Ran Levi

University of Aberdeen | ABDN · Institute of Mathematics

The function of the neocortex is fundamentally determined by its repeating microcircuit motif, but also by its rich, interregional connectivity. We present a data-driven computational model of the anatomy of non-barrel primary somatosensory cortex of juvenile rat, integrating whole-brain scale data while providing cellular and subcellular specifici...

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 fusi...

The function of the neocortex is fundamentally determined by its repeating microcircuit motif, but also by its rich, interregional connectivity. We present a data-driven computational model of the anatomy of non-barrel primary somatosensory cortex of juvenile rat, integrating whole-brain scale data while providing cellular and subcellular specifici...

The function of the neocortex is fundamentally determined by its repeating microcircuit motif, but also by its rich, interregional connectivity. We present a data-driven computational model of the anatomy of non-barrel primary somatosensory cortex of juvenile rat, integrating whole-brain scale data while providing cellular and subcellular specifici...

Neurons in a neural circuit exhibit astonishing diversity in terms of the numbers and targets of their synaptic connections and the statistics of their spiking activity. We hypothesize that this diversity is the result of an underlying tension in the neural code between reliability – highly correlated activity across trials on the single neuron lev...

Persistence modules were introduced in the context of topological data analysis. Generalised persistence module theory is the study of functors from an arbitrary poset, or more generally an arbitrary small category, to some abelian target category. In other words, a persistence module is simply a representation of the source category in the target...

The function of the neocortex is fundamentally determined by its repeating microcircuit motif, but also by its rich, interregional connectivity. We present a data-driven computational model of the anatomy of non-barrel primary somatosensory cortex of juvenile rat, integrating whole-brain scale data while providing cellular and subcellular specifici...

In motor-related brain regions, movement intention has been successfully decoded from in-vivo spike train by isolating a lower-dimension manifold that the high-dimensional spiking activity is constrained to. The mechanism enforcing this constraint remains unclear, although it has been hypothesized to be implemented by the connectivity of the sample...

A binary state on a graph means an assignment of binary values to its vertices. A time dependent sequence of binary states is referred to as binary dynamics. We describe a method for the classification of binary dynamics of digraphs, using particular choices of closed neighbourhoods. Our motivation and application comes from neuroscience, where a d...

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 $${\mathcal {G}}$$ G , we associate with it a “flag tournaplex” which is a tournaplex containing the directed flag co...

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 pr...

In motor-related brain regions, movement intention has been successfully decoded from in-vivo spike train by isolating a lower-dimension manifold that the high-dimensional spiking activity is constrained to. The mechanism enforcing this constraint remains unclear, although it has been hypothesized to be implemented by the connectivity of the sample...

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...

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, o...

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 G, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of G, but also...

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 G, we associate with it a ``flag tournaplex'' which is a tournaplex containing the directed flag complex of G, but...

An injective word over a finite alphabet V is a sequence w=v1v2⋯vt of distinct elements of V. The set 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⊂Inj(V). Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and...

Presentation in Royal Holloway University London 19.02.2020

We present a new computing package FLAGSERmathsizesmall, 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 FLAGSERmathsizesmall is based on the program RIPSERmathsizesmall by...

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 $\Delta(W)$ of a subposet $W \subset \mathrm{inj}(V)$. Complexes of injective words arose recently in ap...

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 fo...

Abstract: The brain is a vast network of neurons, each of which connected to hundreds and sometimes thousands of others in an intricate and highly complex fashion. Each neuron is capable of performing complicated tasks that are expressed in its firing patterns, but all brain functions are achieved by ensembles of neurons operating in a highly coord...

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. The subject is related to many topics in homotopy theory and has applications in physics and in topological robotics. In this paper we give a very general homot...

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 restrictio...

This is a talk I gave in a workshop on topology and neuroscience https://neurotop2018.org/ held in EPFL. I discuss ideas for analysis structure and function in brain networks.

This is a preliminary lecture on some aspects of simplicial topology given in GETCO, OAXACA 2018. The lecture was given as preparation for a research lecture on applications of topology to neuroscience.

This is a report on work in progress with a team from EPFL and the Blue Brain project, involving applications of topology to the study of synaptic plasticity in the brain.

In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod p homology of Ω(BG ∧ p), when G is a finite group, BG ∧ p is the p-completion of its classifying space, and Ω(BG ∧ p) is the loop space of BG ∧ 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...

This is a presentation I gave in the Abel Symposium 2018 held in Geiranger, Norway, on 6th of June.

Many biological systems consist of branching structures that exhibit a wide variety of shapes. Our understanding of their systematic roles is hampered from the start by the lack of a fundamental means of standardizing the description of complex branching patterns, such as those of neuronal trees. To solve this problem, we have invented the Topologi...

10 pages, 5 figures, conference or other essential info

This is a second talk I gave at Kyoto University in July 2017. Following the introductory talk on algebraic topology applicable to neuroscience, this talk is an exposition of my first collaborative work with the Blue Brain Project.

This is a summary of an introductory talk I gave at Kyoto University that is a first in a sequence of two. The topic is basic algebraic topology and the exposition is geared specifically as background for non-topologists who wish to understand my collaborative work with the Blue Brain Project. The followup lecture is also posted.

The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow,...

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...

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 con...

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 complex...

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 gro...

For any prime p, the theory of p-local compact groups is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and $p$-compact groups, and generalises the earlier concept of p-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In th...

A major question in the theory of $p$-local finite groups was whether any saturated fusion system over a finite $p$-group admits an associated centric linking system, and when it does, whether it is unique. Both questions were answered in the affirmative by Chermak, using the theory of partial groups and localities he developed. Using Chermak's ide...

