Arrangements of hyperplanes and conics via real structures(Fostering Joint International Research)

2018 Fiscal Year Final Research Report

• Project/Area Number: 15KK0144
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: Yoshinaga Masahiko 北海道大学, 理学研究院, 教授 (90467647)
• Research Collaborator: Feichtner Eva Maria Bremen大学, 数学科, 教授 ; Dimca Alexandru ; Bailet Pauline ; Liu Ye
• Project Period (FY): 2016 – 2018
• Keywords: 超平面配置

Outline of Final Research Achievements

Hyperplane arrangements appear in various area of mathematics. One of the peculiar points is that a hyperplane arrangement is naturally related with several "discrete structure" (e.g., intersection poset, lattice points, and real structure). The relationship between discrete structures and geometric structures of arrangements is focused in this project. We obtained several results on twisted cohomology and Milnor fibers of arrangements. In particular, we found an example (the Icosidodecahedral arrangement) of 16 planes which first homology of the Milnor fiber contains a torsion. We also introduced G-Tutte polynomial for a list of integral vectors and a commutative Lie group G. G-Tutte polynomial unifies several known "generalized Tutte polynomials" and provide a common framework which enables to treat unified manner.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

離散的な構造を使うことで、幾何学的な構造（図形の形）を扱う新しい手法が得られた。特に今回見つかった２０・１２面体配置は、非常に対称性が高いのみならず、これまで予想されていなかった性質を有しており、今後超平面配置の位相的構造を調べる際には、２０・１２面体配置の分析が重要な役割を果たすと思われる。
数え上げ理論と関連して、G-Tutte多項式というものを導入し、基礎的な研究を行った。Tutte多項式はもともとグラフに対して定義され、結び目理論、統計物理などとも密接に関係している。Tutte多項式の様々な一般化を統一するG-Tutte多項式の、既存の結果との類似点や相違点が明らかになった。