The modal logic GL ("Gödel-Löb"), also known as the logic of provability, has been shown to be arithmetically complete by Solovay, that is the modal operator of GL has precisely the modal properties of the provability predicate of PA in PA. In this context I'm interested in two question. Which of the two we address I would make dependent on the choice of participants of the seminar.
The first question would be whether there are alternative arithmetical completeness results once fixed points for modal formulas are added to the language and logic. For example, it there a modal logic that is arithmetically complete with respect to the truth predicate of FS? Taking up this question would involve working through Solovay's result thoroughly.
The second question or theme would be to investigate how modifying the definition of the provability predicate (whilst remaining extensional equivalent) might affect the modal properties of the provability predicate and to discuss these deviant logics of provability.
Boolos: The Logic of Provability.
Smorynski: Self-reference and Modal Logic.
Feferman: “The Arithmetization of Metamathematics in a General Setting”.
D. Guaspari and R. M. Solovay: “Rosser sentences”.
Czarnecki, Marek, and Konrad Zdanowski. "A modal logic of a truth definition for finite models." [Draft]
Visser: “Peano's Smart Children: A Provability Logical Study of Systems with Built-in Consistency”.