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In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by…
This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point…
By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary…
In this paper, motivated by symplectic topology, we explore categorical entropy and present two main results. The first result establishes a relation between categorical entropies of functors on a category and its localization.…
In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to…
We find that simple neural networks with ReLU activation generate polytopes as an approximation of a unit sphere in various dimensions. The species of polytopes are regulated by the network architecture, such as the number of units and…
We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept…
We introduce the notion of smooth cell complexes and its subclass consisting of gathered cell complexes within the category of diffeological spaces (cf. Definitions 1 and 3). It is shown that the following hold. (1) With respect to the…
The cell complex structure is one of the most fundamental structures in topology and combinatorics, the Morse decomposition of a dynamical system analyzes the global gradient behavior, and the Reeb graph of a function is an elementary tool…
Type families on higher inductive types such as pushouts can capture homotopical properties of differential geometric constructions including connections, curvature, and vector fields. We define a class of pushouts based on simplicial…
The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2)…
Volumetric brain reconstructions provide an unprecedented opportunity to gain insights into the complex connectivity patterns of neurons in an increasing number of organisms. Here, we model and quantify the complexity of the resulting…
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group…
We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $\kappa$) in the intrinsic length metric. The reconstructed spaces are in the form of…
In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model…
Homotopy links have proven to be one of the most powerful tools of stratified homotopy theory. In previous work, we described combinatorial models for the generalized homotopy links of a stratified simplicial set. For many purposes, in…
In 1975 Wegner conjectured that the nerve of every finite good cover in R^d is d-collapsible. We disprove this conjecture. A good cover is a collection of open sets in R^d such that the intersection of every subcollection is either empty or…
The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of $G$. While independence complexes have been studied extensively for many graph classes, including square grid graphs, relatively…
For a commutative ring $R$, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of $R$ (the frame of radical ideals in $R$) and its Hochster dual frame, as lattices in the…
The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar $\phi^3$ theory, from all nestings and chainings of a primitive self-energy…