3d Brownian motion and sets of dimension 2.99999 9999

As you breathe, air eventually reaches the alveoli in your lungs. You might wonder if a particle of air is equally likely to reach all parts of your lungs. The answer is emphatically no! Mathematically one explanation is a theorem of Bourgain, which states that for any closed set in space, there is always a subset of Hausdorff dimension less than 3, which contains all the points of the set where a randomly moving particle can land. In this talk, I will present explicit numerical bounds on the dimension of the subset in Bourgain's theorem, obtained in joint work with Michael Albert and Alyssa Genschaw. I will focus on the story of how we arrived at these results, the role of software in the arguments, and potential for future improvements.