Abstract: In this talk, I will present three current projects, at various stages of completion. The first (a set of questionnaires) focuses at the level of student and instructor perceptions. The second (a course 'flipping' trial) and third (a calculus intervention) on the course and student levels, respectively. Though the projects are seemingly disjoint, I will make the argument that our, and our students', perceptions of mathematics and of each other affect our students' mathematical experiences and ultimately their mathematics learning.

We introduce the approachability ideal I[\kappa] for regular \kappa, make some basic observations, and establish a connection with internally approachable models. Using the trichotomy theorem to guarantee the existence of an exact upper bound for a certain sequence, we proceed to prove the existence in ZFC of a scale of length \aleph_{\omega+1} in a reduced product \omega_k for k \in A, an infinite subset of \omega.

I will discuss the properties of discrete random Schrödinger operators in which the random part of the potential is supported on a sublattice. For the standard Anderson model, no results concerning localization/delocalization transition are rigorously established. For trimmed Anderson model described above, one can trace out the onset of the localization breakup, in the strong disorder regime (for some examples). This is a joint work with Sasha Sodin.