BIMSA > BIMSA-Tsinghua Quantum Symmetry Seminar Application of Operator Theory for the Collatz Conjecture
Application of Operator Theory for the Collatz Conjecture
Organizers
Lin Zhe Huang , Zheng Wei Liu , Sébastien Palcoux , Yi Long Wang , Jin Song Wu
Speaker
Takehiko Mori
Time
Wednesday, March 26, 2025 10:30 AM - 12:00 PM
Venue
A3-3-301
Online
Zoom 242 742 6089 (BIMSA)
Abstract
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