Visualization of the linearity for mapping class group and its relation to topology

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05154
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kochi University of Technology
• Principal Investigator: Kasahara Yasushi 高知工科大学, 共通教育教室, 准教授 (80299370)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 曲面の写像類群 / 線型表現 / 曲線複体 / 幾何的交叉 / Johnson filtration

Outline of Final Research Achievements

We continued to study problems which arise from the viewpoint of the visualization of the linearity of mapping class groups of surfaces, which we had established some years ago, toward building the theoretical setup for solving the linearity problem for the mapping class groups which is considered as one of the fundamental problems in the field. Consequently, we could deepened our understanding of the visualization itself as well as related topics. In particular, we obtained the complete classification of the 2g+1 dimensional linear representations of mapping class groups of surfaces of genus g for g>5 which was the first case of the previously unknown dimensions.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

現在大きな関心を持たれている数学の基礎的研究分野であり, 現実を記述する究極の理論を与えることを目指す理論物理学や, タンパク質の高次構造の記述などとの関連も期待されている, 曲面の写像類群の研究において, 研究代表者の発見した独自の観点である「線型性の視覚化」の観点から研究を継続し,その理解を深め, ある専門的な問題を完全に解決することができた. これにより, 数学の当該研究分野の研究にある程度寄与することができたと考えられる.