Welschinger signs and webs of maximally inflected curves
 
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:  We define a
mixed Wronskian, a function P^1 ->  P^1.  The inverse image of the positive
reals encodes the real geometry of  f and conjecturally is 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.