ncts

Validated numerics and computer-assisted proofs are powerful techniques that, with sufficient set-up, can be used to prove deep mathematical results that might be intractable with solely pencil-and-paper analysis. There has been much development in recent years in applying these techniques to infinite-dimensional problems, and they are now sufficiently mature that, for example, a black box implementation has been developed in MATLAB for computer-assisted proofs of periodic orbits in ordinary and delay differential equations. The incredible effectiveness of these methods sometimes overshadows the fact that, in order for them to be useful, a lot of functional analysis and clever reformulation of the problem at hand is usually needed. In this talk, I will present some recent developments in computer-assisted proofs in nonlinear dynamics. Most of the talk will be at a high level, emphasizing the fundamental ideas of computer-assisted proofs and applications to problems in neural networks and ordinary and delay differential equations. However, we will dig into the analysis a few times to show the work that can be needed to ensure a successful computer-assisted proof.