We made substantial progress on all four Themes of the proposal. To highlight selected substantial developments for each of the four Themes, we accomplished the following:
A. We developed a deep and very satisfactory understanding of all Newton maps for polynomials: these Newton maps are originally motivated by root finding, but they constitute interesting dynamical systems in their own right. Meanwhile, we developed a very precise understanding of these dynamical systems in a precision that is so far known only for polynomial maps (which are generally considered much easier). More precisely, we showed how to extend the fundamental issue of “rigidity” from (unicritical) polynomials to certain rational maps: we established the “Rational Rigidity” Principle to Newton maps; this means that such maps are always rigid unless they contain embedded non-rigid polynomial dynamics. In addition, we provided a complete classification of all postcritically finite Newton maps, which is the largest class of non-polynomial rational maps for which the power of Thurston’s famous characterization theorem has been brought to use. This, in turn, paved the way for one more important result, our “Decomposition Theorem”, that is a key element in using Thurston’s theorem for more general rational maps.
B. We advanced Thurston’s fundamental characterization theorem of rational maps and his lamination theory to the world of transcendental maps, developing a novel way of understanding of spaces of iterated polynomials and transcendental maps. In particular, we developed an extension of Thurston’s theorem to a large class of “structurally finite” transcendental maps, and developed perspectives for extensions in even greater generality. Such extensions have been conjectured for close to 40 years.
C. We developed an extremely efficient polynomial root finder based on Newton’s method that turns the perceived problem of “chaotic dynamics” into an advantage, factorizing polynomials of degree exceeding a billion in a matter of hours, far beyond the feasibility of known algorithms (and for certain families of known polynomials, we accomplished this in a matter of minutes rather than days). In addition, we developed a powerful theory for Newton’s method that makes this method unique in the sense that is works very efficiently in practice, and is supported by a powerful theory (even though the theory does not yet fully support the success in its full efficiency).
In addition, we investigated root finding methods in several variables that are of substantial practical use: the highlight result is that the Weierstrass-Durand-Kerner method is not generally convergent, which disproves practical experience of the numerics community for 130 years, but confirms a conjecture of Smale. This became possible by connecting the problem to computer algebra and exploiting modern computer algebra systems to their current limits. An interesting perspective is to investigate whether there are any generally convergent root finders in several variable (extending McMullen’s fundamental result in one variable).
Moreover, we discovered the new feature that both the Weierstrass and the Ehrlich-Aberth methods have “diverging” orbits that exist forever and converge to infinity or other non-root limit points.
D. We have investigated “Iterated Monodromy Groups”, an innovative concept that connects holomorphic dynamics to geometric group theory and helps solve dynamical questions in terms of their group structure, and that conversely contributes to geometric group theory by providing natural classes of groups with properties that used to be thought of as “exotic”. Here our main result is a complete understanding in which cases iterated monodromy groups of entire transcendental maps are amenable.
