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Ran Levi

Publications and source records attributed to Ran Levi.

At least 19 recordsLinked to original sources
arXiv2026

Representation Cohomology of a Small Category

Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to as representation cohomology, is the object studied in this article. In particular, to any small category $C$, we associate a simplicial object in $Cat$, where for each $n\ge 0$ the objects of the level $n$ category are the simplices of the nerve of $C$. The basic properties of the resulting representation cohomology of these simplicial objects and certain subobjects are then studied in detail. We present some general theoretical computations in favourable cases.

math.RT↗
arXiv2025

Stability of 0-dimensional persistent homology in enriched and sparsified point clouds

We give bounds for dimension 0 persistent homology and codimension 1 homology of Vietoris--Rips, alpha, and cubical complex filtrations from finite sets related by enrichment (adding new elements), sparsification (removing elements), and aligning to a grid (uniformly discretizing elements). For enrichment we use barycentric subdivision, for sparsification we use a minimum separating distance, and for aligning to a grid we take the quotient when dividing each coordinate value by a fixed step size. We are motivated by applications presenting large and irregular datasets, and the development of persistent homology to better work with them. In particular, we consider an application to ecology, in which the state of an observed species is inferred through a high-dimensional space with environmental variables as dimensions. This ``hypervolume'' has geometry (volume, convexity) and topology (connectedness, homology), which are known to be related to the current and potentially future status of the species. We offer an approach for the analysis of hypervolumes with topological guarantees, complementary to current statistical methods, giving precise bounds between persistence diagrams of Vietoris--Rips and alpha complexes, and a duality identity for cubical complexes. Implementation of our methods, called TopoAware, is made available in C++, Python, and R, building upon the GUDHI library.

math.AT↗
arXiv2025

Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding

Graph Neural Networks (GNNs) excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learning (TDL) addresses this limitation by leveraging combinatorial topological spaces. However, existing TDL models are restricted to undirected settings and fail to capture the higher-order directed patterns prevalent in many complex systems, e.g., brain networks, where such interactions are both abundant and functionally significant. To fill this gap, we introduce Semi-Simplicial Neural Networks (SSNs), a principled class of TDL models that operate on semi-simplicial sets -- combinatorial structures that encode directed higher-order motifs and their directional relationships. To enhance scalability, we propose Routing-SSNs, which dynamically select the most informative relations in a learnable manner. We prove that SSNs are strictly more expressive than standard graph and TDL models. We then introduce a new principled framework for brain dynamics representation learning, grounded in the ability of SSNs to provably recover topological descriptors shown to successfully characterize brain activity. Empirically, SSNs achieve state-of-the-art performance on brain dynamics classification tasks, outperforming the second-best model by up to 27%, and message passing GNNs by up to 50% in accuracy. Our results highlight the potential of principled topological models for learning from structured brain data, establishing a unique real-world case study for TDL. We also test SSNs on standard node classification and edge regression tasks, showing competitive performance. We will make the code and data publicly available.

cs.LG↗
arXiv2025

Realizability of fusion systems by discrete groups: II

We compare four different types of realizability for saturated fusion systems over discrete $p$-toral groups. For example, when $G$ is a locally finite group all of whose $p$-subgroups are artinian (hence discrete $p$-toral), we show that it has ``weakly Sylow'' $p$-subgroups and give explicit constructions of saturated fusion systems and associated linking systems associated to $G$. We also show that a fusion system over a discrete $p$-toral group $S$ is saturated if its set of morphisms is closed under a certain topology and the finite subgroups of $S$ satisfy the saturation axioms, and prove a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.

math.GR↗
arXiv2024

Realizability of fusion systems by discrete groups

For a prime $p$, fusion systems over discrete $p$-toral groups are categories that model and generalize the $p$-local structure of Lie groups and certain other infinite groups in the same way that fusion systems over finite $p$-groups model and generalize the $p$-local structure of finite groups. In the finite case, it is natural to say that a fusion system $\mathcal{F}$ is realizable if it is isomorphic to the fusion system of a finite group, but it is less clear what realizability should mean in the discrete $p$-toral case. In this paper, we look at some of the different types of realizability for fusion systems over discrete $p$-toral groups, including realizability by linear torsion groups and sequential realizability, of which the latter is the most general. After showing that fusion systems of compact Lie groups are always realized by linear torsion groups (hence sequentially realizable), we give some new tools for showing that certain fusion systems are not sequentially realizable, and illustrate it with two large families of examples.

math.GR↗
arXiv2023

Foundations of Differential Calculus for modules over posets

Let $k$ be a field and let $C$ be a small category. A $k$-linear representation of $C$, or a $kC$-module, is a functor from $C$ to the category of finite dimensional vector spaces over $k$. When the category $C$ is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. This paper offers a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of $kC$-modules, under some restrictions on the category $C$. As a starting point, for a $kC$-module $M$ we define its gradient \emph{gradient} \(\nabla[M]\) as a virtual module in the Grothendieck group of isomorphism classes of $kC$-modules. Pushing the analogy with ordinary differential calculus and discrete calculus on graphs, we define left and right divergence via the appropriate left and right Kan extensions and two bilinear pairings on modules and study their properties, specifically with respect to adjointness relations between the gradient and the left and right divergence. The left and right divergence are shown to be rather easily computable in favourable cases. Having set the scene, we concentrate specifically on the case where the category $C$ is a finite poset. Our main result is a necessary and sufficient condition for the gradient of a module $M$ to vanish under certain hypotheses on the poset. We next investigate implications for two modules whose gradients are equal. Finally we consider the resulting left and right Laplacians, namely the compositions of the divergence with the gradient, and study an example of the relationship between the vanishing of the Laplacians and the gradient.

