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Holomorphic Dynamics connecting Geometry, Root-Finding, Algebra, and the Mandelbrot set

Periodic Reporting for period 3 - HOLOGRAM (Holomorphic Dynamics connecting Geometry, Root-Finding, Algebra, and the Mandelbrot set)

Reporting period: 2018-12-01 to 2019-09-30

Dynamical systems play an important role all over science, from celestial mechanics, evolution biology and economics to mathematics. Specifically holomorphic dynamics has been credited as “straddling the traditional borders between pure and applied mathematics”. Activities of numerous top-level mathematicians, including Fields medalists and Abel laureates, demonstrate the attractivity of holomorphic dynamics as an active and challenging research field.
We propose to work on a research project based in holomorphic dynamics that actively connects to adjacent mathematical fields. We work on four closely connected Themes:
A. we develop a classification of holomorphic dynamical systems and a Rigidity Principle, proposing the view that many of the additional challenges of non-polynomial rational maps are encoded in the simpler polynomial setting;
B. we advance 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;
C. we develop 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 several million in a matter of minutes rather than months – and providing a family of rational maps that are highly susceptible to combinatorial analysis, leading the way for an understanding of more general maps;
D. and we connect this to geometric group theory via “Iterated Monodromy Groups”, an innovative concept that helps solve dynamical questions in terms of their group structure, and that contributes to geometric group theory by providing natural classes of groups with properties that used to be thought of as “exotic”.

The importance of this research project, like much of fundamental research, lies less on its immediate impact for practical problems, but on the development of a fundamental understanding of underlying mathematical issues on an deep and abstract level.
"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).

B. we are extending the fundamental ""Thurston theory"" of holomorphic dynamics from the well known setting of rational maps on the Riemann sphere to interesting families of transcendental functions, i.e. from maps of finite degree to those of infinite degree. This is an extension that has been conjectured for more than 30 years.

C. We are developing and fine-tuning Newton's well known root finding method, to a setting that has completely found all roots of certain polynomials of degree greater than one billion, and that advances this method both in practice and in its theoretical understanding.

D. We are advancing the theory of Iterated Monodromy Groups, an active and vibrant link between dynamical systems and group theory, in particular in the direction of extending it from rational to transcendental dynamics, and in enhancing the understanding of the properties of the resulting groups."
All the progress mentioned goes substantially beyond the state of art, in all four Themes. We expect further profound contributions to the field of holomorphic dynamics, as well as adjacent fields such as geometry group theory and numerical analysis, in all the directions mentioned.