Related papers: The Dowker theorem via discrete Morse theory
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 show that a derived bi-duality dg-module is quasi-isomorphic to the homotopy limit of a certain tautological functor. This is a simple observation, which seems to be true in wider context. From the view point of derived Gabriel topology,…
We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a…
We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped…
In 1962, H. de Vries proved a duality theorem for the category {\bf HC} of compact Hausdorff spaces and continuous maps. The composition of the morphisms of the dual category obtained by him differs from the set-theoretic one. Here we…
Liouville's theorem says that in dimension greater than two, all conformal maps are M\"obius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally…
We show that, up to Morita equivalence, any finite-dimensional algebra with a suitable homological system, admits an exact Borel subalgebra. This generalizes a theorem by Koenig, K\"ulshammer and Ovsienko, which holds for quasi-hereditary…
We propose a duality between quiver gauge theories and the combinatorics of dimer models. The connection is via toric diagrams together with multiplicities associated to points in the diagram (which count multiplicities of fields in the…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
The hammock localization provides a model for a homotopy function complex in any Quillen model category. We prove that a homotopy between a pair of morphisms induces a homotopy between the maps induced by taking the hammock localization. We…
Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over…
We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular…
In this paper, we summarize a general method of transforming DG structures into higher structures on the various complexes related to the reduced bar resolution of a given quiver algebra using algebraic Morse theory. As an application, we…
Robin Forman's highly influential 2002 paper A User's Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract…
The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing…
We consider all possible dynamical theories which evolve two transverse vector fields out of a three-dimensional Euclidean hyperplane, subject to only two assumptions: (i) the evolution is local in space, and (ii) the theory is invariant…
We construct a model structure on the category of ordered simplicial complexes, Quillen equivalent to the standard model structure on simplicial sets. This shows that simplicial complexes, which are fully combinatorial in nature, provide a…
Generalizing Duality Theorem of H. de Vries, we define a category which is dually equivalent to the category of all locally compact Hausdorff spaces and all perfect maps between them.
We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics, in which the Schr\"{o}dinger equation is a difference equation. It reproduces all the known ones whose…
We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of $C_2$, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside…