Rigidity of non-isometric actions of discrete groups and non-linear spectral gap

2022 Fiscal Year Final Research Report

• Project/Area Number: 17H02840
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: Nayatani Shin 名古屋大学, 多元数理科学研究科, 教授 (70222180)
• Co-Investigator(Kenkyū-buntansha): 井関 裕靖 慶應義塾大学, 理工学部(矢上), 教授 (90244409)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: スペクトルギャップ最大化 / 最適埋め込み / 離散群の剛性 / 球面内の極小曲面

Outline of Final Research Achievements

We studied the optimization problems concerning embeddings into Euclidean spaces and the linear spectral gap of a finite graph and were able to find optimal solutions for a distance-regular graph. We studied the problem of finding an edge-length function maximizing the linear spectral gap and proved a Nadirashvili type theorem. We studied a new optimization problem concerning embeddings and the linear spectral gap of a manifold. We presented some examples where the problems can be solved and proved a Nadirashvili type theorem. We showed that a discrete equivariant harmonic map form a finitely generated group equipped with a random walk induces a boundary map under appropriate assumptions.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究は, ラプラシアン第1固有値のように, それ自身変分問題の最適値であるものをリーマン計量をすべて動かしてさらに最大化するという, 高次の変分問題を扱っており, 数学研究の新たな発展に関わるものと考えている. 新たにNash等長埋め込みと関連する双対最適化問題を設定したことも意義があろう. 離散と連続にまたがる研究であり, 材料科学への示唆も期待できよう.