Topological applications of Stanley-Reisner rings of simplicial complexes

Айзенберг, Антон Андреевич

doi:10.1090/s0077-1554-2013-00200-9

Cited by 3 publications

(3 citation statements)

References 21 publications

(68 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…cdim Z K J for flag K. In fact, this identity holds for any simplicial complex. It can be obtained by applying the combinatorial Alexander duality (see [BP,Corollary 2.4.6]) to the results of Aizenberg [Ai,Proposition 3.2]. He proved that the s-link-acyclic simplicial complexes are exactly s-subcomplexacyclic complexes.…”

Section: Multigraded Poincaré Series Of Pontryagin Algebras and Lie S...mentioning

confidence: 99%

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

Вылегжанин¹

2022

Preprint

View full text Add to dashboard Cite

For any flag simplicial complex K, we describe the multigraded Poincaré series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex Z K . We compute the LS-category of Z K for flag complexes and give a lower bound in the general case. The key observation is that the Milnor-Moore spectral sequence collapses at the second sheet for flag K.We also show that the results of Panov and Ray about the Pontryagin algebras of Davis-Januszkiewicz spaces are valid for arbitrary ring coefficients, and introduce the Z × Z m ≥0 -grading on the Pontryagin algebras similar to the multigrading on the cohomology of Z K .

show abstract

Section: Multigraded Poincaré Series Of Pontryagin Algebras and Lie S...mentioning

confidence: 99%

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

Вылегжанин¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The alternative "a simplicial complex is contractible or homotopy equivalent to a sphere" can be decided by computing its simplicial homology. Proposition 4.3 was further used in [4] to show that the depth of the Stanley-Reisner algebra krK P s (i.e. the maximal length of regular sequences in this algebra) coincides with dim P .…”

Section: 3mentioning

confidence: 99%

Topology of nerves and formal concepts

Ayzenberg

2019

Preprint

View full text Add to dashboard Cite

The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial simplification of simplicial complexes and allows to see a certain duality on a class of simplicial complexes. This duality generalizes Poincare duality on cell subdivisions of manifolds. On the other hand, with the notion of a topological formal context, we review the classical proofs of two basic theorems of homotopy topology: Alexandrov Nerve theorem and Quillen-McCord theorem, which are both important in the applications. A brief overview of the applications of the Nerve theorem in brain studies is given. The focus is made on the task of the external stimuli space reconstruction from the activity of place cells. We propose to use the combination of FCA and topology in the analysis of neural codes. The lattice of formal concepts of a neural code is homotopy equivalent to the nerve complex, but, moreover, it allows to analyse certain implication relations between collections of neural cells.

show abstract

“…Book [10] shows the classification of the systems using HTL (Figure 7). In the late 20 -early 21 centuries, HTL were used to improve the conditions of the natural lighting.…”

Section: Classification Of the Guide Lightsmentioning

confidence: 99%

Solar exposure condition improvement in urban area using light guide

Gritsiyenko

Kas'yanov

2017

MATEC Web Conf.

View full text Add to dashboard Cite

Abstract. This article provides an overview of the light guides and lighting systems used to improve the natural lighting conditions of the premises with insufficient lighting comfort. The retrospective of occurrence of the light guides from the 19th century to the present time is shown. The relationship between the modern light guides and light-guide and other lighting systems that have been developed since the middle of the 19th century is demonstrated. The relevance of the use of light guides during the construction and reconstruction of urban buildings is considered. The article describes the classification of light guides developed by the author, and shows the classification of light guides proposed by Aizenberg J.B. For the first time it was proposed to use light guides to improve the conditions of solar exposure of the shaded buildings. The example of two buildings shows a scheme for the use of light guides to improve the solar exposure conditions of the shaded buildings of the urban area. Setting of the tasks to be carried out to consider the use of the light guides in order to improve insolation conditions is completed.

show abstract

Topological applications of Stanley-Reisner rings of simplicial complexes

Cited by 3 publications

References 21 publications

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

Topology of nerves and formal concepts

Solar exposure condition improvement in urban area using light guide

Contact Info

Product

Resources

About