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Ariki-Koike Algebras # Cluster Algebras # Group Rings # Hecke Algebras # KLR Algebras # Schur Algebras o Commutative Algebra o Fusion Systems o Group Theory # Abelian Groups # Algebraic Groups # Combinatorial Group Theory # Computational Group Theory # Finite Groups # Geometric Group Theory # Infinite Groups # Lie Groups # Permutation Groups o Lie Theory # Algebraic Groups # Lie Algebras # Lie Groups # p-Adic Groups # Representations of Algebraic Groups o Non-Associative Algebras # Jordan and Axial Algebras # Lie Algebras # Other Non-Associative Rings o Representation Theory # Geometric Representation Theory # Representations of Algebraic Groups # Representations of Finite Groups # Representations of Lie Algebras # Representations of Symmetric Groups o Semigroups + Analysis o Complex Analysis o Functional Analysis o Harmonic Analysis + Category Theory o Derived Categories o Higher Category Theory o Triangulated Categories + Combinatorics o Additive Combinatorics o Algebraic Combinatorics o Extremal Combinatorics o Graph Theory + Geometry o Algebraic Geometry # Birational Geometry o Differential Geometry o Discrete Geometry o Non-Commutative Geometry o Symplectic Geometry + Logic + Mathematical Finance + Mathematical Physics + Number Theory o Additive Number Theory o Analytic Number Theory o Algebraic Number Theory o Computational Number Theory o Diophantine Equations o Diophantine Geometry o Number Theory and Forms o Probabilistic Number Theory + Optimization + Probability Theory + Topology o Algebraic Topology o Analytic Topology o Differential Topology o Dynamical Systems o Geometric Topology * Short Courses + Algebra + Analysis + Category Theory + Combinatorics + Geometry + Mathematical Physics + Number Theory + Probability Theory + Topology * Lecture Courses + Algebra + Analysis + Category Theory + Combinatorics + Geometry + Mathematical Physics + Number Theory + Probability Theory + Topology Terence Tao: Almost all Collatz orbits attain almost bounded values 13th March, 2023 Watch LaterRemove Cinema Mode Define the Collatz map Col on the natural numbers by setting Col(n) to equal 3n+1 when n is odd and n/2 when n is even. The notorious Collatz conjecture asserts that all orbits of this map eventually attain the value 1. This remains open, even if one is willing to work with almost all orbits rather than all orbits. We show that almost all orbits n, Col(n), Col^2(n), ... eventually attain a value less than f(n), for any function f that goes to infinity (no matter how slowly). A key step is to obtain an approximately invariant (or more precisely, self-similar) measure for the (accelerated) Collatz dynamics. This video is part of the Institute for Advanced Study's Members' colloquium. 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