arXiv subjects

Dmitry N. Kozlov

Publications and source records attributed to Dmitry N. Kozlov.

At least 19 recordsLinked to original sources

arXiv2024

Homology and Euler characteristic of generalized anchored configuration spaces of graphs

In this paper we consider the generalized anchored configuration spaces on $n$ labeled points on a~graph. These are the spaces of all configurations of $n$ points on a~fixed graph $G$, subject to the condition that at least $q$ vertices in some pre-determined set $K$ of vertices of $G$ are included in each configuration. We give a non-alternating formula for the Euler characteristic of such spaces for arbitrary connected graphs, which are not trees. Furthermore, we completely determine the homology groups of the generalized anchored configuration spaces of $n$ points on a circle graph.

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arXiv2023

Stirling complexes

In this paper we study natural reconfiguration spaces associated to the problem of distributing a fixed number of resources to labeled nodes of a tree network, so that no node is left empty. These spaces turn out to be cubical complexes, which can be thought of as higher-dimensional geometric extensions of the combinatorial Stirling problem of partitioning a set of named objects into non-empty labeled parts. As our main result, we prove that these Stirling complexes are always homotopy equivalent to wedges of spheres of the same dimension. Furthermore, we provide several combinatorial formulae to count these spheres. Somewhat surprisingly, the homotopy type of the Stirling complexes turns out to depend only on the number of resources and the number of the labeled nodes, not on the actual structure of the tree network.

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arXiv2023

Configuration spaces of labeled points on a circle with two anchors

In this paper we calculate the homology of configuration spaces of $n$ points on a circle, subject to the condition that two pre-determined points are included in the configuration. We make use of discrete Morse theory both to determine the Betti numbers, as well as to provide an explicit combinatorial description of the bases both for homology and cohomology.

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arXiv2018

Quantitative aspects of acyclicity

We study several aspects of the $k$-th Cheeger constant of a complex X, a parameter that quantifies the distance of $X$ from a complex $Y$ with nontrivial $k$-th cohomology over $\mathbb{Z}_2$. Our results include general methods for bounding the cosystolic norm of a cochain and for bounding the Cheeger constant of a complex, a discussion of expansion of pseudomanifolds and geometric lattices, probabilistic upper bounds on Cheeger constants, and application of non-Abelian expansion to random complexes.

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arXiv2018

Topology of scrambled simplices

In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a~standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a $Δ$-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a large family of both $Δ$-complexes, as well as simplicial complexes, which are contractible, but not collapsible.

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arXiv2017

Witness structures and immediate snapshot complexes

In this paper we introduce and study a new family of combinatorial simplicial complexes, which we call immediate snapshot complexes. Our construction and terminology is strongly motivated by theoretical distributed computing, as these complexes are combinatorial models of the standard protocol complexes associated to immediate snapshot read/write shared memory communication model. In order to define the immediate snapshot complexes we need a new combinatorial object, which we call a witness structure. These objects are indexing the simplices in the immediate snapshot complexes, while a special operation on them, called ghosting, describes the combinatorics of taking simplicial boundary. In general, we develop the theory of witness structures and use it to prove several combinatorial as well as topological properties of the immediate snapshot complexes.

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arXiv2017

Structure theory of flip graphs with applications to Weak Symmetry Breaking

This paper is devoted to advancing the theoretical understanding of the iterated immediate snapshot (IIS) complexity of the Weak Symmetry Breaking task (WSB). Our rather unexpected main theorem states that there exist infinitely many values of n, such that WSB for n~processes is solvable by a certain explicitly constructed 3-round IIS protocol. In particular, the minimal number of rounds, which an IIS protocol needs in order to solve the WSB task, does not go to infinity, when the number of processes goes to infinity. Our methods can also be used to generate such values of n. We phrase our proofs in combinatorial language, while avoiding using topology. To this end, we study a~certain class of graphs, which we call flip graphs. These graphs encode adjacency structure in certain subcomplexes of iterated standard chromatic subdivisions of a simplex. While keeping the geometric background in mind for an additional intuition, we develop the structure theory of matchings in flip graphs in a purely combinatorial way. Our bound for the IIS complexity is then a corollary of this general theory. As an afterthought of our result, we suggest to change the overall paradigm. Specifically, we think, that the bounds on the IIS complexity of solving WSB for n processes should be formulated in terms of the size of the solutions of the associated Diophantine equation, rather than in terms of the value n itself.

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arXiv2016

All binomial identities are orderable

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on the round complexity of the weak symmetry breaking task. Furthermore, we introduce the notion of a fundamental binomial identity and find an infinite family of values, other than the prime powers, for which no fundamental binomial identity can exist.

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arXiv2016

Topology of the immediate snapshot complexes

The immediate snapshot complexes were introduced as combinatorial models for the protocol complexes in the context of theoretical distributed computing. In the previous work we have developed a formal language of witness structures in order to define and to analyze these complexes. In this paper, we study topology of immediate snapshot complexes. It was known that these complexes are always pure and that they are pseudomanifolds. Here we prove two further independent topological properties. First, we show that immediate snapshot complexes are collapsible. Second, we show that these complexes are homeomorphic to closed balls. Specifically, given any immediate snapshot complex $P(\tr)$, we show that there exists a homeomorphism $φ:\da^{|\supp\tr|-1}\ra P(\tr)$, such that $φ(σ)$ is a subcomplex of $P(\tr)$, whenever $σ$ is a simplex in the simplicial complex $\da^{|\supp\tr|-1}$.

