Many mathematical questions can be solved in multiple ways, each with its own advantages. Short, conceptual proofs can avoid complicated calculations, but sometimes cannot provide the detailed quantitative information those calculations would reveal. A recent insight is that, sometimes, mathematics can have it both ways. By studying the structure of mathematical proof itself, techniques from the field known as proof theory make it possible to take abstract proofs and to extract detailed calculations from them. Used in the opposite direction, these techniques can take certain kinds of lengthy calculations and replace them with short, abstract arguments which can then be generalized to prove new results. The focus of this project is to both further extend these techniques to new areas, particularly recently discovered applications in statistics, as well as continue the application of known techniques to new problems, especially in areas where probability and randomness play a central role.
In this project, Towsner will build on previous applications of ultraproducts to studying the way mathematical objects can be separated into structured and random parts. An explicit quantitative approach to such dichotomies has long been central to extremal graph theory, but recent work has shown that these can also be viewed in measure-theoretic terms by using ultraproducts to mediate between the finitary and infinitary perspectives, leading to new results in the area. Towsner will study this connection systematically, both developing new tools in the infinitary setting and using the proof-theoretic functional interpretation to translate these tools back to the classical setting, extracting explicit calculations from these infinitary arguments.
