Recent frontiers in physics education research include systematic investigations in the upper division. I have been involved in several collaborative efforts to conduct research on student learning in thermal and statistical physics. The focus in thermodynamics has been on student ideas about the First and Second Laws and the associated concepts (e.g., work, heat, entropy); several studies yield additional insights about broader ideas, such as state functions. Research in statistical physics has focused on the concepts underlying multiplicity and related ideas in probability. Our research interests have included aspects of more advanced physics thinking, including connections between physics and relevant mathematics concepts in many of these areas, in order to explore the interaction of the mathematics and the physics in student understanding; examples include student interpretation of canonical representations, such as pressure-volume (P-V) diagrams, partial derivatives, and Taylor series expansions. We have recently extended our work to investigate student understanding in analogous mechanical and chemical engineering courses. Results from research are guiding the development of curricular materials designed for the upper division.