Papers by Anton Ayzenberg
Математический сборник, 2021
Матрицей-стрелкой называется матрица с нулями вне главной диагонали, первой строки и первого стол... more Матрицей-стрелкой называется матрица с нулями вне главной диагонали, первой строки и первого столбца. В работе исследуется пространство $M_{\operatorname{St}_n,\lambda}$ всех эрмитовых матриц-стрелок размера $(n+1)\times (n+1)$, имеющих заданный простой спектр $\lambda$. Доказано, что это пространство - гладкое $2n$-мерное многообразие с локально стандартным действием тора, описана топология и комбинаторика его пространства орбит. При $n\geqslant 3$ пространство орбит $M_{\operatorname{St}_n,\lambda}/T^n$ не является многогранником, а значит, $M_{\operatorname{St}_n,\lambda}$ не является квазиторическим многообразием. Тем не менее на $M_{\operatorname{St}_n,\lambda}$ имеется действие полупрямого произведения $T^n\rtimes\Sigma_n$ и его пространство орбит диффеоморфно специальному простому многограннику $\mathscr B^n$, который получается из куба срезкой граней коразмерности 2. При $n=3$ пространство орбит $M_{\operatorname{St}_3,\lambda}/T^3$ является полноторием, граница которого разбита регулярным образом на шестиугольники, что позволило описать кольца когомологий и эквивариантных когомологий шестимерного многообразия $M_{\operatorname{St}_3,\lambda}$ и еще одного многообразия - его двойника. Библиография: 32 названия.
arXiv (Cornell University), Jan 28, 2015
We consider the orbit type filtration on a manifold X with locally standard action of a compact t... more We consider the orbit type filtration on a manifold X with locally standard action of a compact torus and the corresponding homological spectral sequence pE X q r ,˚. If all proper faces of the orbit space Q " X{T are acyclic and free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms are equal to the h 1-numbers of the Buchsbaum simplicial poset S Q dual to Q. Betti numbers of X depend only on the orbit space Q but not on the characteristic function. If X is a slightly different object, namely the model space X " pPˆT n q{ " where P is a cone over Buchsbaum simplicial poset S, we prove that dimpE X q 8 p,p " h 2 p pSq. This gives a topological evidence for the fact that h 2-numbers of Buchsbaum simplicial posets are nonnegative.
arXiv (Cornell University), Apr 12, 2022
In this paper we prove that there exists an asymptotical diagonalization algorithm for a class of... more 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.
arXiv (Cornell University), Oct 10, 2010
Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial co... more Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial complex K P which carries complete information about the combinatorial type of P. In the case when P is simple, K P is the same as ∂P * , where P * is a polar dual polytope. Using the canonical embedding of a polytope P into nonnegative orthant R m , where m is a number of its facets, we introduce a moment-angle space Z P for a polytope P. It is known, that in the case of a simple polytope P the space Z P is homeomorphic to the moment-angle complex (D 2 , S 1) KP. When P is not simple, we prove that the space Z P is homotopically equivalent to the space (D 2 , S 1) KP. This allows to introduce bigraded Betti numbers for any convex polytope. A Stanley-Reisner ring of a polytope P can be defined as a Stanley-Reisner ring of a simplicial complex K P. All these considerations lead to a natural question: which simplicial complexes arise as K P for some polytope P ? We have proceeded in this direction by introducing a notion of a polytopic simplicial complex. It has the following property: link of each simplex in a polytopic complex is either contractible, or retractible to a subcomplex, homeomorphic to a sphere. The complex K P is a polytopic simplicial complex for any polytope P. Links of so called face simplices in a polytopic complex are polytopic complexes as well. This fact is sufficient enough to connect face polynomial of a simplicial complex K P to the face polynomial of a polytope P , giving a series of inequalities on certain combinatorial characteristics of P. Two of these inequalities are equalities for each P and represent Euler-Poincare formula and one of Bayer-Billera relations for flag f-numbers. In the case when P is simple all inequalities turn out to be classical Dehn-Sommerville relations.
Sbornik Mathematics, Sep 1, 2017
We consider a sheaf of exterior algebras on a simplicial poset S and introduce a notion of homolo... more We consider a sheaf of exterior algebras on a simplicial poset S and introduce a notion of homological characteristic function. Two objects are associated with these data: a graded sheaf I and a graded cosheaf p Π. When S is a homology manifold, we prove the isomorphism H n´1´p pS; Iq-H p pS; p Πq which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence E 2 p,q-H n´1´p pS; U n´1`q b Iq ñ H p`q pS; p Πq, where U˚is the local homology stack on S. This spectral sequence, in turn, extends Zeeman's spectral sequence in interpretation of McCrory. We apply these results to toric topology. Let X be an orientable manifold with locally standard action of a compact torus and acyclic proper faces of the orbit space. A principal torus bundle Y is associated with X and the orbit type filtration on X is covered by a topological filtration on Y. Then the second pages of homological spectral sequences associated with these two filtrations are isomorphic in many positions.
arXiv (Cornell University), Feb 4, 2015
Let X be a 2n-manifold with a locally standard action of a compact torus T n. If the free part of... more Let X be a 2n-manifold with a locally standard action of a compact torus T n. If the free part of action is trivial and proper faces of the orbit space Q are acyclic, then there are three types of homology classes in X: (1) classes of face submanifolds; (2) k-dimensional classes of Q swept by actions of subtori of dimensions ă k; (3) relative k-classes of Q modulo BQ swept by actions of subtori of dimensions ě k. The submodule of H˚pXq spanned by face classes is an ideal in H˚pXq with respect to the intersection product. It is isomorphic to pZrS Q s{Θq{W , where ZrS Q s is the face ring of the Buchsbaum simplicial poset S Q dual to Q; Θ is the linear system of parameters determined by the characteristic function; and W is a certain submodule, lying in the socle of ZrS Q s{Θ. Intersections of homology classes different from face submanifolds are described in terms of intersections on Q and T n .
