arXiv subjects

Anton Ayzenberg

Publications and source records attributed to Anton Ayzenberg.

At least 19 recordsLinked to original sources

arXiv2026

Homotopy types of finite étale spaces and generalized inflations

Inflation of a simplicial complex $K$ is a construction well known in combinatorial topology. It replaces each vertex $i$ of $K$ with a finite number $n_i$ of its copies, and each simplex $\{i_0,\ldots,i_k\}$ with $n_{i_0}n_{i_1}\cdots n_{i_k}$ many copies so that the collection of vertex-copies is spanned by a simplex in the inflation if and only if their originals were spanned by a simplex in the original complex. The celebrated poset fiber theorem of Björner, Wachs, and Welker describes the homotopy type of such inflation in terms of homotopy types of $K$ and its links. In the current paper, we introduce more general inflations over simplicial posets: we replace each simplex with an arbitrary finite set of copies. The way how these sets patch together is specified by a commutative diagram, or, equivalently, a sheaf on the corresponding finite topology. The generalized inflation can be understood as étale space of such sheaf. We prove that, whenever this inflation sheaf is flabby, the poset fiber theorem still applies. We prove all results similar to those known for vertex inflations. We also cover the previous result of the first author about homotopy types of clique complexes of multigraphs.

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arXiv2025

Sheaf theory: from deep geometry to deep learning

This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory accessible to those with modest mathematical familiarity. We describe intuitions and motivations underlying sheaf theory shared by both theoretical researchers and practitioners, bridging classical mathematical theory and its more recent implementations within signal processing and deep learning. We observe that most notions commonly considered specific to cellular sheaves translate to sheaves on arbitrary posets, providing an interesting avenue for further generalization of these methods in applications, and we present a new algorithm to compute sheaf cohomology on arbitrary finite posets in response. By integrating classical theory with recent applications, this work reveals certain blind spots in current machine learning practices. We conclude with a list of problems related to sheaf-theoretic applications that we find mathematically insightful and practically instructive to solve. To ensure the exposition of sheaf theory is self-contained, a rigorous mathematical introduction is provided in appendices which moves from an introduction of diagrams and sheaves to the definition of derived functors, higher order cohomology, sheaf Laplacians, sheaf diffusion, and interconnections of these subjects therein.

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arXiv2023

Coverings by open and closed hemispheres

In this paper we study the nerves of two types of coverings of a sphere $S^{d-1}$: (1) coverings by open hemispheres; (2) antipodal coverings by closed hemispheres. In the first case, nerve theorem implies that the nerve is homotopy equivalent to $S^{d-1}$. In the second case, we prove that the nerve is homotopy equivalent to a wedge of $(2d-2)$-dimensional spheres. The number of wedge summands equals the Möbius invariant of the geometric lattice (or hyperplane arrangement) associated with the covering. This result explains some observed large-scale phenomena in topological data analysis. We review the particular case, when the coverings are centered in the root system $A_d$. In this case the nerve of the covering by open hemispheres is the space of directed acyclic graphs (DAGs), and the nerve of the covering by closed hemispheres is the space of non-strongly connected directed graphs. The homotopy types of these spaces were described by Björner and Welker, and the incarnation of these spaces appeared independently as "the poset of orders" and "the poset of preorders" respectively in the works of Bouc. We study the space of DAGs in terms of Gale and combinatorial Alexander dualities, and propose how this space can be applied in automated machine learning.

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arXiv2023

On actions of tori and quaternionic tori on products of spheres

In this paper we study the actions of tori (standard compact tori, as well as their quaternionic analogues) on products of spheres. It is proved that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for a real torus $\mathbb{Z}_2^n$ was proved by the second author in 2019. We also provide a statement about arbitrary compact topological groups, generalizing the mentioned results, as well as the results of the first author about the actions of a compact torus of complexity one.

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arXiv2023

Orbit spaces of equivariantly formal torus actions of complexity one

Let a compact torus $T=T^{n-1}$ act on an orientable smooth compact manifold $X=X^{2n}$ effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If $H^{odd}(X)=0$ and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space $Q=X/T$ is a homology $(n+1)$-sphere. If, in addition, $π_1(X)=0$, then $Q$ is homeomorphic to $S^{n+1}$. We introduce the notion of $j$-generality of tangent weights of torus action. For any action of $T^k$ on $X^{2n}$ with isolated fixed points and $H^{odd}(X)=0$, we prove that $j$-generality of weights implies $(j+1)$-acyclicity of the orbit space $Q$. This statement generalizes several known results for actions of complexity zero and one. In complexity one, we give a criterion of equivariant formality in terms of the orbit space. In this case, we give a formula expressing Betti numbers of a manifold in terms of certain combinatorial structure that sits in the orbit space.

