• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.391-405
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• 1996

In this article, we calculate the cohomology groups of flat vector bundles on some manifolds.

• Communications of Mathematical Education
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• v.16
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• pp.1-65
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• 2003

The focus of this article is on the process of cultivating students' interests so that they come to view mathematics as an activity worthy of their engagement. We first define and operationalize the notion of interests, in the process developing a perspective in which they are seen to be generative, to evolve, and to be deeply cultural. We concretize this perspective by presenting an analysis of a classroom design experiment that documents both the process by which the students' interests evolved and the means by which these developments were supported. We then frame the analysis as a case in which to tease out the implications for a nascent design theory of mathematical interests and in doing so give particular attention to the issue of equity in students' access to significant mathematical ideas

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.623-669
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• 2018

In Donaldson-Thomas theory, moduli spaces are locally the critical locus of a holomorphic function defined on a complex manifold. In this paper, we develop a theory of critical virtual manifolds which are the gluing of critical loci of holomorphic functions. We show that a critical virtual manifold X admits a natural semi-perfect obstruction theory and a virtual fundamental class $[X]^{vir}$ whose degree $DT(X)=deg[X]^{vir}$ is the Euler characteristic ${\chi}_{\nu}$(X) weighted by the Behrend function ${\nu}$. We prove that when the critical virtual manifold is orientable, the local perverse sheaves of vanishing cycles glue to a perverse sheaf P whose hypercohomology has Euler characteristic equal to the Donaldson-Thomas type invariant DT(X). In the companion paper, we proved that a moduli space X of simple sheaves on a Calabi-Yau 3-fold Y is a critical virtual manifold whose perverse sheaf categorifies the Donaldson-Thomas invariant of Y and also gives us a mathematical theory of Gopakumar-Vafa invariants.

• Journal of the Korea Convergence Society
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• v.12 no.5
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• pp.291-301
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• 2021

The purposes of this paper were to provide the basic segments of qualitative research to assist the understanding of qualitative research and to provide implications of indicators (i.e., criteria) to determine the rigor of the qualitative research in the area of Adapted Physical Education (APE). This paper was divided into five sub-categories to facilitate understanding the qualitative research, which were (a) the epistemological stances, (b) the data collection methods (c) the data analysis (d) the trustworthiness and (e) the implications of Adapted Physical Activity Taxonomy (APAT) in APE. Qualitative researchers deliver their understanding of human experience and knowledge by means of text rather than number so their explanations and understanding tend be subjective and distinct from otherwise interpretaotherwise (Crotty, 1998; Pitney& Parker, 2009). Examining the rigor of qualitative research is therefore concerned due to the complexity of understanding of reality (Goodwin 2020; Zitomer & Goodwin, 2014). In the meantime the roles of qualitative research have been highlighted in educational research including the area of APE because qualitative research enables researchers to examine different voices of school members (e.g., students with and without disabilities parents teachers Goodwin, 2020; Hodge et al., 2019; Zitomer & Goodwin, 2014).