Stratified Spaces Research Articles

Foliated stratified spaces and a De Rham complex describing intersection space cohomology

The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with stratification depth 1 and flat link bundles, the present paper provides a de Rham-theoretic description of the theory HI by a complex of global smooth differential forms on the top stratum. We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. Flat link bundles arise for example in foliated stratified spaces and in reductive Borel-Serre compactifications of locally symmetric spaces. A precise topological definition of the notion of a stratified foliation is given.

Monotone covering properties and properties they imply

We study properties of spaces that were proven in an earlier paper of Chase and Gruenhage (2013) [5] to follow from monotonic metacompactness. We show that all of the results of that earlier paper that follow from the monotonic covering property follow just from these weaker properties. The results we obtain are either strengthenings of earlier results or are new even for the monotonic covering property. In particular, some corollaries are that monotonically countably metacompact spaces are hereditarily metacompact, and separable monotonically countably metacompact spaces are metrizable. It follows that the well-known examples of stratifiable spaces given by McAuley and Ceder are not monotonically countably metacompact; we show that they are not monotonically meta-Lindelöf either. Finally, we answer a question of Gartside and Moody by exhibiting a stratifiable space which is monotonically paracompact in the locally finite sense, but not monotonically paracompact in the sense of Gartside and Moody.

THE PROVISION OF VERTICAL SOCIAL POCKETS FOR BETTER SOCIAL INTERACTION IN HIGH-RISE LIVING

High-rise living makes it difficult for social communities to form despite the provision of several dedicated social spaces at ground, podium, mid- and roof-levels, partially due to the sheer number of residents per block. The pull of easily-accessed, solitary recreational activities such as surfing the Internet and watching television further exacerbates the problem. Social spaces need to be brought closer to residents to promote social interaction. This paper proposes to improve social interaction by supplementing currently stratified social spaces with vertically-connected social hubs using existing transitional spaces in high-rise living, such as the lift lobby, to create micro-communities comprising the residents of each respective floor in a cost-effective way. Previous research indicated that strong communities look after each other, indicating that the creation of microcommunities will create an effect not unlike the defensible space theory. Therefore, this paper analyses residential high-rise layouts based on chosen case studies in Malaysia, then suggests several possible design outcomes that turn the lift lobby into social spaces at every floor level.

THE PROVISION OF VERTICAL SOCIAL POCKETS FOR BETTER SOCIAL INTERACTION IN HIGH-RISE LIVING

High-rise living makes it difficult for social communities to form despite the provision of several dedicated social spaces at ground, podium, mid- and roof-levels, partially due to the sheer number of residents per block. The pull of easily-accessed, solitary recreational activities such as surfing the Internet and watching television further exacerbates the problem. Social spaces need to be brought closer to residents to promote social interaction. This paper proposes to improve social interaction by supplementing currently stratified social spaces with vertically-connected social hubs using existing transitional spaces in high-rise living, such as the lift lobby, to create micro-communities comprising the residents of each respective floor in a cost-effective way. Previous research indicated that strong communities look after each other, indicating that the creation of microcommunities will create an effect not unlike the defensible space theory. Therefore, this paper analyses residential high-rise layouts based on chosen case studies in Malaysia, then suggests several possible design outcomes that turn the lift lobby into social spaces at every floor level.

Completely prime L-filters, irreducible L-filters and sobriety

In this paper, for a frame L, we characterize modified sobriety in stratified L-topological spaces and strong L-topological spaces internally using certain classes of stratified L-filters. While the first characterization using completely prime L-filters is trivial and applies to all stratified L-topological spaces, the other one using irreducible L-filters generalizes an approach of R.E. Hoffmann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strong L-topological spaces.