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arXiv2023

Cluster-permutohedra and submanifolds of flag varieties with torus actions

In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del R\'ıo-Francos, López-Dudet, Oliveros, and Schulte in 2010, and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain finite poset defined for a simple graph $Γ$. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $Γ$. This poset is similar to the graphicahedron of $Γ$, in particular, 1-skeleta of both posets are isomorphic to Cayley graphs of the symmetric group. We describe the relation between cluster-permutohedron and graphicahedron using Galois connection and the notion of a core of a finite topology. We further prove that the face poset of the natural torus action on the manifold of isospectral $Γ$-shaped Hermitian matrices is isomorphic to the cluster-permutohedron. Using recent results in toric topology, we show that homotopy properties of graphicahedra may serve an obstruction to equivariant formality of isospectral matrix manifolds. We introduce a generalization of a cluster-permutohedron and describe the combinatorial structure of a large family of manifolds with torus actions, including Grassmann manifolds and partial flag manifolds.

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arXiv2023

How is a graph not like a manifold?

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of $j$-independency of tangent weights at each fixed point implies $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of $(n-1)$-dimensional torus, under certain colorability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. Such observation underlines certain similarity between actions of complexity one and torus manifolds.

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arXiv2023

Toric orbit spaces which are manifolds

We characterize the actions of compact tori on smooth manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza--Klein model of Dirac's monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.

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arXiv2022

Topology of cognitive maps

In present paper we discuss several approaches to reconstructing the topology of the physical space from neural activity data of CA1 fields in mice hippocampus, in particular, having Cognitome theory of brain function in mind. In our experiments, animals were placed in different new environments and discovered these moving freely while their physical and neural activity was recorded. We test possible approaches to identifying place cell groups out of the observed CA1 neurons. We also test and discuss various methods of dimension reduction and topology reconstruction. In particular, two main strategies we focus on are the Nerve theorem and point cloud-based methods. Conclusions on the results of reconstruction are supported with illustrations and mathematical background which is also briefly discussed.

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arXiv2022

Topology and geometry of data manifold in deep learning

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and substantiate the geometric and topological view of the learning process of neural networks. Our attention is focused on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of the data manifold on different layers. We also propose a method for assessing the generalizing ability of neural networks based on topological descriptors. In this paper, we use the concepts of topological data analysis and intrinsic dimension, and we present a wide range of experiments on different datasets and different configurations of convolutional neural network architectures. In addition, we consider the issue of the geometry of adversarial attacks in the classification task and spoofing attacks on face recognition systems. Our work is a contribution to the development of an important area of explainable and interpretable AI through the example of computer vision.

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arXiv2022

Topological approach to diagonalization algorithms

In this paper we prove that there exists an asymptotical diagonalization algorithm for a class of sparse Hermitian (or real symmetric) matrices if and only if the matrices become Hessenberg matrices after some permutation of rows and columns. The proof is based on Morse theory, Roberts' theorem on indifference graphs, toric topology, and computer-based homological calculations.

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arXiv2022

The second cohomology of regular semisimple Hessenberg varieties from GKM theory

We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group $\mathfrak{S}_n$ by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an $\mathfrak{S}_n$-module. Our formula is not exactly the same as the known formula by Chow or Cho-Hong-Lee but they are equivalent. We also discuss its higher degree generalization.

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arXiv2022

Matroids in toric topology

In this paper we study general torus actions on manifolds with isolated fixed points from combinatorial point of view. The main object of study is the poset of face submanifolds of such actions. We introduce the notion of a locally geometric poset -- the graded poset locally modelled by geometric lattices, and prove that for any torus action, the poset of its faces is locally geometric. Next we discuss the relations between posets of faces and GKM-theory. In particular, we define the face poset of an abstract GKM-graph and show how to reconstruct the face poset of a manifold from its GKM-graph.

