I am (jointly) running the following seminars and reading groups.
1) Differential Geometry Reading Group. Closely following some nice course lecture notes and problem sets by Zuoqin Wang (can be found here and here) with a goal of covering differential forms, integration on manifolds, Stokes theorem, vector bundles, and a fast-paced survey of essentials of Riemannian geometry.
2) Mathematical and Theoretical Physics Seminar. Reading Brian C. Hall's "Quantum Theory for Mathematicians" with a mixed audience of mathematicians and physicists. Here and there, we might throw some real physics into the mix and approach classical (and quantum) field theories.
3) Harmonic Analysis and Operator Algebras Seminar. Reading Deitmar-Echterhoff's "Principles of Harmonic Analysis". While we have started out with a goal of covering the first 8 chapters of the book, we may not do so linearly. Both general aspects of operator algebras/theory and specific topics of interest aligning with quantum information and computing can be considered in parallel.
I am also participating in a seminar on symmetric spaces with an eye toward representation theory.