arXiv subjects

Felix Wierstra

Publications and source records attributed to Felix Wierstra.

16 recordsLinked to original sources

arXiv2023

Beyond Signal and Noise: Unraveling Scale Invariance in Neuroscience and Financial Networks with Topological Data Analysis

Topological Data Analysis (TDA) is increasingly crucial in investigating the shape of complex data structures across scientific fields, particularly in neuroscience and finance. This study delves into persistent homology, a TDA component initially aimed at differentiating between signal and noise. We explore two methodologies: the conventional cycle length approach and the novel death-birth ratio method proposed by Bobrowski and Skraba. Analyzing rs-fMRI data from the Human Connectome Project and daily $S\&P 500$ financial networks, our study compares these methods in identifying significant cycles. A key discovery is a robust relationship between z-score thresholds applied to bar lengths or ratios and behavioural traits in brain networks and market volatility in financial networks. In the brain, this is evident in the strong correlation between significant 1-cycles, brain volumes, and sex-based differences. In financial markets, a fractal pattern emerges, where market volatility negatively correlates with the number of significant cycles, indicating that more complex market topologies are associated with increased stability. Our findings also imply a fractal nature of 1-cycles at both population levels and across multiple days in the stock market. The distribution of significant loops, marked by high z-scores, remains consistent across various z-score thresholds, revealing a scale-invariant, fractal structure in both data sets. Given the scale invariance in these fractal structures, the traditional TDA distinction between signal and noise becomes less meaningful. This suggests that all scales of cycle length are relevant, challenging the conventional approach of segregating signal from noise and broadening the scope of TDA to reveal intricate, scale-invariant relationships in complex systems.

physics.soc-ph↗

arXiv2023

A Lie theoretic approach to the twisting procedure and Maurer-Cartan simplicial sets over arbitrary rings

The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial set of an $L_\infty$-algebra plays an important role in deformation theory and many other areas of mathematics. Unfortunately, this construction only works over a field of characteristic $0$. The goal of this paper is to show that the notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for a much larger number of operads than just the $L_\infty$-operad. More precisely, we show that the Koszul dual of every unital Hopf cooperad (a cooperad in the category of unital associative algebras) with an arity $0$ operation admits a twisting procedure, a natural notion of Maurer-Cartan equation and under some mild additional assumptions can also be integrated to a Maurer-Cartan simplicial set. In particular, we show that the Koszul dual of the Barratt-Eccles operad and its $E_n$-suboperads admit Maurer-Cartan simplicial sets. In this paper, we will work over arbitrary rings.

math.AT↗

arXiv2022

On the homotopy fixed points of Maurer-Cartan spaces with finite group actions

We develop the basic theory of Maurer-Cartan simplicial sets associated to (shifted complete) $L_\infty$ algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant simplicial set into the homotopy fixed points is a homotopy equivalence of Kan complexes, provided the $L_\infty$ algebra is concentrated in non-negative degrees. As an application, and under certain connectivity assumptions, we provide rational algebraic models of the fixed and homotopy fixed points of mapping spaces equipped with the action of a finite group.

math.AT↗

arXiv2022

Commutative homotopical algebra embeds into non-commutative homotopical algebra

Over a field of characteristic zero, we show that the forgetful functor from the homotopy category of commutative dg algebras to the homotopy category of dg associative algebras is faithful. In fact, the induced map of derived mapping spaces gives an injection on all homotopy groups at any basepoint. We prove similar results both for unital and non-unital algebras, and also Koszul dually for the universal enveloping algebra functor from dg Lie algebras to dg associative algebras. An important ingredient is a natural model for these derived mapping spaces as Maurer-Cartan spaces of complete filtered dg Lie algebras (or curved Lie algebras, in the unital case).

math.AT↗

arXiv2022

A recognition principle for iterated suspensions as coalgebras over the little cubes operad

Our main result is a recognition principle for iterated suspensions as coalgebras over the little disks operads. Given a topological operad, we construct a comonad in pointed topological spaces endowed with the wedge product. We then prove an approximation theorem that shows that the comonad associated to the little $n$-cubes operad is weakly equivalent to the comonad $\Sigma^n \Omega^n$ arising from the suspension-loop space adjunction. Finally, our recognition theorem states that every little $n$-cubes coalgebra is homotopy equivalent to an $n$-fold suspension. These results are the Eckmann--Hilton dual of May's foundational results on iterated loop spaces.

math.AT↗

arXiv2021

The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time

We prove that the simplicial cocommutative coalgebra of singular chains on a connected topological space determines the homotopy type rationally and one prime at a time, without imposing any restriction on the fundamental group. In particular, the fundamental group and the homology groups with coefficients in arbitrary local systems of vector spaces are completely determined by the natural algebraic structure of the chains. The algebraic structure is presented as the class of the simplicial cocommutative coalgebra of chains under a notion of weak equivalence induced by a functor from coalgebras to algebras coined by Adams as the cobar construction. The fundamental group is determined by a quadratic equation on the zeroth homology of the cobar construction of the normalized chains which involves Steenrod's chain homotopies for cocommutativity of the coproduct. The homology groups with local coefficients are modeled by an algebraic analog of the universal cover which is invariant under our notion of weak equivalence. We conjecture that the integral homotopy type is also determined by the simplicial coalgebra of integral chains, which we prove when the universal cover is of finite type.

math.AT↗

arXiv2020

Rational homotopy equivalences and singular chains

Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of connected and pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps $π_1$-rational homotopy equivalences. In this paper, we compare these two notions and show that $π_1$-rational homotopy equivalences correspond to maps that induce $Ω$-quasi-isomorphisms on the rational singular chains, i.e. maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalgebra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different algebraic notions of weak equivalences: quasi-isomorphism and $Ω$-quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative.

