2022 Fiscal Year Final Research Report
Comprehensive study of algebraic varieties and singularities and their applications to engineering centered on tropical geometry
| Project/Area Number |
17K05206
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Keywords | トロピカル幾何 |
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| Outline of Final Research Achievements |
Basic facts such as the formulation by pointed monoids for toric tropical manifolds are summarized, and F1-algebras and unipotent commutative semirings in algebraic geometry are discussed with an awareness of the relation to ordered additive groups, and further research is conducted for future applications to singularity theory. The connection with the results of Shustin et al. on the tropicalization of hypersurface families of fixed Newtonian polyhedra is investigated. For applications to scheduling problems, we continued our research on extracting information on the original network structure from tropical polynomials. Assisted in organizing a meeting in this field by young researchers.
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
これまで代数幾何においては当然とされてきた可換環の枠組みについて反省し，概型理論が成立する最小の代数系で理論構築を行うことで，証明の簡略化や本質の抽出ができる知識伝授ができるようになった．また国内でのトロピカル幾何の研究集会の当面の常設化ができた．工程計画問題等，応用数学・工学において，ニュートン多面体を介した幾何的手法が導入できる新たな方向性を提示した．
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