Type 4 impossibility results

Many important and classical results in mathematics show that certain tasks are impossible. Examples are “you can’t square the circle” or that “the square root of two cannot be expressed as a fraction of natural numbers.” Important recent applications of these types of results are in cryptocurrency and cryptography . There are certain problems in mathematics that were posed about a century ago that are still not solved. The research in this grant explains why: the current techniques for solution can be proved to be not strong enough to solve them. The project involves graduate students. Between problems that can be solved with inherently finite (i.e., recursive) information and problems that require the uncountable Axiom of Choice to solve (such as the existence of a non-Lebesgue measurable set), there are problems that cannot be solved using inherently countable information. This project shows that classical problems of von Neumann from the 1930s about ergodic theory and of Poincaré about qualitative dynamics at the turn of the 20th century cannot be solved using inherently countable information. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.