Every finite planar tree can be drawn as the pre-image of the line segment \$[0,1]\$ by a polynomial with algebraic coefficients. There is a unique smallest-degree extension of \$\mathbb{Q}\$ in which one can take the coefficients, thus associating this field to the tree. All finite extensions of \$\mathbb{Q}\$ arise this way.
Thus begins the tale of Shabat polynomials, Belyi functions and the world of Grothendieck’s dessins d’enfants. We sketch the contours of this story, weaving together strands from complex analysis and conformal mappings, Riemann surfaces, group theory, Galois theory, algebraic topology, and even Brownian motion. We hope the audience member will share in our surprise that such simple objects could be the locus of such a rich intersection of mathematics.
The discussion will often be quite elementary and there will be many pictures. We will conclude with several simple-to-state open questions.