Classical Sturm-Liouville theory tells us that second-order ODEs on an interval behave a lot like sine and cosine: the n-th solution has n-1 roots, the roots interlace, and so on.  The story is much more exciting than that -- Sturm originally proved a much stronger result that seems to have been forgotten over time (it's unclear why; one theory is that Courant & Hilbert didn't put it in their book).  The theory is completely open in higher dimensions, we survey the topological version (featuring Courant, Gelfand, Arnold, Bourgain, ...) and discuss some recent advances in the metric theory (featuring Yau, Logunov, ...) -- in particular, we discuss a new approach that is based on a new type of geometric inequality in optimal transport.

Stefan Steinerberger received his Ph.D. from the University of Bonn in 2013 and has since been at Yale, first as a postdoc and now as an assistant professor. His work is in analysis with emphasis on harmonic analysis, spectral theory, partial differential equations and applications to other fields.