Related papers: Homology Representations Arising from a Hypersimpl…
We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational…
While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising…
To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW…
Polytope complexes are the generalisation of polygon meshes in geo-information systems (GIS) to arbitrary dimension, and a natural concept for accessing spatio-temporal information. Complexes of each dimension have a straight-forward…
This is an expository article. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via…
The lattice of intersections of reflecting hyperplanes of a complex reflection group W may be considered as the poset of 1-eigenspaces of the elements of W. In this paper we replace 1 with an arbitrary eigenvalue and study the topology and…
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our…
Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the…
In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a…
Persistent homology theory is a relatively new but powerful method in data analysis. Using simplicial complexes, classical persistent homology is able to reveal high dimensional geometric structures of datasets, and represent them as…
In this note we connect the language of Bessis's Garisde categories with Salvetti's metrical-hemisphere complexes in order to find new examples of weak Garside groups. As our main example, we show that the fundamental group of the…
Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they…
Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by…
Consider the quotient $G/B$ of a simple matrix Lie group $G$ by a subgroup $B$ isomorphic to a direct product of some of $S^1$s and $S^3$s such that its adjoint representation can be extended over $G$. Then it naturally inherits a stable…
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…
We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large…
The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the…
We derive an obstruction to representing a homology class of a symplectic 4-manifold by an embedded, possibly disconnected, symplectic surface.
To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study…
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…