Geometry analysis on discrete spaces under a lower Ricci curvature bound

2022 Fiscal Year Final Research Report

• Project/Area Number: 21K20315
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Saitama University
• Principal Investigator: Yohei Sakurai 埼玉大学, 理工学研究科, 准教授 (90907958)
• Project Period (FY): 2021-08-30 – 2023-03-31
• Keywords: Riemann幾何学 / 離散幾何解析 / Ricci曲率 / 幾何学流

Outline of Final Research Achievements

Ricci curvature is one of the most fundamental objects in Riemannian geometry, and it has been studied from various perspectives. Hamilton has introduced a geometric flow called Ricci flow via Ricci curvature, and Perelman has solved the well-known Poincare conjecture based on the Ricci flow theory. In this project, we focus on a super solution to the Ricci flow called super Ricci flow, and obtain several geometric and analytic results for the heat equation along super Ricci flow. On the other hand, in recent years, there are some attempts to introduce the notion of Ricci curvature for non-smooth spaces. In this project, we provide a notion of super Ricci flow for discrete spaces from the viewpoint of the Bakry-Emery theory. We construct some examples, and conclude characterization results via functional inequalities.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

Ricci流はPerelmanによる3次元Poincare予想の解決において重要な役割を果たした．最近Bamlerにより高次元Ricci流の収束理論が発展しており，Perelmanの3次元の場合の結果を包括する理論が確立されつつある．そこでは優Ricci流に沿った熱方程式の幾何解析が鍵となっている．本研究で得られた諸結果は，それらの理論の今後の更なる発展に寄与しうるものであると考えられる．また離散空間上の幾何解析は純粋数学のみならず応用数学の観点からも注目を集めている．本研究で得られたグラフに対する優Ricci流に関する諸結果についても，材料科学や機械学習など他の分野への還元が期待される．