The study of canonical inner models for large cardinal hypotheses, with particular attention to their fine structure theory, and iterability issues.
An inner model is a transitive class model of ZF(C). The first nontrivial inner model, Gödel's constructible universe L, showed that the axiom of choice and the generalised continuum hypothesis were consistent relative to the axioms of Zermelo–Fraenkel set theory ZF.
However, only weak large cardinal properties survive in L. The principal goal of inner model theory is to define (at least under appropriate anti-large cardinal assumptions) a core model K that reflects the large cardinal structure of V (this structure may be present in V not in the form of actual large cardinals, but through strong combinatorial properties, while in K we expect to see the actual large cardinals "needed" for the presence of these properties).
The core model should be like L, in that its construction should be canonical (in several precise ways, for example, it should be invariant under set forcing extensions of V) and accomplished "from below" via definability. It should also be close to V, meaning that we require some form of covering lemma.
Recent developments in the theory are intimately tied up with the study of natural models of determinacy and their class HOD of hereditarily ordinal definable sets. This recent approach is called descriptive inner model theory.
