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KITAZAWA, Naoki/ Post-doctoral Researcher



KITAZAWA, Naoki KITAZAWA, Naoki(Post-doctoral Researcher) The theory of Morse functions and higher dimensional versions and its application to geometry of manifolds is one of important studies of geometry, mathematics and recently, applications to various branches of science and technology such as data analysis or the analysis of datasets is expected. Related to the study, I have studied algebraic and differential topological properties of fold maps, which are simplest generalizations of Morse functions, and more general maps with good geometric properties. Morse functions always exist plentifully on manifolds and from singular points, appearing discretely, we can know information of fundamental topological invariants such as Euler numbers and homology groups of the manifolds and from fold maps and more general maps, we can sometimes know more precise information on the manifolds. I have introduced and studied several classes of fold maps satisfying good algebraic and differential topological properties and geometric properties of source manifolds: more precisely, under appropriate conditions, I have discovered restrictions not only on homology groups, but also more precise information such as topological types and differentiable structures of manifolds. As a related work, I have succeeded in explicit constructions of several fold maps, which are fundamental, important and difficult. Recently, I am interested in developing new methods in these studies powerful in developing various branches of geometry of manifolds and also interested in applications of the studies to problems of other branches of science and technology.

Keyword the singularity theory of smooth maps; Morse functions and fold maps, algebraic topology of manifolds, differential topology of manifolds
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