• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1323-1333
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• 2020

In 2009, Borg [2] suggested a conjecture concerning the size of a t-intersecting k-uniform family of faces of an arbitrary simplicial complex. In this paper, we give a strengthening of Borg's conjecture for shifted simplicial complexes using algebraic shifting.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.265-276
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• 2017

Let K be a fan-like simplicial sphere of dimension n-1 such that its associated complete fan is strongly polytopal, and let v be a vertex of K. Let K(v) be the simplicial wedge complex obtained by applying the simplicial wedge operation to K at v, and let $v_0$ and $v_1$ denote two newly created vertices of K(v). In this paper, we show that there are infinitely many strongly polytopal fans ${\Sigma}$ over such K(v)'s, different from the canonical extensions, whose projected fans ${Proj_v}_i{\Sigma}$ (i = 0, 1) are also strongly polytopal. As a consequence, it can be also shown that there are infinitely many projective toric varieties over such K(v)'s such that toric varieties over the underlying projected complexes $K_{{Proj_v}_i{\Sigma}}$ (i = 0, 1) are also projective.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1357-1366
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• 2018

It is shown that the Mayer-Vietoris sequence holds for the cohomology of complexes of Lie algebroids which are defined on simplicial complexes and satisfy the compatibility condition concerning restrictions to the faces of each simplex. The Mayer-Vietoris sequence will be obtained as a consequence of the extension lemma for piecewise smooth forms defined on complexes of Lie algebroids.

• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1613-1639
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• 2019

Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure strongly shellable complexes. Meanwhile, pure strongly shellable complexes can be characterized by the corresponding codimension one graphs. In addition, we show that the facet ideals of pure strongly shellable complexes have linear quotients.

• Journal of Korea Multimedia Society
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• v.19 no.2
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• pp.146-154
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• 2016

This paper presents methods to analyze the topological structures of the spaces composed of patches extracted from waveform signals, which can be applied to the classification of signals. Commute time embedding is performed to transform the patch sets into the corresponding geometries, which has the properties that the embedding geometries of periodic or quasi-periodic waveforms are represented as closed curves on the low dimensional Euclidean space, while those of aperiodic signals have the shape of open curves. Persistent homology is employed to determine the topological invariants of the simplicial complexes constructed by randomly sampling the commute time embedding of the waveforms, which can be used to discriminate between the groups of waveforms topologically.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.175-183
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• 2010

The classical group completion theorem states that under a certain condition the homology of ${\Omega}BM$ is computed by inverting ${\pi}_0M$ in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category ${\mathcal{A}}$. The 1-morphisms of ${\mathcal{A}}$ are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of ${\mathcal{A}}$, which is shown to be homotopy equivalent to ${\mathbb{Z}}{\times}BAut^+_{\infty}$.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1265-1283
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• 2019

We develop a framework to construct geometric representations of finite groups G through the correspondence between real toric spaces $X^{\mathbb{R}}$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the G-module structure of the homology of $X^{\mathbb{R}}$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type A and B, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representation geometrically as the homology of real toric varieties.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.16 no.1
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• pp.291-299
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• 2016

This paper raises a question of whether a simple periodic phenomenon is associated with the topology and provides the convincing answers to it. A variety of music instrumental sound signals are used to prove our assertion, which are embedded in Euclidean space to analyze their topologies by computing the homology groups. A commute time embedding is employed to transform segments of waveforms into the corresponding geometries, which is implemented by organizing patches according to the graph-based metric. It is shown that commute time embedding generates the intrinsic topological complexities although their geometries are varied according to the spectrums of the signals. This paper employs a persistent homology to determine the topological invariants of the simplicial complexes constructed by randomly sampling the commute time embedding of the waveforms, and discusses their applications.