Finite dimensionality of motives, Conservativity, and beyond

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03514
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Hiroshima University
• Principal Investigator: Kimura Shun-ichi 広島大学, 先進理工系科学研究科(理), 教授 (10284150)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 組合せゲーム / K理論 / Enforce Operator / Comply/Constrain / 超現実数 / 有限次元性 / モチーフ理論

Outline of Final Research Achievements

We considered the K-theory of Combinatorial Game theory as a generalization of the notion of Numbers, and (re)-discovered that this generalization gives counter-intuitive notion of infinity for usual mathematicians, and started up to make a regorous and understandable explanation of this notion of numbers.
As for the theory of Combinatorial games, (1) we studied Yama-Nims and Triangluar-Nims, which are 2 piles Nims, where the player takes some tokens from one pile, and returns a strictly smaller number to the other pile. In particular, when combined with Wythoff type variation, very interesting winning strategy appears. (2) For Subtraction Nims, we combine them with Enforce operator and Carry on operator, and defined Grundy numbers for each of these extensions. but when we combine both of them, entailing phenomena appears and Grundy numbers are not defined any more.

Free Research Field

組合せゲーム理論

Academic Significance and Societal Importance of the Research Achievements

無限を含む数概念の拡張は、例えば物理学の繰り込み理論の新しい解釈など、驚くべき応用につながる可能性がある。Enforce Operator や Entailing Phenomena は、本研究は組合せゲーム理論の枠組みで行われているが、隠された情報や確率・不確定性などが現れるより一般のゲーム理論の脈絡にも応用できる可能性があり、そうなれば経済学での意思決定に対する新しい提案につながる可能性がある。単純に組合せゲームとして対人（あるいは対コンピューターで）遊ぶゲームとしても面白いゲームの提案につながる可能性がある。