- BIMSA-Tsinghua量子对称讨论班

Application of Operator Theory for the Collatz Conjecture

组织者

黄林哲 , 刘正伟 , 塞巴斯蒂安·帕尔库 , 王亦龙 , 吴劲松

演讲者

Takehiko Mori

时间

2025年03月26日 10:30 至 12:00

地点

A3-3-301

线上

Zoom 242 742 6089 (BIMSA)

摘要

The Collatz map (or the 3n+1-map) f is defined on positive integers by setting f(n) equal to 3n+1 when n is odd and n/2 when n is even. The Collatz conjecture states that starting from any positive integer n, some iterate of f takes value 1. In this study, we discuss formulations of the Collatz conjecture by C∗-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the C∗-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated C∗-algebras.

References: arXiv:2411.08084