Algebraic Theta Functions and Rational Solutions to the KP Equation

The theta function of a smooth algebraic curve provides solutions to the KP equation in mathematical physics. The theta function is highly transcendental function, but this can change when the curve becomes singular. I will present a classification of those singular curves whose theta function is polynomial, and prove that give rational solutions to the KP equation. In particular, I'll try to explain how everything essentially follows from Abel's theorem. This is joint work with John B. Little and Türkü Çelik.