14 citations · 109 across the 30 of their papers we have counts for
33 papers
Homotopy types of finite étale spaces and generalized inflations
Anton Ayzenberg, Nadya Khoroshavkina
Inflation of a simplicial complex is a construction well known in combinatorial topology. It replaces each vertex of with a finite number of its copies, and each…
On actions of tori and quaternionic tori on products of spheres
Anton Ayzenberg, Dmitry Gugnin
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…
Toric orbit spaces which are manifolds
Anton Ayzenberg, Vladimir Gorchakov
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 resul…
Topology of cognitive maps
Konstantin Sorokin, Anton Ayzenberg, Konstantin Anokhin +4
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, havi…
Topology and geometry of data manifold in deep learning
German Magai, Anton Ayzenberg
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 q…
Topological approach to diagonalization algorithms
Anton Ayzenberg, Konstantin Sorokin
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 H…
The second cohomology of regular semisimple Hessenberg varieties from GKM theory
Anton Ayzenberg, Mikiya Masuda, Takashi Sato
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 H…
Matroids in toric topology
Anton Ayzenberg, Vladislav Cherepanov
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…
Cluster-permutohedra and submanifolds of flag varieties with torus actions
Anton Ayzenberg, Victor Buchstaber
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 topo…
How is a graph not like a manifold?
Anton Ayzenberg, Mikiya Masuda, Grigory Solomadin
For an equivariantly formal action of a compact torus on a smooth manifold with isolated fixed points we investigate the global homological properties of the graded poset $…
Clique complexes of multigraphs, edge inflations, and tournaplexes
Anton Ayzenberg, Alexey Rukhovich
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 b…
Orbit spaces of equivariantly formal torus actions of complexity one
Anton Ayzenberg, Mikiya Masuda
Let a compact torus act on an orientable smooth compact manifold effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all p…
Topology of misorientation spaces
Anton Ayzenberg, Dmitry Gugnin
Let and be discrete subgroups of . The double quotients of the form were introduced in material science under the name misor…
Topology of nerves and formal concepts
Anton Ayzenberg
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. FC…
Torus actions of complexity one in non-general position
Anton Ayzenberg, Vladislav Cherepanov
Let the compact torus act on a smooth compact manifold effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orb…
Torus action on quaternionic projective plane and related spaces
Anton Ayzenberg
For an action of a compact torus on a smooth compact manifold~ with isolated fixed points the number is called the complexity of the action. In th…
Space of isospectral periodic tridiagonal matrices
Anton Ayzenberg
A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space of Hermitian periodic tridiagonal -matrice…
Manifolds of isospectral matrices and Hessenberg varieties
Anton Ayzenberg, Victor Buchstaber
We study the space of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized To…
Torus actions of complexity one and their local properties
Anton Ayzenberg
We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topo…
Dimensions of multi-fan algebras
Anton Ayzenberg
Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data.…
Toric manifolds over 3-polytopes
Anton Ayzenberg
In this note we gather and review some facts about existence of toric spaces over 3-dimensional simple polytopes. First, over every combinatorial 3-polytope there exists a quasitor…
Volume polynomials and duality algebras of multi-fans
Anton Ayzenberg, Mikiya Masuda
We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan determines a volume polynomial whose value…
Locally standard torus actions and sheaves over Buchsbaum posets
Anton Ayzenberg
We consider a sheaf of exterior algebras on a simplicial poset and introduce a notion of homological characteristic function. Two natural objects are associated with these data…
Topological model for h"-vectors of simplicial manifolds
Anton Ayzenberg
Any manifold with boundary gives rise to a Poincare duality algebra in a natural way. Given a simplicial poset whose geometric realization is a closed orientable homology manif…