Related papers: Boundary complexes and weight filtrations
We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…
In this paper, we take advantage of a reinterpretation of differential modules admitting a flag structure as a special class of perturbations of complexes. We are thus able to leverage the machinery of homological perturbation theory to…
In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a cohomology of a log Calabi--Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual…
We provide lower bounds on the connectivity of the independence complexes of hypergraphs. Additionally, we compute the homotopy types of the independence complexes of $d$-uniform properly-connected triangulated hypergraphs.
We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal…
This paper is a synthesis and extension of three earlier papers on $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Our goal is to show that the homotopy types of such complexes are determined…
On conformally compact manifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and extend aspects of already known…
We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact…
We study canonical filtrations of finite-dimensional associative algebras and Lie algebras. These filtrations are defined via optimal destabilizing one-parameter subgroups in the sense of geometric invariant theory (GIT), and appear to be a…
Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial…
We describe the notion of a \emph{weighting} along a submanifold $N\subset M$, and explore its differential-geometric implications. This includes a detailed discussion of weighted normal bundles, weighted deformation spaces, and weighted…
We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.
Factorization homology theories of topological manifolds, after Beilinson, Drinfeld and Lurie, are homology-type theories for topological $n$-manifolds whose coefficient systems are $n$-disk algebras or $n$-disk stacks. In this work we…
We study Tate-Vogel and relative cohomologies of complexes by applying the model structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) of modules. We show first that the class of complexes admitting a complete $\A$…
A challenge in computational topology is to deal with large filtered geometric complexes built from point cloud data such as Vietoris-Rips filtrations. This has led to the development of schemes for parallel computation and compression…
We study open-closed discrete mappings that satisfy the weighted estimate of the distortion of modulus of families of paths. It is proved that the mappings mentioned above have a continuous extension into the isolated point of the boundary,…
In this paper we study a new notion of category weight of homology classes developing further the ideas of E. Fadell and S. Husseini. In the case of closed smooth manifolds the homological category weight is equivalent to the cohomological…
This paper explores the irreducible uniformly bounded weight modules of map extended Special Lie algebras and map extended Hamiltonian Lie algebras under some condition on the action of the Laurent polynomial ring A_{N}.
We define and construct mixed Hodge structures on real schematic homotopy types of complex projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on…
We introduce new techniques to study the differential complexes associated to tube structures on $M \times \mathbb{T}^m$ of corank $m$, in which $M$ is a compact manifold and $\mathbb{T}^m$ is the $m$-torus. By systematically employing…