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arXiv2014

Topology of the view complex

In this paper we consider a family of simplicial complexes, which we call the view complexes. Our choice of objects of study is motivated by theoretical distributed computing, since the view complex is a key simplicial construction used for protocol complexes in the snapshot computational model. We show that the view complex $\view$ can be collapsed to the well-known complex $χ(Δ^n)$, called standard chromatic subdivision of a simplex, and that $χ(Δ^n)$ is itself collapsible. Furthermore, we show that the collapses can be performed simultaneously in entire orbits of the natural symmetric group action. Our results yield a purely combinatorial and constructive understanding of the topology of view complexes, at the same time as they enhance our knowledge about the standard chromatic subdivision of a simplex.

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arXiv2014

Weak symmetry breaking and abstract simplex paths

Motivated by questions in theoretical distributed computing, we develop the combinatorial theory of abstract simplex path subdivisions. Our main application is a short and structural proof of the theorem of Castaneda and Rajsbaum. This theorem in turn implies the solvability of the weak symmetry breaking task in the immediate snapshot wait-free model in the case when the number of processes is not a power of a prime number.

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arXiv2014

Standard protocol complexes for the immediate snapshot read/write model

In this paper we consider a family of abstract simplicial complexes which we call immediate snapshot complexes. Their definition is motivated by theoretical distributed computing. Specifically, these complexes appear as protocol complexes in the general immediate snapshot execution model. In order to define and to analyze the immediate snapshot complexes we use the novel language of witness structures. We develop the rigorous mathematical theory of witness structures and use it to prove several combinatorial as well as topological properties of the immediate snapshot complexes. In particular, we prove that these complexes are simplicially homeomorphic to simplices.

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arXiv2009

The threshold function for vanishing of the top homology group of random $d$-complexes

For positive integers $n$ and $d$, and the probability function $0\leq p(n)\leq 1$, we let $Y_{n,p,d}$ denote the probability space of all at most $d$-dimensional simplicial complexes on $n$ vertices, which contain the full $(d-1)$-dimensional skeleton, and whose $d$-simplices appear with probability $p(n)$. In this paper we determine the threshold function for vanishing of the top homology group in $Y_{n,p,d}$, for all $d\geq 1$.

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arXiv2008

Topology of moduli spaces of tropical curves with marked points

In this paper we study topology of moduli spaces of tropical curves of genus $g$ with $n$ marked points. We view the moduli spaces as being imbedded in a larger space, which we call the {\it moduli space of metric graphs with $n$ marked points.} We describe the shrinking bridges strong deformation retraction, which leads to a substantial simplification of all these moduli spaces. In the rest of the paper, that reduction is used to analyze the case of genus 1. The corresponding moduli space is presented as a quotient space of a torus with respect to the conjugation ${\mathbb Z}_2$-action; and furthermore, as a homotopy colimit over a simple diagram. The latter allows us to compute all Betti numbers of that moduli space with coefficients in ${\mathbb Z}_2$.

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arXiv2008

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces $MG_{1,n}^v$, which parametrize the isometry classes of metric graphs of genus 1 with $n$ marks on vertices are homotopy equivalent to the spaces $TM_{1,n}$, which are the moduli spaces of tropical curves of genus 1 with $n$ marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.

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arXiv2008

Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits

The main characters of this paper are the moduli spaces $TM_{g,n}$ of rational tropical curves of genus $g$ with $n$ marked points, with $g\geq 2$. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces $X_{g,n}$, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes $Δ_g$, with the latter being interesting on their own right. We use these homotopy colimit representations to describe a CW complex decomposition for each $X_{g,n}$. Furthermore, we use these developments, coupled with some standard tools for working with homotopy colimits, to perform an in-depth analysis of special cases of genus 2 and 3, gaining a complete understanding of the moduli spaces $X_{2,0}$, $X_{2,1}$, $X_{2,2}$, and $X_{3,0}$, as well as a partial understanding of other cases, resulting in several open questions and in further conjectures.

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arXiv2006

Homology tests for graph colorings

We describe a simple homological test for obstructions to graph colorings. The main idea is to combine the framework of Hom-complexes with the following general fact: an arbitrary Z_2-space has nontrivial homology with Z_2-coefficients in the dimension equal to its Stiefel-Whitney height. Actually, as a result we have a whole family of homology tests, one for each test graph. In general, these tests will give different answers, depending heavily on the choice of the test graph. We illustrate this phenomenon with some examples.

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arXiv2006

Cohomology of colorings of cycles

We compute the cohomology groups of the spaces of colorings of cycles, i.e., of the prodsimplicial complexes Hom(C_m,K_n). We perform the computation first with Z_2, and then with integer coefficients. The main technical tool is to use spectral sequences in conjunction with a detailed combinatorial analysis of a family of cubical complexes, which we call torus front complexes. As an application of our method, we demonstrate how to collapse each connected component of Hom(C_m,C_n) onto a garland of cubes.

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