giotto-ai/giotto-tda.

cs.LG↗

arXiv2020

From trees to barcodes and back again: theoretical and statistical perspectives

Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise.

math.AT↗

arXiv2020

Invariance properties of coHochschild homology

The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in arXiv:0711.1023 by Hess, Parent, and Scott as a tool to study free loop spaces. In this article we prove "agreement" for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra $C$ is isomorphic to the Hochschild homology of the dg category of appropriately compact $C$-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space $X$ is equivalent to the suspension spectrum of the free loop space on $X$, as long as $X$ is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established by the authors in arXiv:1402.4719, we prove that "agreement" holds for coTHH as well.

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arXiv2019

Topological exploration of artificial neuronal network dynamics

One of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resorted to graph theory, statistics, and statistical mechanics to describe the spatiotemporal structure of such network dynamics. Our novel approach employs tools from algebraic topology to characterize the global properties of network structure and dynamics. We propose a method based on persistent homology to automatically classify network dynamics using topological features of spaces built from various spike-train distances. We investigate the efficacy of our method by simulating activity in three small artificial neural networks with different sets of parameters, giving rise to dynamics that can be classified into four regimes. We then compute three measures of spike train similarity and use persistent homology to extract topological features that are fundamentally different from those used in traditional methods. Our results show that a machine learning classifier trained on these features can accurately predict the regime of the network it was trained on and also generalize to other networks that were not presented during training. Moreover, we demonstrate that using features extracted from multiple spike-train distances systematically improves the performance of our method.

q-bio.NC↗

arXiv2018

The homotopy theory of coalgebras over simplicial comonads

We apply the Acyclicity Theorem of Hess, Kerdziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of coalgebras over comonads arising from simplicial adjunctions, under mild conditions on the adjunction and the associated comonad. We study three concrete examples of such adjunctions where the left adjoint is comonadic and show that in each case the component of the derived counit of the comparison adjunction at any fibrant object is an isomorphism, while the component of the derived unit at any 1-connected object is a weak equivalence. To prove this last result, we explain how to construct explicit fibrant replacements for 1-connected coalgebras in the image of the canonical comparison functor from the Postnikov decompositions of their underlying simplicial sets. We also show in one case that the derived unit is precisely the Bousfield-Kan completion map.

math.AT↗

arXiv2018

Configuration spaces of products

We show that the configuration spaces of a product of parallelizable manifolds may be recovered from those of the factors as the Boardman-Vogt tensor product of right modules over the operads of little cubes of the appropriate dimension. We also discuss an analogue of this result for manifolds that are not necessarily parallelizable, which involves a new operad of skew little cubes.

math.AT↗

arXiv2017

Two-Tier Mapper: a user-independent clustering method for global gene expression analysis based on topology

There is a growing need for unbiased clustering methods, ideally automated. We have developed a topology-based analysis tool called Two-Tier Mapper (TTMap) to detect subgroups in global gene expression datasets and identify their distinguishing features. First, TTMap discerns and adjusts for highly variable features in the control group and identifies outliers. Second, the deviation of each test sample from the control group in a high-dimensional space is computed and the test samples are clustered in a global and local network using a new topological algorithm based on Mapper. Validation of TTMap on both synthetic and biological datasets shows that it outperforms current clustering methods in sensitivity and stability; clustering is not affected by removal of samples from the control group, choice of normalization nor subselection of data. There is no user induced bias because all parameters are data-driven. Datasets can readily be combined into one analysis. TTMap reveals hitherto undetected gene expression changes in mouse mammary glands related to hormonal changes during the estrous cycle. This illustrates the ability to extract information from highly variable biological samples and its potential for personalized medicine.

q-bio.GN↗

arXiv2017

A necessary and sufficient condition for induced model structures

A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary "acyclicity" condition. We show that for a broad class of "accessible model structures" - a generalization introduced here of the well-known combinatorial model structures - this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model structures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differential graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of "bialgebras" in the sense considered here.

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