Geometric and constructive studies of higher dimensional manifolds and applications to higher dimensional data

Reference No.	20200027
Type/Category	Grant for Young Researchers-Short-term Joint Research
Title of Research Project	Geometric and constructive studies of higher dimensional manifolds and applications to higher dimensional data
Principal Investigator	Naoki Kitazawa（Institute of Mathematics for Industry, Kyushu University・Postdoctoral researcher）
Research Period	July 12,2021. ～ July 15,2021.
Keyword(s) of Research Fields	Singularity theory of differentiable maps. Algebraic topology of manifolds, Differential topology of manifolds. Fiber topology. Machine learning. Visualizations.
Abstract for Research Report	Recent developments in Web services and IoT devices produce lots of "big data ". Among them there are various strong methods to analyze data in subspaces of dimensions at most 3 of high dimensional spaces. Currently we are expecting mathematical tools are strong in data analysis and machine-learning and actually methods based on sophisticated modern geometry such as manifold learning are strong. In product designs etc., we often encounter with data in subspaces of higher dimensions. Developing strong methods to analyze such data or so-called "high dimensional data" are important problems. We mathematicians and experts of data analysis who are knowledgeable about geometry and other branches of mathematics gather and discuss applications of “studies of higher dimensional manifolds from geometric and combinatorial viewpoints via smooth maps into lower dimensional spaces”, which we have pioneered, to higher-dimensional data analysis. We mathematicians present mathematical tools and notions such as Morse functions, fold maps, which are higher dimensional versions of Morse functions, Reeb spaces etc.. Experts of data analysis who use several theory of geometry present explicit studies on visualizations related to higher dimensional objects. Through these stuffs, we discuss what we need for applications of this mathematical theory to analysis of higher dimensional datasets, discover explicit programs, formulate problems mathematically and study these programs further and this is the main purpose. As more explicit problems, we expect new developments on the following problems which are challenging problems in so-called “Fiber Topology”, which concerns topological theory of preimages of mappings and applications to visualizations of datasets etc.. 1 Global and geometric structures of multi-functions in (multi-objective) optimizations. 2 Visualizations of higher dimensional datasets via preimages of mappings into lower dimensional spaces. We will continue collaborations based on research results.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Naoki Kitazawa（Institute of Mathematics for Industry, Kyushu University・Post-doctoral researcher） Osamu Saeki（Institute of Mathematics for Industry, Kyushu University・Professor (Director)） Daisuke Sakurai（Kyushu University・Associate Professor） Shigeo Takahashi（Aizu University・Professor） Naoki hamada（KLab Inc.・Researcher (Engineer)）
Adviser	Naoki Hamada (KLab Inc.)