Lecture notes

Problem set - it's enough to solve 1 of the exercises, and for the open problems you can instead write an essay on one of these.

Please submit the problem set by end of October 2025. Instructions on how to submit will follow soon!

This minicourse begins with elementary questions about zero sets of real polynomials in the high-degree regime and concludes with problems in analysis and PDE. The potential topics if the minicourse may include:

Bézout’s theorem; Harnack’s inequality; and the first part of Hilbert’s 16th problem on topological configurations of real algebraic curves.
Gudkov’s correspondence with Arnold and Rokhlin; Hilbert weights.
Analogs and breakdowns of algebraic properties in PDE: Courant’s nodal domain theorem; counterexamples to the Courant–Herrmann Conjecture and “transcendental” Bézout theorem.
Sodin's question: How different are nodal sets from algebraic curves—are there more or fewer restrictions?
The work of D. Elboim and A. Sly on "Drawing Random trees" and its applications to Sodin's question.
Y. Krauz's article "Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions".
One-dimensional questions on zero counts: Chebyshev systems; Sturm’s “lost” theorem; fewnomials; and a question of Noga Alon.
Gauss map and minimal surfaces; Osserman’s analog of Schottky’s theorem.

Selected elementary open questions:

How many points on the plane can be connected by an algebraic curve of degree ?
How many equilibrium points can an electrostatic field generated by point charges have (Maxwell's conjecture)?
Do biharmonic functions have double nests?
Is there a crude estimate (up to a multiplicative constant) or an algorithm for finding the minimal degree of a polynomial that realizes a prescribed topological configuration of loops on the plane?