math.AT↗

arXiv2020

Homotopy morphisms between convolution homotopy Lie algebras

In previous works by the authors, a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of $\infty$-morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such $\infty$-morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently extended to accept $\infty$-morphisms in both slots simultaneously. We apply this theory to rational models for mapping spaces.

math.AT↗

arXiv2019

Iterated suspensions are coalgebras over the little disks operad

We study the Eckmann-Hilton dual of the little disks algebra structure on iterated loop spaces: With the right definitions, every $n$-fold suspension is a coalgebra over the little $n$-disks operad. This structure induces non-trivial cooperations on the rational homotopy groups of an $n$-fold suspension. We describe the Eckmann-Hilton dual of the Browder bracket, which is a cooperation that forms an obstruction for an $n$-fold suspension to be an $(n+1)$-fold suspension, i.e. if this cooperation is non-zero then the space is not an $(n+1)$-fold suspension. We prove several results in equivariant rational homotopy theory that play an essential role in our results. Namely, we prove a version of the Sullivan conjecture for the Maurer-Cartan simplicial set of certain $L_\infty$-algebras equipped with a finite group action, and we provide rational models for fixed and homotopy fixed points of (mapping) spaces under some connectivity assumptions in the context of finite groups. We further show that by using the Eckmann-Hilton dual of the Browder operation we can use the rational homotopy groups to detect the difference between certain spaces that are rationally homotopy equivalent, but not homotopy equivalent.

math.AT↗

arXiv2019

Lie, associative and commutative quasi-isomorphism

Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homotopy theory, showing that the rational homotopy type of a space is determined by its associative dg algebra of rational cochains. We also show a Koszul dual statement, under an additional completeness hypothesis: two homotopy complete dg Lie algebras whose universal enveloping algebras are quasi-isomorphic as associative dg algebras must themselves be quasi-isomorphic. The latter result applies in particular to nilpotent Lie algebras (not differential graded), in which case it says that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic.

math.RA↗

arXiv2019

The functor of singular chains detects weak homotopy equivalences

The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\infty}$-coalgebra $\mathbf{C}_*(X)$ with the property that the $0^{\text{th}}$ homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode, i.e. it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces $f: X\to Y$ is a weak homotopy equivalence if and only if $\mathbf{C}_*(f): \mathbf{C}_*(X)\to \mathbf{C}_*(Y)$ is an $\mathbfΩ$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying the cobar functor $\mathbfΩ$ to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains.

math.AT↗

arXiv2018

Algebraic Hopf invariants and rational models for mapping spaces

In this paper we will define an invariant $mc_{\infty}(f)$ of maps $f:X \rightarrow Y_{\mathbb{Q}}$ between a finite CW-complex and a rational space $Y_{\mathbb{Q}}$. We prove that this invariant is complete, i.e. $mc_{\infty}(f)=mc_{\infty}(g)$ if an only if $f$ and $g$ are homotopic. We will also construct an $L_{\infty}$-model for the based mapping space $Map_*(X,Y_{\mathbb{Q}})$ from a $C_{\infty}$-coalgebra and an $L_{\infty}$-algebra.

math.AT↗

arXiv2018

Hopf invariants and differential forms

Let $f,g:M \rightarrow N$ be two maps between simply-connected smooth manifolds $M$ and $N$, such that $M$ is compact and $N$ is of finite $\mathbb{R}$-type. The goal of this paper is to use integration of certain differential forms to obtain a complete invariant of the real homotopy classes of the maps $f$ and $g$.

math.AT↗

arXiv2018

$E_n$-Hopf invariants

The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1} $ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $E_n$-operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. In this paper we will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the $E_n$-bar construction on the cochains of $X$ and the homotopy groups of $X$. This pairing gives us a set of invariants of homotopy classes of maps from $S^m$ to a simplicial set $X$, this pairing can detect more homotopy classes of maps than the classical Hopf invariant. The second part of the paper is devoted to combining the $E_n$-Hopf invariants with the Koszul duality theory for $E_n$-operads to get a relation between the $E_n$-Hopf invariants of a space $X$ and the $E_{n+1}$-Hopf invariants of the suspension of $X$. This is done by studying the suspension morphism for the $E_\infty$-operad, which is a morphism from the $E_{\infty}$-operad to the desuspension of the $E_\infty$-operad. We show that it induces a functor from $E_\infty$-algebras to $E_\infty$-algebras, which has the property that it sends an $E_\infty$-model for a simplicial set $X$ to an $E_\infty$-model for the suspension of $X$. We use this result to give a relation between the $E_n$-Hopf invariants of maps from $S^m$ into $X$ and the $E_{n+1}$-Hopf invariants of maps from $S^{m+1}$ into the suspension of $X$. One of the main results we show here, is that this relation can be used to define invariants of stable homotopy classes of maps.

math.AT↗

arXiv2018

Lie Theory for Complete Curved $A_\infty$-algebras

In this paper we develop the $A_\infty$-analog of the Maurer-Cartan simplicial set associated to an $L_\infty$-algebra and show how we can use this to study the deformation theory of $\infty$-morphisms of algebras over non-symmetric operads. More precisely, we define a functor from the category of (curved) $A_\infty$-algebras to simplicial sets which sends an $A_\infty$-algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. We also show that this functor can be used to study deformation problems over a field of characteristic greater or equal than $0$. As a specific example of such a deformation problem we study the deformation theory of $\infty$-morphisms over non-symmetric operads.

math.QA↗

arXiv2018

Convolution algebras and the deformation theory of infinity-morphisms

Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.

math.QA↗