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

Objective

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”.

Call for proposal

ERC-2015-AdG
See other projects for this call

Host institution

TECHNISCHE UNIVERSITAT BERLIN
Address
Strasse Des 17 Juni 135
10623 Berlin
Germany
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 342 867,25

Beneficiaries (2)

TECHNISCHE UNIVERSITAT BERLIN
Germany
EU contribution
€ 1 342 867,25
Address
Strasse Des 17 Juni 135
10623 Berlin
Activity type
Higher or Secondary Education Establishments
JACOBS UNIVERSITY BREMEN GGMBH
Germany
EU contribution
€ 969 613,75
Address
Campus Ring 1
28759 Bremen
Activity type
Higher or Secondary Education Establishments