The birth of modern trends on commutative algebra and convex polytopes with statistical and computational strategies

2019 Fiscal Year Final Research Report

• Project/Area Number: 26220701
• Research Category: Grant-in-Aid for Scientific Research (S)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Hibi Takayuki 大阪大学, 情報科学研究科, 教授 (80181113)
• Project Period (FY): 2014-05-30 – 2019-03-31
• Keywords: グレブナー基底 / 二項式イデアル / 凸多面体 / 実験計画 / Ａ超幾何系 / ホロノミック勾配法 / 分割表

Outline of Final Research Achievements

The present research project holds official events such as international conferences on commutative algebra, Groebner basis projects and workshops on convex polytopes with inviting 127 foreign researchers, including postdocs and graduate students, from overseas. It conducts the solid framework for prospective international joint research in this research field. Furthermore, cultivation of human resources of domestic and overseas young researchers is promoted. From the viewpoint of research, a huge amount of original results on the topics, including reflexive polytopes, experimental design and convex polytopes, contingency tables and polyomino ideals, A-hypergeometric systems and convex polytopes, holonomic gradient methods, binomial ideals and syzygy theory, have been obtained. It turns out that our project succeeded in inviting new trends of commutative algebra and convex polytopes.

Free Research Field

計算可換代数と組合せ論

Academic Significance and Societal Importance of the Research Achievements

可換代数も凸多面体論も、伝統的な純粋数学の研究分野である。両者の相互関係は、1980年以降、著しい発展を遂げ、可換代数と組合せ論と呼ばれる境界分野が誕生した。本基盤研究（Ｓ）は、「統計」、及び、「計算」の観点から、可換代数、及び、凸多面体論の現代的な潮流を誘うことを目的とし、その研究を展開した。その結果、可換代数と統計と凸多面体論の華麗なる三重奏を誕生させることに成功し、今後の１０年間の、可換代数と凸多面体論の輝かしい展望を切り開いた。