math.AT↗
arXiv2021

An application of neighbourhoods in digraphs to the classification of binary dynamics

A binary state on a graph means an assignment of binary values to its vertices. For example, if one encodes a network of spiking neurons as a directed graph, then the spikes produced by the neurons at an instant of time is a binary state on the encoding graph. Allowing time to vary and recording the spiking patterns of the neurons in the network produces an example of binary dynamics on the encoding graph, namely a one-parameter family of binary states on it. The central object of study in this article is the closed neighbourhood of a vertex $v$ in a graph $\mathcal{G}$, namely the subgraph of $\mathcal{G}$ that is induced by $v$ and all its neighbours in $\mathcal{G}$. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. As a test case we demonstrate an application of the method to binary dynamics that arises from sample activity on the Blue Brain Project reconstruction of cortical tissue of a rat.

math.AT↗
arXiv2021

Loop space homology of a small category

In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $Ω(BG^\wedge_p)$, when $G$ is a finite group, $BG^\wedge_p$ is the $p$-completion of its classifying space, and $Ω(BG^\wedge_p)$ is the loop space of $BG^\wedge_p$. The main purpose of this work is to shed new light on Benson's result by extending it to a more general setting. As a special case, we show that if $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is the geometric realization of its nerve, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a "plus construction" for $|\mathcal{C}|$ in the sense of Quillen (taken with respect to $R$-homology), then $H_*(Ω(|\mathcal{C}|^+_R);R)$ can be described as the homology of a chain complex of projective $R\mathcal{C}$-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson's theorem is now the case where $\mathcal{C}$ is the category of a finite group $G$, $R=\mathbb{F}_p$ for some prime $p$, and $|\mathcal{C}|^+_R=BG^\wedge_p$.

math.AT↗
arXiv2021

Complexes of Tournaments, Directionality Filtrations and Persistent Homology

Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph $\mathcal{G}$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of $\mathcal{G}$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.

math.AT↗
arXiv2020

Characterisation of polyhedral products with finite generalised Postnikov decomposition

A generalised Postnikov tower for a space $X$ is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to $X$. In this paper we give a characterisation of a polyhedral product $Z_K(X,A)$ whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include $p$-local and rational versions of the theorem. We end with a group theoretic application.

math.AT↗
arXiv2019

On the topology of complexes of injective words

An injective word over a finite alphabet $V$ is a sequence $w=v_1v_2\cdots v_t$ of distinct elements of $V$. The set $\mathrm{inj}(V)$ of injective words on $V$ is partially ordered by inclusion. A complex of injective words is the order complex $Δ(W)$ of a subposet $W \subset \mathrm{inj}(V)$. Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and are of independent interest in topology and combinatorics. In this article we mainly study Permutation Complexes, i.e. complexes of injective words $Δ(W)$, where $W$ is the downward closed subposet of $\mathrm{inj}(V)$ generated by a set of permutations of $V$. In particular, we determine the homotopy type of $Δ(W)$ when $W$ is generated by two permutations, and prove that any stable homotopy type is realizable by a permutation complex. We describe a homotopy decomposition for the complex of injective words $Γ(K)$ associated with a simplicial complex $K$, and point out a connection to a result of Randal-Williams and Wahl. Finally, we discuss some probabilistic aspects of random permutation complexes.

math.AT↗
arXiv2019

Computing persistent homology of directed flag complexes

We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser [Bau18a], but is optimized specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelization by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an Approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.

math.AT↗
arXiv2019

Polyhedral products over finite posets

Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset $\calp$, that include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over $\calp$ of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that include face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley-Reisner ring of a polyhedral poset and show that, like in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset $\calp$ we construct a simplicial poset $s(\calp)$, and show that if $\calp$ is a polyhedral poset then polyhedral products over $\calp$ coincide up to homotopy with the corresponding polyhedral products over $s(\calp)$.

math.AT↗
arXiv2016

Quantifying topological invariants of neuronal morphologies

Nervous systems are characterized by neurons displaying a diversity of morphological shapes. Traditionally, different shapes have been qualitatively described based on visual inspection and quantitatively described based on morphometric parameters. Neither process provides a solid foundation for categorizing the various morphologies, a problem that is important in many fields. We propose a stable topological measure as a standardized descriptor for any tree-like morphology, which encodes its skeletal branching anatomy. More specifically it is a barcode of the branching tree as determined by a spherical filtration centered at the root or neuronal soma. This Topological Morphology Descriptor (TMD) allows for the discrimination of groups of random and neuronal trees at linear computational cost.

q-bio.NC↗
arXiv2016

Groups of unstable Adams operations on p-local compact groups

A $p$-local compact group is an algebraic object modelled on the homotopy theory associated with $p$-completed classifying spaces of compact Lie groups and p-compact groups. In particular $p$-local compact groups give a unified framework in which one may study $p$-completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and p-compact groups, $p$-local compact groups admit unstable Adams operations - self equivalences that are characterised by their cohomological effect. Unstable Adams operations on $p$-local compact groups were constructed in a previous paper by F. Junod and the authors. In the current paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of operations.

math.AT↗
arXiv2016

Topological analysis of the connectome of digital reconstructions of neural microcircuits

A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices Ð groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science.

q-bio.NC↗
arXiv2015

Automorphisms of $p$-local compact groups

Self equivalences of classifying spaces of $p$-local compact groups are well understood by means of the algebraic structure that gives rise to them, but explicit descriptions are lacking. In this paper we use a construction of Robinson of an amalgam $G$, realizing a given fusion system, to produce a split epimorphism from the outer automorphism group of $G$ to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding $p$-local compact group.

math.AT↗