I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
Some multilinear algebra including the Kernel Theorem and Volterra composition,
Some Fourier analysis including the Bochner-Schwartz Theorem,
An introduction to wavelets with a view to structure theorems for spaces of distributions or function spaces,
Probability theory on spaces of distributions including the Lévy Continuity Theorem,
A study of homogeneous distributions and elementary solutions to linear PDEs.
My question is: What "cool topics/applications" would it be nice to include in such a course? I am particularly interested in examples with a high return on investment, i.e., that would not take too long to cover yet would provide the students with valuable tools for eventually a future research career in analysis. Please provide references where I can learn more about your suggestions. I would like some variety if possible. I got suggestions pertaining to probability, PDEs and mathematical physics, but it would be nice to get apps related to other areas of math.