Georgia Institute of Technology College of Sciences

The Chirikov-Taylor standard map is a prototype of chaos in Hamiltonian systems, exhibiting the stretch and fold mechanism expected to give rise to chaotic behavior in a wide variety of physical contexts, from kinetic theory to fluid and celestial mechanics. Despite the general expectation that tools from smooth ergodic theory describe disordered dynamcial behavior, checking whether these tools actually apply for given models is a notoriously hard problem. For instance, it remained open for decades (the so-called Standard Map Conjecture) whether the standard map admits a positive Lyapunov exponent on a positive area set, despite clear numerical evidence and strong heuristic arguments in favor. A Lean certificate regarding a solution of this conjecture was released by OpenAI on October 6th, along with a putative AI-written writeup of the proof. 

The purpose of this talk is to discuss some of the history and motivation behind the standard map, along with an account of the known mathematical results regarding the standard map — work of myself and many others. I will attempt to make cases for two assertions, somewhat at odds with each other: (i) the resolution of the standard map conjecture is of immense importance for smooth ergodic theory; and (ii) the manner in which the conjecture is resolved leaves a tremendous amount of work to come. While I will touch on the OpenAI proof, the primary focus will be on settled mathematical results.