Fiber Topology Meets Applications

整理番号 20200011
種別 一般研究-研究集会(II)
研究計画題目 Fiber Topology Meets Applications
研究代表者 櫻井大督(九州大学 汎オミクス計測・計算科学センター・准教授)
研究実施期間 2021年1月6日(水)~ 2021年1月8日(金)
研究分野のキーワード 特異点論,ファイバーのトポロジー,可視化,地球温暖化問題,半導体,オペレーションズリサーチ
本研究で得られた成果の概要 In this forum, we gathered researchers from academia and the industry to discuss the future directions of fiber topology. This is a proudly unique communication where pure mathematicians, computer scientists and data analysis practitioners exchange ideas from the different ends of the broad scientific arena. As it followed, the speakers brought themes from broad spectra, including those in mathematics, topological analysis in visualization, the game industry, environmental sciences and chemistry. The goal of this forum is to develop fiber topology to solve problems in applications. This occasion was the first step for it.

The forum has viewed it important to investigate the application problems before constructing theories and algorithms for fiber topology. In fact, the entire planning of collaboration is driven by the applications, rather than by the theories. This approach has been favored as a means to keep this forum mission-oriented. It is also a favored approach of the Pan-Omics Data-Driven Research Innovation Center, who continues to sponsor this forum.

As has been visible in the program document, the applications of fiber topology in this forum include computational chemistry, eco-friendly aviation regulation and operational research. The key application in computational chemistry is to understand the conformation of molecules. We formulate this problem as the analysis of high-dimensional fields. The primary source of data is the simulations run by Yoshihiro Kangawa (Kyushu University).

The aviation regulation project led by Bastian Kern and Hiroshi Yamashita (both German Aerospace Center) investigates the tradeoff between costs for finance and various factors of air pollution such as CO2 and NOx. Atmospheric simulations are conducted to estimate the varying costs for variations of aviation path. The tradeoffs conflicting costs in the considered paths are investigated through the Pareto analysis. The task for fiber topology in the coming years is to analyze such tradeoffs in the form of a high-dimensional field. Naoki Hamada joined the discussion to provide his expertise on Pareto analysis.

The Pareto analysis is itself an application of fiber topology as presented by Reiya Hagiwara (Kyushu University) and Hamad et al. The plan to treat benchmark problems using fiber topology was discussed. It is also an interesting inverse problem for the fiber topology computation. We are going to proceed to apply the upcoming results for the purpose of game developments, an industrial application important as a mission especially for this forum and the Institute of Mathematics for Industry.

Given these requirements by the applications, computational studies are going to focus on firstly the development of a Reeb space algorithm that works for arbitrary dimensional domains and is robust (Sakurai and Carr). Another focus is the scalability of related computation (Carr, Akito Fujii at Kyushu University) and simplification (Petár Hristov at University of Leeds), as introduced in their talks.

For this to work, a few key theoretical understanding of the topology of fibers were lacking in the computer scientists. The talks from pure mathematics therefore had an exceptional importance for all of us. Takahiro Yamamoto introduced the elimination of singularity at the boundary of the domain. Osamu Saeki explained his joint-work on simplifying the indefinite fibrations on 4-manifolds. These indications from the theorists are now being adapted by the computer scientists. These mathematical developments will also be continued in further support of this new community. Early results are already published [1].

Last but not least, we are grateful to all the participants including the audience, KLab, the Pan-Omics Center and IMI.

1. D Sakurai & T Yamamoto. “Visually Evaluating the Topological Equivalence of Bounded Bivariate Fields,” Topological Methods in Visualization - Applications and Software. Springer, in print.
高橋 成雄(会津大学・教授)
濱田 直希((株)富士通研究所・研究員)
櫻井 大督(汎オミクス計測・計算科学センター・准教授)
佐伯 修(IMI・教授,部局長)
Hamish Carr(英国 Univeristy of Leeds・Senior Lecturer (准教授相当))