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arXiv2021

Space of isospectral periodic tridiagonal matrices

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,λ}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $λ$. Using the discretized Shrödinger operator we describe all spectra $λ$ for which $X_{n,λ}$ is a topological manifold. The space $X_{n,λ}$ carries a natural effective action of a compact $(n-1)$-torus. We describe the topology of its orbit space and, in particular, show that whenever the isospectral space is a manifold, its orbit space is homeomorphic to $S^4\times T^{n-3}$. There is a classical dynamical system: the flow of the periodic Toda lattice, acting on $X_{n,λ}$. Except for the degenerate locus $X_{n,λ}^0$, the Toda lattice exhibits Liouville--Arnold behavior, so that the space $X_{n,λ}\setminus X_{n,λ}^0$ is fibered into tori. The degenerate locus of the Toda system is described in terms of combinatorial geometry: its structure is encoded in the special cell subdivision of a torus, which is obtained from the regular tiling of the euclidean space by permutohedra. We apply methods of commutative algebra and toric topology to describe the cohomology and equivariant cohomology modules of $X_{n,λ}$.

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arXiv2020

Clique complexes of multigraphs, edge inflations, and tournaplexes

In this paper we introduce and study the topology of clique complexes of multigraphs without loops. These clique complexes generalize tournaplexes, which were recently introduced by Govc, Levi, and Smith for the topological study of brain functional networks. We study a general construction of edge-inflated simplicial posets, which generalize clique complexes of multigraphs. The poset fiber theorem of Björner, Wachs, and Welker is applied to obtain the homotopy wedge decomposition of an edge-inflated simplicial poset. The homological corollary of this result allows to parallelize the homology computations for edge inflated complexes, in particular, for clique complexes of multigraphs and tournaplexes. We provide functorial versions of some results to be used in computations of persistent homology. Finally, we introduce a general notion of simplex inflations and prove homotopy wedge decompositions for this class of spaces.

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arXiv2020

Topology of misorientation spaces

Let $G_1$ and $G_2$ be discrete subgroups of $SO(3)$. The double quotients of the form $X(G_1,G_2)=G_1\backslash SO(3)/G_2$ were introduced in material science under the name misorientation spaces. In this paper we review several known results that allow to study topology of misorientation spaces. Neglecting the orbifold structure, all misorientation spaces are closed orientable topological 3-manifolds with finite fundamental groups. In case when $G_1,G_2$ are crystallography groups, we compute the fundamental groups $π_1(X(G_1,G_2))$, and apply Thurston's elliptization conjecture to describe these spaces. Many misorientation spaces are homeomorphic to $S^3$ by Poincaré conjecture. The sphericity in these examples is related to the theorem of Mikhailova--Lange, which constitutes a certain real analogue of Chevalley--Shephard--Todd theorem. We explicitly describe topological types of several misorientation spaces avoiding the reference to Poincaré conjecture. Classification of misorientation spaces allows to introduce new $n$-valued group structures on $S^3$ and $\mathbb{R}P^3$. Finally, we outline the connection of the particular misorientation space $X(D_2,D_2)$ to integrable dynamical systems and toric topology.

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arXiv2019

Topology of nerves and formal concepts

The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial simplification of simplicial complexes and allows to see a certain duality on a class of simplicial complexes. This duality generalizes Poincare duality on cell subdivisions of manifolds. On the other hand, with the notion of a topological formal context, we review the classical proofs of two basic theorems of homotopy topology: Alexandrov Nerve theorem and Quillen--McCord theorem, which are both important in the applications. A brief overview of the applications of the Nerve theorem in brain studies is given. The focus is made on the task of the external stimuli space reconstruction from the activity of place cells. We propose to use the combination of FCA and topology in the analysis of neural codes. The lattice of formal concepts of a neural code is homotopy equivalent to the nerve complex, but, moreover, it allows to analyse certain implication relations between collections of neural cells.

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arXiv2019

Torus actions of complexity one in non-general position

Let the compact torus $T^{n-1}$ act on a smooth compact manifold $X^{2n}$ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space $X^{2n}/T^{n-1}$ if the action is cohomologically equivariantly formal (which essentially means that $H^{odd}(X^{2n};\mathbb{Z})=0$). It happens that homology of the orbit space can be arbitrary in degrees $3$ and higher. For any finite simplicial complex $L$ we construct an equivariantly formal manifold $X^{2n}$ such that $X^{2n}/T^{n-1}$ is homotopy equivalent to $Σ^3L$. The constructed manifold $X^{2n}$ is the total space of the projective line bundle over the permutohedral variety hence the action on $X^{2n}$ is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of the action in $j$-general position and prove that, for any simplicial complex $M$, there exists an equivariantly formal action of complexity one in $j$-general position such that its orbit space is homotopy equivalent to $Σ^{j+2}M$.

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