Logic
and
Set Theory

Computability, Complexity and Randomness

Algorithmic randomness, Effective descriptive set theory, Computable analysis, Computable algebra and model theory

Combinatorial set theory

As Walter Deuber said, "complete disorder is impossible". Combinatorial set theory searches out swaths of uniform behavior, especially in various types of partition relations, of the cardinals, ordinals, linear orders, and partial orders that underpin many transfinite arguments.

Descriptive set theory

Descriptive set theory is the study of definability of various sets of real numbers. It turns out that we can infer many properties of such a set just from the syntax of its definition. One famous tool of this theory is the determinacy of infinite games, stating that one player must necessarily have a winning strategy in a certain type of two-player game of infinite duration. One popular application of this theory is the rating of many key problems in pure mathematics according to their complexity.

Forcing with ideals

Shelah's powerful method of proper forcing, used for decades for independence results regarding the structure of the real line can be sharpened quite a bit under the assumption that the forcing is in a certain sense definable. This leads to the study of various σ-ideals on Polish spaces and the quotient algebras of Borel sets modulo the ideal. The forcing properties of the quotient are closely connected with the analytic and descriptive properties of the ideal, providing a strong link between proper forcing and such fields as descriptive set theory, measure theory, dynamical systems and Ramsey theory.

Inner model theory

Inner models, large cardinals, notions of forcing, descriptive set theory.