Fiber Topology Meets Applications

Reference No.	20200011
Type/Category	Grant for General Research-Workshop（Ⅱ）
Title of Research Project	Fiber Topology Meets Applications
Principal Investigator	Daisuke Sakurai（Pan-Omics Data-Driven Research Innovation Center・Associate Professor）
Research Period	January 6,2021. ～ January 8,2021.
Keyword(s) of Research Fields	Singularity theory, fiber topology, visualization, global warming, semi-conductors, operations research
Abstract for Research Report	PURPOSE In differential topology, the study of the topology of fibers, known as fiber topology, extends the Morse theory, of functions, to that of maps (Saeki, 2004). This establishes a thorough exploration of topological transitions of inverse images. It has been proven powerful for understanding today’s data whose size and complexity are overwhelming (Takahashi, 2004; Hamada & Chiba, 2017). In the proposed meeting, we gather researchers from academia and the industry. The meeting aides the interaction among mathematics, computation, and applications. For this, participants gather from a broad spectra of research disciplines, including not only mathematics and computer science, but also operational research, environmental sciences and semiconductors. The operational research, conducted by Naoki Hamada (Fujitsu Research Institute), Daisuke Sakurai (Kyushu University Pan-Omics Data-Driven Research Innovation Center) et al., is a project with assembly scheduling workers in Fujitsu. Together, we have been designing software products that offer optimal schedules based on characterization implied by fiber topology. Environmental scientists from Hiroshi Yamashita and Bastian Kern (German Aerospace Center, Germany) have some history of joint-research with Sakurai while he was in Germany at Zuse Institute Berlin (Germany). Their current research project by these environmental scientists focus on proposing a new tax incentive for balancing the environmental costs against the operational cost in the aviation industry. Fiber topology is employed for understanding the high-dimensional space spanned by different kinds of subcosts. Finally, the application in semiconductor research employs fiber topology for understanding the potential fields of atomic configurations. For simulation outputs done with Japan’s next-generation supercomputer Fugaku, we extract the landscape of the high-dimensional potential fields. Fiber topology gives a concise and parameter-free representation of the landscape, which is also robust against numerical errors. All of the applications pose a challenge to the current state-of-the-art analysis of fiber topology. Especially, the high spatial dimensionality in the data is something that requires an update to the numerical methods. The current, rather loose, mathematics found in the computer science literature meets a more rigorous treatment by Takahiro Yamamoto (Tokyo Gakugei University). OUTCOME In this first meeting that gathers all of our collaboration teams, we set interdisciplinary milestones of progress in the coming years. Sakurai has so far held discussions with each member face-to-face, and this meeting establishes a coherent grand strategy among the international team. This builds a network of researchers who talk to different fields from the point of view of theories, computation, and problems in the society. The participants include senior researchers experienced in their fields and junior researchers who are building new research areas. Theorists not only provide solutions, but get feedback about desirable expansions. We discuss how fiber topology should let us extract information about the energy potential of atomic configurations, robustness of the assembly schedule against errors, and the impact of the inter-governmental decisions towards the aviation industry and air pollution. While our progress in computation has seen a success, there are theoretical questions left, such as how the mathematical results translate to and from numerical representations. What directions in mathematical theories benefit numerical ones, and vice versa, is another question. The collaboration is also a part of the interdisciplinary data-driven activities taking place at Kyushu University. Sakurai and Osamu Saeki are with Pan-Omics Data-Driven Research Innovation Center, which co-managed by the Institute of Mathematics for Industry together with other four research centers in the university. Further gatherings will take place in the coming years in order to evaluate and surpass the milestones set this time. --- In recent years, fiber topology (Saeki, 2004) has been actively pushed forward from mathematical studies towards computation. While the computational approach for the Morse-case (Takahashi et al., 2004) was studied separately, the two research domains started communication after some years. For more general cases, theoretical insights have been transferred into numerical algorithms through direct collaboration with computer scientists (Sakurai et al., 2015). This brought