Abstract: A 3-dimensional subspace f of real polynomials defines a map f : P^1 -> P^2 whose image is a rational plane curve. It is maximally inflected when all of its flexes are real, equivalently, when its Wronski determinant has only real roots. We associate two a priori distinct signs (\pm 1) to f: the Welschinger invariant of the rational curve and the degree of the Wronski map at f. Extensive computation suggests that these signs coincide. While studying this conjecture we were led to a deeper conjecture: From f, we define a a function W : CP^1 -> CP^1 which encodes some real geometry of f and conjecturally gives an object called a web. We conjecture that known bijections between webs and standard Young tableaux and between tableaux and maximally inflected curves recovers the curve.

This talk will explain this picture with compelling evidence and beautiful pictures. It is joint work with Brazelton, Karp, Le, Levinson, McKean, Peltola, and Speyer.