Grant: Discovery Grants Program - Individual

In the world of the infinite, there are many strange and wondrous things, from ways of carving up a ball and rearranging the pieces to make two balls, both the same size as the first one, to election methods which always pick winners fairly, no matter the number of candidates, if only we had infinitely many voters. But the mathematical methods used to create such things are entirely non-constructive; they exist in an idealized, mathematical universe, but we can never describe them in a simple or concrete way. In other words, there is a gap between the things that mathematics allows us to create and those which we may explicitly describe. It is exactly this gap, and the objects which fall just on either side of it, that I propose to study over the course of my Discovery Grant. In doing so, I hope to shed new light on the limits of certain, more constructive, formal methods. Those objects which fall outside of their reach represent hard limits on what those methods can accomplish. Meanwhile, for those objects that are within that reach, we can often exploit this fact, as having simple descriptions ensures they have will additional structure. By discovering both new examples of these phenomena and developing general principles which characterize when certain paradoxical objects can and, just as importantly, cannot, have simple descriptions, we will better chart this murky divide for future explorers of the infinite.