a bi-directional impacts where (i) mathematics boosts the progress in computation (Tierny & Carr, 2016) and (ii) computer science also helps mathematics (Sakurai et al., 2015). Simultaneously, fiber topology has been applied for decision making with multiple optimization problems (Hamada, N., & Chiba, 2017). The organizing board is now confident in adapting the achieved theories and techniques for industrial and inter-governmental decision making. As our section above (PURPOSE) explains, we have conducted discussions with world-leading research groups to identify mutual interests that would advance both the society and techniques. We have started gathering preliminary results. With the proposed meeting, we finish the initial goal setting phase, and move on to the mission of providing solutions to application problems. --- The meeting starts with a survey of relevant problems in the application fields. Computer scientists, then, proceed to present their solutions their fiber topology-based algorithms can offer and what mathematical improvements are necessary. Finally, differential topologists show what would complement the current computational algorithms. The schedule is as follows: SCHEDULE ---------------- SESSION 1 (DAY1): MESSAGE FROM APPLICATIONS 10:00-11:00 Fiber Topology Meets Applications: Where We are Heading Daisuke Sakurai (Kyushu University) 11:00-12:00 Modeling semiconductor epitaxy for next generation power device application Yoshihiro Kangawa (Kyushu University) 12:00-13:30 Lunch 14:30-15:30 Fiber-Topological Information Extraction for Assembly Scheduling Naoki Hamada (Fujitsu Research Institute) 13:30-14:30 Balancing Air Traffic and Pollution Hiroshi Yamashita and Bastian Kern (German Aerospace Center, Germany) 15:30-16:30 Data-Driven Collaboration Around Kyushu University Kenji Ono and Daisuke Sakurai (Kyushu University) 16:30-17:00 Discussion of the Next Steps, Part 1 (Applications) SESSION 2 (DAY 2): PROGRESS IN COMPUTER SCIENCE 11:00-12:00 Topological Analysis of Level Sets and Fibers Hamish Carr (University of Leeds, Britain) 12:00-13:30 Lunch 13:30-14:30 Simplification of Fiber Topology Petar Hristov and Hamish Carr (University of Leeds, Britain) 14:30-15:30 Extracting the Topological Transition of Contours and Fibers Shigeo Takahashi (University of Aizu) 15:30-16:30 Triangulating Firmly Extruded Prisms Akito Fujii, Daisuke Sakurai and Kenji Ono (Kyushu University) 16:30-17:00 Discussion of the Next Steps, Part 2 (Computation) SESSION 3 (DAY 3): SUPPORT FROM THEORIES 10:00-11:00 Mathematical Review of Reeb Spaces and Reeb Graphs Takahiro Yamamoto (Tokyo Gakugei University) 11:00-12:00 Next Steps for Reeb Spaces (temporary title) Osamu Saeki (Kyushu University) 12:00-12:30 Discussion of the Next Steps, Part 3 (Theories) Concluding Remarks Daisuke Sakurai BUDGET ------------- Together with additional budgets available at the hands of the organizing board, we pay the travel cost of organizing board and the invited speakers. The travel cost we cover for the domestic participants is 30,000 from this budget for each, although we combine other budgets when necessary. That of the participants from overseas is 200,000 yen for each from this budget. Again, we combine other budgets when necessary. Bastian Kern and Petar Hristov may have to delegate the presentation to Hiroshi Yamashita and Hamish Carr, respectively, due to the budget amount. AFTER YEAR 2020 -------------------------- We plan to continue applying for your research budget in the coming years. Given the rather large scale of the collaboration, we hope to apply to 研究集会(I) in a year or two. --- REFERENCES -------------- Saeki, O. (2004). Topology of singular fibers of differentiable maps. (Lecture Notes in Mathematics). Springer Heidelberg. Takahashi, S., Takeshima, Y., & Fujishiro, I. (2004). Topological volume skeletonization and its application to transfer function design. Graphical Models, 66(1), 24-49. Tierny, J., & Carr, H. (2016). Jacobi fiber surfaces for bivariate Reeb space computation. IEEE transactions on visualization and computer graphics, 23(1), 960-969. Sakurai, D., Saeki, O., Carr, H., Wu, H. Y., Yamamoto, T., Duke, D., & Takahashi, S. (2015). Interactive Visualization for Singular Fibers of Functions f: R 3→ R 2. IEEE transactions on visualization and computer graphics, 22(1), 945-954. Hamada, N., & Chiba, K. (2017). Knee point analysis of many-objective Pareto fronts based on Reeb graph. In 2017 IEEE Congress on Evolutionary Computation (CEC) (pp. 1603-1612). IEEE.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Shigeo Takahashi（University of Aizu・Professor） Naoki Hamada（Fujitsu research institute・Research Staff） Daisuke Sakurai（Kyushu University, Pan-Omics Data-Driven Research Innovation Center in Research Institute for Information Technology・Associate Professor） Osamu Saeki（Kyushu University, Institute of Math for Industry・Professor, Head of Institute） Hamish Carr（University of Leeds, UK・Senior Lecturer） Takahiro Yamamoto（Tokyo Gakugei University・Associate Professor）