arXiv (Cornell University), Mar 28, 2018
An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. W... more An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space M Stn,λ of Hermitian arrow pn`1qˆpn`1qmatrices with fixed simple spectrum λ. We prove this space to be a smooth 2n-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If n ě 3, the orbit space M Stn,λ {T n is not a polytope, hence M Stn,λ is not a quasitoric manifold. However, there is a natural permutation action on M Stn,λ which induces the combined action of a semidirect product T n¸Σ n. The orbit space of this large action is a simple polytope B n. The structure of this polytope is described in the paper. In case n " 3, the space M St3,λ {T 3 is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold M St3,λ using the general theory developed by the first author. This theory is also applied to a certain 6-dimensional manifold called the twin of M St3,λ. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.
arXiv (Cornell University), Dec 14, 2020
In this paper we introduce and study the topology of clique complexes of multigraphs without loop... more 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 Bjorner, 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, 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.
arXiv (Cornell University), Mar 11, 2022
In this paper we study general torus actions on manifolds with isolated fixed points from combina... more 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.
arXiv (Cornell University), Apr 18, 2022
Despite significant advances in the field of deep learning in applications to various fields, exp... more 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.
arXiv (Cornell University), Dec 25, 2019
Let a compact torus T " T n´1 act on an orientable smooth compact manifold X " X 2n effectively, ... more 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 pXq " 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 pn`1q-sphere. If, in addition, π 1 pXq " 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 pXq " 0, we prove that j-generality of weights implies pj`1q-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.
Transactions of the Moscow Mathematical Society, Apr 9, 2014
For a simplicial complex K on m vertices and simplicial complexes K 1 ,. .. , K m , we introduce ... more For a simplicial complex K on m vertices and simplicial complexes K 1 ,. .. , K m , we introduce a new simplicial complex K(K 1 ,. .. , K m), called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari, M. Bendersky, F. R. Cohen, and S. Gitler. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes P (P 1 ,. .. , P m), as introduced by G. Agnarsson. The substitution gives rise to an operad structure on the set of finite simplicial complexes in which a simplicial complex on m vertices is considered as an m-ary operation. We prove the following main results: (1) the complex K(K 1 ,. .. , K m) is a simplicial sphere if and only if K is a simplicial sphere and the K i are the boundaries of simplices, (2) the class of spherical nerve-complexes is closed under substitution, (3) multigraded betti numbers of K(K 1 ,. .. , K m) are expressed in terms of those of the original complexes K, K 1 ,. .. , K m. We also describe connections between the obtained results and the known results of other authors.
arXiv (Cornell University), Feb 15, 2014
Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus ac... more Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus actions on moment-angle complexes. In the paper we study the relation between Buchstaber invariants and classical invariants of simplicial complexes such as bigraded Betti numbers and chromatic invariants. The following two statements are proved. (1) There exists a simplicial complex U such that s(U) = s R (U). (2) There exist two simplicial complexes with equal bigraded Betti numbers and chromatic numbers, but different Buchstaber invariants. To prove the first theorem we define Buchstaber number as a generalized chromatic invariant. This approach allows to guess the required example. The task then reduces to a finite enumeration of possibilities which was done using GAP computational system. To prove the second statement we use properties of Taylor resolutions of face rings.
arXiv (Cornell University), Jan 18, 2013
For a simplicial complex K on m vertices and simplicial complexes K 1 ,. .. , Km a composed simpl... more For a simplicial complex K on m vertices and simplicial complexes K 1 ,. .. , Km a composed simplicial complex K(K 1 ,. .. , Km) is introduced. This construction generalizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of polytopes P (P 1 ,. .. , Pm) defined by G. Agnarsson in most important cases. The composition defines a structure of an operad on a set of finite simplicial complexes, in which a complex on m vertices is viewed as an m-adic operation. We prove the following: (1) a composed complex K(K 1 ,. .. , Km) is a simplicial sphere iff K is a simplicial sphere and K i are the boundaries of simplices; (2) a class of spherical nerve-complexes is closed under the operation of composition (3) finally, we express multigraded Betti numbers of K(K 1 ,. .. , Km) in terms of multigraded Betti numbers of K, K 1 ,. .. , Km using a composition of generating functions.
Proceedings of the Steklov Institute of Mathematics, 2015
We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorp... more We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties.
arXiv (Cornell University), May 12, 2019
Let the compact torus T n´1 act on a smooth compact manifold X 2n effectively with nonempty finit... more 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 pX 2n ; Zq " 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 3 L. 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 .
arXiv (Cornell University), Mar 20, 2022
For an equivariantly formal action of a compact torus T on a smooth manifold X with isolated fixe... more 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 SpXq of face submanifolds. We prove that the condition of j-independency of tangent weights at each fixed point implies pj`1q-acyclicity of the skeleta SpXq r for r ą 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 pn´1q-independent action of pn´1q-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.
arXiv (Cornell University), Mar 26, 2022
In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-P... more In this paper we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del Río-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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Papers by Anton Ayzenberg