A saturated fusion system consists of a finite p-group S, together with a category which encodes “conjugacy” relations among subgroups of S, and which satisfies certain axioms which are motivated by properties of the fusion in a Sylow p-subgroup of a finite group. We describe here new ways of constructing abstract saturated fusion systems, first as...

The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the ea...

A p-local compact group is an algebraic object modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. In the study of these objects unstable Adams operations, are of fundamental importance. In this paper we define unstable Adams operations within the theory of p-local compact groups, and show that...

The spaces BG_2 and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has for q odd the classifying spaces of the finite groups BG_2(q)and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article compute the mod 2 loop...

We study cohomology for $p$-local finite groups with non-constant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences.

This paper gives an overview of Fred Cohen's work and is a summary of the talk which I gave during his 60th birthday conference, held at the University of Tokyo in July 2005.

We construct a homotopy theoretic setup for homology decompositions of classifying spaces of p-compact groups. This setup is then used to obtain a subgroup decomposition for p-compact groups which generalizes the subgroup decomposition with respect to p-stubborn subgroups for a compact Lie group constructed by Jackowski, McClure and Oliver. Homolog...

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifyi...

We define and study a certain class of spaces which includes p-completed classifying spaces of compact Lie groups, classifying spaces of p-compact groups, and p-completed classifying spaces of certain locally finite discrete groups. These spaces are determined by fusion and linking systems over "discrete p-toral groups" - extensions of (ℤ/p ∞)r by...

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In our paper (2), we constructed a family of 2-local finite groups which are "exotic" in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin7(q) (q...

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifyi...

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construc...

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ‘conjugacy’ relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying...

A “p-local finite group” is an algebraic object which consists of a system of fusion (conjugacy) data in a finite p-group S, as formalized by Ll. Puig, extended by some extra information contained in a category which allows rigidification of the fusion data. Such objects have “classifying spaces” which satisfy many of the homotopy theoretic propert...

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category ℒ p c (G) with the following properties. Two p-completed classifying spaces BG p ∧ and BG’ p ∧ have the same homotopy type if and only if the associated categories ℒ p c (G) and ℒ p c...

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an...

Fix a prime p. A mod-p homotopy group extension of a group π by a group G is a fibration with base space Bπp∧ and fibre BGp∧. In this paper we study homotopy group extensions for finite groups. We observe that there is a strong analogy between homotopy group extensions and ordinary group extensions. The study involves investigating the space of sel...

Consider the space X, = ℝP∞/ℝPn-1 together with the boundary map in the Barratt-Puppe sequence $${X_n} \to \Sigma \mathbb{R}{{\text{P}}^{n - 1}}.

Let be a commutative ring with a unit and let h be a homology theory taking value in the category of cocommutative coalgebras over . Then, restricted to the category of homotopy associativeH spaces, the theoryh takes value in the category of cocommutative Hopf algebras over . Throughout this note by an H space we mean a homotopy associative H space...

L etS2n 1 k denote the fiber of the degree k map on the sphere S2n 1 .I fk = pr ,w here p is an odd prime and n divides p 1, then S2n 1 k is known to be a loop space. It is also known that S3 2r is a loop space for r 3. In this paper we study the possible loop structures on this family of spaces for all primes p. In particular we show that S 3 4 is...

It is shown that for certain discrete p-perfect groups G, in particular for all p-perfect finite groups and certain groups of finite virtual cohomological dimension, the loop space on the p-completed classifying space is a retract of the loop space on the p-completion of a certain finite complex. For those finite virtual cohomological dimension gro...

It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the p-completed classifying space BGandp has the property that ΩBGandp is a retract of the loop space on a simply-connected, [open face F]p-finite, p-complete space. In this note we consider a particular example where this theor...

It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the p-completed classifying space BGp∧ has the property that ΩBGp∧ is a retract of the loop space on a simply-connected, double struck F signp-finite, p-complete space. In this note we consider a particular example where this theorem...

We consider the homotopy type of classifying spaces BG ,w here Gis a nite p-group, and we study the question whether or not the mod p cohomology of BG, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of BG. This article is devoted to producing some families of nite 2-grou...

Let G be a finite p-perfect group. We show that the mod-p homology of ΩBG ∧ p grows either polynomially or semi-exponentially. A conjecture due to F. Cohen states that ΩBG ∧ p for such groups G is spherically resolvable of finite weight. We show that any space X, which satisfies the conclusion of Cohen's conjecture has the property that its homolog...

Let R be a torsion free principal ideal domain. We study the growth of torsion in loop space homology of simply-connected R-coalgebras C, whose homology admits an exponent r in R. Here by loop space homology we mean the homology of the loop algebra construction on C. We compute a bound on the growth of torsion in such objects and show that in gener...

Let G be a finite p-superperfect group. A conjecture of F. Cohen suggests that \( \Omega BG_{P}^{\Lambda } \) is resolvable by finitely many fibrations over spheres and iterated loop spaces on spheres, where \( ( - ){\text{ }}_{P}^{\Lambda } \) denotes the p-completion functor of Bousfield and Kan. We produce a counter-example to this conjecture an...

We study a comparison criterion for loop spaces onp-localized classifying spaces of certain nite p-perfect groups G. In particular we show that, under certain hypotheses, the homotopy type of those spaces is determined by the mod-p cohomology of G together with a nite Postnikov system.

The purpose of this note is to record natural ltered simplicial group models for iterated loop spaces. The models are derived by classical methods for simplicial groups along with the identication for certain group theoretic kernels. The main content of the article is the study of some useful properties of these models. One feature of these models...