Factorization homology of stratified spaces

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg–Steenrod axioms; it can also be viewed as an analogue of the Baez–Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topological quantum field theory. Using these axioms, we extend the nonabelian Poincare duality of Salvatore and Lurie to the setting of stratified spaces—this is a nonabelian version of the Poincare duality given by intersection homology. We pay special attention to the simple case of singular manifolds whose singularity datum is a properly embedded submanifold, and give a further simplified algebraic characterization of these homology theories. In the case of 3-manifolds with one-dimensional submanifolds, these structures give rise to knot and link homology theories.

Analysis on singular spaces: Lie manifolds and operator algebras

We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds--called "Lie manifolds"--that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here--work that spans over close to two decades--was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.

Subcategories of the category of L-convex spaces

In this paper, several types of L-convex spaces are introduced, including stratified L-convex spaces, convex-generated L-convex spaces, weakly induced L-convex spaces and induced L-convex spaces. Their relations are discussed category-theoretically. Firstly, it is shown that there is a Galois correspondence between the category SL-CS of stratified L-convex spaces (resp. the category WIL-CS of weakly induced L-convex spaces) and the category L-CS of L-convex spaces. In particular, SL-CS and WIL-CS are both coreflective subcategories of L-CS. Secondly, it is proved that there is a Galois correspondence between the category CS of convex spaces and the category SL-CS (resp. WIL-CS). Specially, CS can be embedded into SL-CS and WIL-CS as a coreflective subcategory. Finally, it is shown that the category CGL-CS of convex-generated L-convex spaces, the category IL-CS of induced L-convex spaces and CS are isomorphic.

The spaces between: Children, teachers, researchers, artists

Abstract Direct engagement with children, teachers and the curriculum that brings them together in pedagogical relationships figure prominently in my work with beginning teachers and researchers. A dialogical approach to teaching, celebrating the contingencies of each class and each situation, leads to a vital and constantly challenging engagement with the process of educating as reciprocal and open-ended. Grounded in philosophies of reciprocity and relationship, this way of being in the university respects the rhizomatic nature of knowing and becoming, eschewing the overly determined and stratified spaces of pre-ordained and standardized curriculum in favour of the emergent and unpredictable.

L-quantum spaces

In this paper, based on a complete residuated lattice L, we introduce the definitions of L-quantum spaces and continuous mappings. Then we establish an adjunction between the category of stratified L-quantum spaces and the opposite category of two-sided L-quantales. We also prove that the category of sober L-quantum spaces is dually equivalent to the category of spatial two-sided L-quantales.

Semi-stratifiable spaces with monotonically normal compactifications

In this paper we use Mary Ellen Rudin's solution of Nikiel's problem to investigate metrizability of certain subsets of compact monotonically normal spaces. We prove that if H is a semi-stratifiable space that can be covered by a σ-locally-finite collection of closed metrizable subspaces and if H embeds in a monotonically normal compact space, then H is metrizable. It follows that if H is a semi-stratifiable space with a monotonically normal compactification, then H is metrizable if it satisfies any one of the following: H has a σ-locally finite cover by compact subsets; H is a σ-discrete space; H is a scattered; H is σ-compact. In addition, a countable space X has a monotonically normal compactification if and only if X is metrizable. We also prove that any semi-stratifiable space with a monotonically normal compactification is first-countable and is the union of a family of dense metrizable subspaces. Having a monotonically normal compactification is a crucial hypothesis in these results because R.W. Heath has given an example of a countable non-metrizable stratifiable (and hence monotonically normal) group. We ask whether a first-countable semi-stratifiable space must be metrizable if it has a monotonically normal compactification. This is equivalent to “If X is a first-countable stratifiable space with a monotonically normal compactification, must H be metrizable?”

Fruit Flies and Moduli: Interactions between Biology and Mathematics

Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.

