arXiv subjects

Denis Lyskov

Publications and source records attributed to Denis Lyskov.

5 recordsLinked to original sources

arXiv2025

Graphical configuration spaces, Contractads and Formality

Given a finite simple connected graph $\Gamma$, the graphical configuration space $\mathrm{Conf}_{\Gamma}(X)$ is the space of collections of points in $X$ indexed by the vertices of $\Gamma$, where points corresponding to adjacent vertices must be distinct. When $X=\mathbb{R}^d$ and the points are replaced by small disks, the resulting spaces for all possible graphs fit together into an algebraic structure that extends the little disks operad, called the little disks contractad $\mathcal{D}_d$. In this paper, we investigate the homotopical and algebraic properties of the little disks contractad $\mathcal{D}_d$. We construct and study Fulton-MacPherson compactifications of graphical configuration spaces, which provide a convenient model for $\mathcal{D}_d$ within the class of compact manifolds with boundary. Using these and wonderful compactifications, we prove that $\mathcal{D}_d$ is formal in the category of (Hopf) contractads for $d=1$, $d=2$, and for chordal graphs for any $d$. We also identify the first obstructions to coformality in the case of cyclic graphs. In addition, we give a combinatorial description of the cell structure of $\mathcal{D}_2$ and present applications to the study of graphical configuration spaces $\mathrm{Conf}_{\Gamma}(X)$ using the language of twisted algebras.

math.AT↗

arXiv2024

Operadic structure on Hamiltonian paths and cycles

We study Hamiltonian paths and cycles in undirected graphs from an operadic viewpoint. We show that the graphical collection $\mathsf{Ham}$ encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a contractad. Similarly, we construct the graphical collection of Hamiltonian cycles $\mathsf{CycHam}$ that forms a right module over the contractad $\mathsf{Ham}$. We use the machinery of contractad generating series for counting Hamiltonian paths/cycles for particular types of graphs.

math.CO↗

arXiv2024

Hilbert series for contractads and modular compactifications

Contractads are operadic-type algebraic structures well-suited for describing configuration spaces indexed by a simple connected graph $\Gamma$. Specifically, these configuration spaces are defined as $\mathrm{Conf}_{\Gamma}(X):=X^{|V(\Gamma)|}\setminus \cup_{(ij)\in E(\Gamma)} \{x_i=x_j\}$. In this paper, we explore functional equations for the Hilbert series of Koszul dual contractads and provide explicit Hilbert series for fundamental contractads such as the commutative, Lie, associative and the little discs contractads. Additionally, we focus on a particular contractad derived from the wonderful compactifications of $\mathrm{Conf}_{\Gamma}(\mathbb{k})$, for $\mathbb{k}=\mathbb{R},\mathbb{C}$. First, we demonstrate that for complete multipartite graphs, the associated wonderful compactifications coincide with the modular compactifications introduced by Smyth. Second, we establish that the homology of the complex points and the homology of the real locus of the wonderful contractad are both quadratic and Koszul contractads. We offer a detailed description of generators and relations, extending the concepts of the Hypercommutative operad and cacti operads, respectively. Furthermore, using the functional equations for the Hilbert series, we describe the corresponding Hilbert series for the homology of modular compactifications.

math.QA↗

arXiv2023

A generalization of operads based on subgraph contractions

We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\overline{\mathcal{M}}_{0,n+1}$ admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gr\"obner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.

math.AT↗

arXiv2022

Reconnectads

We introduce a new operad-like structure that we call a reconnectad; the ``input'' of an element of a reconnectad is a finite simple graph, rather than a finite set, and ``compositions'' of elements are performed according to the notion of the reconnected complement of a subgraph. The prototypical example of a reconnectad is given by the collection of toric varieties of graph associahedra of Carr and Devadoss, with the structure operations given by inclusions of orbits closures. We develop the general theory of reconnectads, and use it to study the ``wonderful reconnectad'' assembled from homology groups of complex toric varieties of graph associahedra.

math.CT↗