Logic is the study of reasoning; and mathematical logic is the study of the type of reasoning done by mathematicians.
-(Shoenfield)
Logic became a subject in its own right toward the end of the nineteenth century at which time its primary application was toward the foundations of mathematics. Today mathematical logic is a thriving part of the mainstream of mathematics itself, pursuing its own goals but also interacting heavily with algebra, analysis, geometry and number theory. Our logic group is made up of:
- Barbara Csima (Computability theory)
- Rahim Moosa (Model theory)
- Ross Willard (Universal algebra)
- Andy Zucker (Set theory)
What follows is a brief description of these areas.
Computability theory studies the relative computational complexity of sets of natural numbers. Professor Csima's research focuses both on looking at structural properties of the class of all sets with respect to various reducibilities that compare information content, the most famous of which is Turing reducibility, and on applying notions from computability theory to learn about the (relative) complexity of mathematical structures and their attributes.
Model theory is a way of doing mathematics: given a mathematical object (such as a ring or a manifold) model theory makes explicit the structure that we are equipping the object with, and then studies the "geometry" of the sets that are definable therein. Professor Moosa's research has focused on the interactions of model theory with algebraic geometry. This includes the model theory of fields equipped with additional operators as well as the model theory of compact complex manifolds.
Universal algebra is the study of algebraic structures, that is, structures in the sense of logic having operations only (no relations), focusing especially on the models of their equational theories using approaches abstracted from algebra or borrowed from logic. Professor Willard's research has focussed on questions of representability of models, finite axiomatizability of theories, computability/uncomputability, and computational complexity.
Set theory is the study of sets, which in the formalism of the Zermelo-Fraenkel axioms with Choice (ZFC) can be used as a foundation for all of mathematics. Within set theory, Professor Zucker's interests include descriptive set theory (the study of the set-theoretic aspects of real analysis) and infinite combinatorics, with applications to areas outside of set theory such as topological dynamics and ergodic theory.