Mini-Workshop: Asymptotic Statistics on Stratified Spaces

Statistical analysis of non-Euclidean data such as data on manifolds is an active and established topic of research, for instance, in the statistical analysis of shape. However, many types of data naturally reside in metric spaces which are not smooth manifolds as a whole, rather they are unions of manifold strata of varying dimensions. These spaces form a key general family of geometric spaces for data analysis. Statistics in stratified spaces has recently found great interest in applications and mathematical theory building. While the fundamental theory is still in its beginnings, as a centerpiece the derivation and investigation of statistics and their asymptotics has materialized. Only a few basic results are known, but it is clear that the geometric constraints imposed by stratified spaces lead to unexpected asymptotic behavior of standard statistical properties, such as “stickiness” of means, see [4]. It is the scope of the proposed workshop to better understand fundamental relations between asymptotic behavior of statistical descriptors and global as well as local geometric and topological structures. This investigation calls for an intense collaboration of the fields involved: statistics & stochastics; geometry & topology; combinatorics, algorithms & numerics. This workshop sought to bring together world-leading scientists and high-potential early career researchers working in this field to collaborate on a focused set of fundamental questions.

On induced I-fuzzy topogenous spaces generated by Lowen factors

Three Lowen factors, namely, ω ,()a ndι, are extended into I-fuzzy topogenous spaces and their properties are studied. Then concepts of induced I-fuzzy topogenous spaces, weakly induced I-fuzzy topogenous spaces and stratified I-fuzzy topogenous spaces are introduced and characterized by Lowen factors. Also, relations among these spaces can also be preserved in their quotient spaces and subspaces.

Some Properties of Topological Spaces Related to the Local Density and the Local Weak Density

In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for stratifiable spaces the local density and the local weak density coincide, these cardinal numbers are preserved under open mappings, are inverse invariant of a class of closed irreducible mappings. Moreover, it is showed that the functor of probability measures of finite supports preserves the local density of compacts.

Stratified LMN-convergence tower spaces

We develop a general theory of convergence for lattice-valued spaces based on the concept of s-stratified LM-filters. For different choices of the frames L and M, different kinds of filters arise, and suitable choices of the lattice N allow to view many existing fuzzy and probabilistic convergence spaces as special examples of our stratified LMN-convergence tower spaces. The stratification requires a certain relation between the frames L and M and we use a so-called stratification mapping to this end. The stratification condition is used to show that the resulting category is fiber-small and hence Cartesian closedness is equivalent to the existence of natural function spaces. We give several examples for our spaces for different choices of the lattices L, M and N with a special focus on enriched LM-fuzzy topological spaces. Finally, we study diagonal axioms and regularity.

Hölder regularity of solutions for Schrödinger operators on stratified spaces

We study the regularity properties for solutions of a class of Schrödinger equations ( Δ + V ) u = 0 on a stratified space M endowed with an iterated edge metric. The focus is on obtaining optimal Hölder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential.

On the power-set $$Q$$ Q -algebras

$$Q$$ -algebras form an important class of ordered algebraic structures, which can be regarded as a generalization of quantales and $$Q$$ -modules, and play an important role in the study of lattice-valued quantales, lattice-valued frames and stratified lattice-valued topological spaces. Every $$Q$$ -algebra is isomorphic to a quotient $$Q$$ -algebra of some power-set $$Q$$ -algebra. We investigate some properties of power-set $$Q$$ -algebras, and, by means of the relations between ordered semigroups, give a general characterization for the strong homomorphisms between power-set $$Q$$ -algebras.

Totally normal cellular stratified spaces and applications to the configuration space of graphs

The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced in \cite{Bas+}, \cite{Tama} by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's theorem on the homotopy dimension of the configuration space of graphs. We also make sample calculations of the fundamental group of ordered and unordered configuration spaces of two points for small graphs.

Sharp parabolic regularity and geometric flows on singular spaces

This is a brief survey about regularity expansions for solutions of elliptic and parabolic problems on spaces with conic singularities. The results themselves are closely related to classical results about elliptic boundary problems, and analogues of these are expected to hold on quite general stratified spaces with incomplete iterated edge metrics. The emphasis here is on the interpretation and application of these expansions to geometric problems.