The first part of the talk will deal with spectral problems for scalar partial differential operators and, in particular, with the so-called Weyl Conjecture (existence of a two-term asymptotic formula for the counting function). This will be, essentially, a popular overview of the subject, charting its development from the non-rigorous work of physicists to the eventual rigorous proof of the Weyl Conjecture. This part of the talk will be based on the Safarov & Vassiliev book [1].
The second part of the talk will deal with spectral problems for systems. The general consensus has always been that spectral analysis of systems is similar to that of scalar problems, only somewhat more complicated technically. I will show that systems are fundamentally different in that as soon as you are working with a pair of unknown functions instead of one, you get a very rich geometric structure emerging from spectral analysis. In particular, I will show that geometric concepts such as metric, teleparallel connection, torsion, spinor, Dirac Lagrangian and electromagnetic covector potential appear naturally as a result of spectral analysis of first order systems. This gives a new perspective on the origins of the main equations of theoretical physics in which all geometric concepts are, traditionally, introduced in an axiomatic fashion.
[1] Yu.Safarov and D.Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, 1997 (hardcover), 1998 (softcover).