Speaker: Takehiko Mori (Chiba University)

Title: Application of Operator Theory for the Collatz Conjecture

Time/Date: 4:45-6:15pm, June 3 (Tue), 2025

Room: 117 Math Sci Building (It will be also online. The Zoom link is the same as before. If you don't have one, please ask Kawahigashi.)

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: Mori, T. Application of operator theory for the collatz conjecture. Adv. Oper. Theory 10, 37 (2025). doi:10.1007/s43036-025-0042501 arXiv:2411.08084v2