Algebro-geometric method for singularity criteria in kinematics

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13452
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Hokkaido University
• Principal Investigator: Ohmoto Toru 北海道大学, 理学研究院, 教授 (20264400)
• Co-Investigator(Renkei-kenkyūsha): Saji Kentaro 神戸大学, 大学院理学研究科, 准教授 (70451432)
• Research Collaborator: Kabata Yutaro 九州大学, IMI研究所, 特任助教
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 写像の特異点論 / 特異点判定法 / 射影微分幾何 / 幾何的代数 / 実代数幾何 / コンピュータ・ヴィジョン

Outline of Final Research Achievements

We developed effective criteria of singularities of differentiable maps in the following three topics. 1. We have newly developed a local theory originated by Darborx and Wilcynski in projective differential geometry of surfaces in relation with applications to computer vision. 2. Combining Geometric Algebra and classification theory of map-germs, we studied singularities arising in differential line geometry (ruled surfaces, line congruences, etc), that suggests a new method in Applied Geometry. 3. Using classification of map-germs and universal polynomials of singularities, we developed classical enumerative geometry of curves and surfaces, that proposes a new algebro-geometric approach to computer vision. 4. By means of `singularity criteria', we studied A-tangent spaces and logarithmic tangents to discriminants (free divisors).

Free Research Field

幾何学