The aims of this project are to investigate and make progress on various questions in inner model theory. An overview with some details can be seen in this project description. The project will focus on parts of the following 3 main topics:

Long extenders: Inner model theory has primarily focused on mice constructed from short extenders, but Woodin and others have investigated to some extent the generalization to long extenders. We aim to consider both the internal theory of long extender mice, and also the question of normalization of iteration trees for long extenders.

Varsovian models: This is a recently developing field, analysing self-iterability in mice, and its connection to HODs and mantles associated to mice (in particular, the set-theoretic geology of mice). There are various kinds of open questions.

Combinatorics: There is both a rich theory analysing the combinatorial principles which hold in mice, such as Diamond and Square principles, and also using inner models to derive consistency strength from combinatorial principles in V.