Interaction between randomness and geometric structures in simplicial complexes

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K21833
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Kyoto University
• Principal Investigator: Hino Masanori 京都大学, 理学研究科, 教授 (40303888)
• Co-Investigator(Kenkyū-buntansha): 平岡 裕章 京都大学, 高等研究院, 教授 (10432709)
• Project Period (FY): 2019-06-28 – 2023-03-31
• Keywords: 単体複体 / 確率論 / 幾何構造

Outline of Final Research Achievements

This study mainly focuses on the geometric structure of a family of simplicial complexes and its relation to probability theory. To this end, the relationship between Betti numbers of simplicial complexes and eigenvalues of the corresponding graph Laplacian was discussed. Furthermore, we proved limit theorems, such as the law of large numbers and the large deviation principle, in relation to the persistent homology for a family of random simplicial complexes. These studies can be regarded as higher-dimensional analogues of the study of random graphs.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

ランダムグラフの研究が長い歴史を持つことに比較して，その高次元版と見なされるランダム単体複体の研究はまだ発展途上といえる．対象の高次元化を行うことで，空間の幾何構造とランダムネスとの結びつきがより顕になることが期待され，数学理論としての理論展開に興味が持たれるものである．本研究課題においては，そのような問題意識に基づいた研究を行った．