Geometry for Materials based on discrete surface theory

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13432
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Kotani Motoko 東北大学, 理学研究科, 教授 (50230024)
• Co-Investigator(Renkei-kenkyūsha): Naito Hisashi 名古屋大学, 多元数理科学研究科, 准教授 (40211411)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 離散幾何学 / スペクトル幾何 / 物性物理

Outline of Final Research Achievements

We call a trivalent graph in the 3-dimensional Euclidean space a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. We develop a discrete surface theory on trivalent graphs, and define several geometric notions such as area, area variation formula, curvature, mean curvature. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision, and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric objects associated with the given discrete surface.

Free Research Field

幾何学、離散幾何解析学