Slaman will investigate the effective, and more generally definable, aspects of mathematical phenomena such as genericity, compactness, and randomness. Jointly with Veronica Becher and Pablo Heiber, both at the University of Buenos Aires, Slaman will apply methods from Computability and Descriptive Set Theory to normality of real numbers, the property that the digits in their representations occur with equal asymptotic frequency, especially considering representations in varying bases. Will also collaborate with Laurent Bienvenu, University of Paris Diderot-Paris 7, and Kelty Allen, an advanced graduate student working with Slaman, on the effective theory of Brownian motion. In parallel, Slaman will investigate the more foundational question, "What are the number theoretic consequences of familiar infinitary principles?" as formalized in first and second order arithmetic. The phrase "first order arithmetic" refers to the structure of the natural numbers N={0,1,2,...} with the operations of addition and multiplication. ``Second order arithmetic'' refers to the expansion of N to include all of the subsets of N and allowing for reference to and quantification over infinite sets. Stated more precisely, Slaman will investigate the extent to which second order principles, such as the existence of a random sequence or the assertion of a frequently applied infinitary combinatorial principle such as Ramsey's Theorem, have non-trivial consequences in first order arithmetic.
This project comes from the perspective of Mathematical Logic, that understanding the means by which one can work with mathematical objects can be as important as, or even equivalent to, understanding those objects themselves. In one of the more interdisciplinary parts of the proposal, this point of view will be invoked to study the problem of constructing real numbers so as to control the behaviors of their representations relative to all integer bases. One goal is to exhibit a fast-running algorithm to output an absolutely normal number, which means that for any integer base b the digits in the base b representation of this number occur with equal frequency over time. Absolute normality is often interpreted as an indicator of randomness, but such an accessible and predictable example would refute that view. Another part of the project puts Mathematical Logic in the foreground by asking for exact information concerning the extent that the properties of the real numbers, in the form of combinatorics seen within infinite subsets of the natural numbers, have consequences among the finite sets.
