Professor Steve Jackson invites you to attend the PhD dissertation defense of Rachid Atmai today, June 15^th, at 2:00 pm in GAB 438. Cake and coffee will be served in GAB 472 following this event.

"Contributions to descriptive set theory"

Abstract:

In this dissertation we study closure properties of pointclasses, scales on sets of reals and the models L[T_{2n}]. We first characterize projective-like hierarchies by their associated ordinals. This solves a conjecture of Steel and a conjecture of Kechris, Solovay, and Steel. The solution to the first conjecture allows us to reprove a strong partition property result on the ordinal of a Steel pointclass. We then develop new methods which produce lightface scales on certain sets of reals. The methods are inspired by Jackson's proof of the Kechris-Martin theorem. We then generalize the Kechris-Martin Theorem to all the Pi^1_{2n+1} pointclasses using Jackson's theory of descriptions. This in turns allows us to characterize the sets of reals of a certain initial segment of the models L[T_{2n}]. We then use this characterization and the generalization of Kechris-Martin theorem to show that the L[T_{2n}] are unique. This generalizes previous work of Hjorth. We then characterize the L[T_{2n}] in term of inner models theory, showing that they actually are constructible models over direct limit of mice with Woodin cardinals, and finally show that the generalized continuum hypothesis holds in these models, solving a conjecture of Woodin.