Related papers: Combinatorics of combinatorial topology
We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated…
There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and…
For a discrete poset $\mathcal X$ McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$ from the order complex to the poset equipped with the Up topology is a weak homotopy equivalence. Much later, Zivaljevi\'{c} defined the…
This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and…
Hypertrees and noncrossing trees are well-established objects in the combinatorics literature, but the hybrid notion of a noncrossing hypertree has received less attention. In this article I investigate the poset of noncrossing hypertrees…
We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Forman's…
This thesis presents new applications of Gale duality to the study of polytopes, point configurations and oriented matroids with extremal combinatorial properties. The first part of the thesis explores construction techniques for neighborly…
Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…
We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a…
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the…
We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary…
We study classes of objects whose combinatorics are closely related to those of posets. The framework of operads and operad algebras allows us to make this relationship precise and provides tools for a deeper understanding of their…
We use model theoretic techniques to construct explicit first-order axiomatizations for the classes of posets that can be represented as systems of sets, where the order relation is given by inclusion, and existing meets and joins of…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we…
A simple convex polytope $P$ is \emph{cohomologically rigid} if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $P$. Not every $P$ has this property, but some important polytopes such as…
We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general…
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like ``holonomy'', ``parallel transport'', ``bundles'', ``combinatorial curvature'' etc. in the context of simplicial (polyhedral) complexes, posets,…