Courant, Bézout and persistence

It is possible to measure oscillations of functions by means of the theory of persistence modules and barcodes. I will explain how Sobolev norms can control such measurements. Applications include generalizations of the Courant nodal domain theorem and the Bézout theorem. The talk is based on joint works with Jordan Payette, Iosif Polterovich, Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević, and on a joint work with Aleksandr Logunov and Mikhail Sodin.