### ERC Project “HOLOGRAM”

Our team is working on the major project "HOLOGRAM" (**HOLO**morphic dynamics connecting **G**eometry, **R**oot finding, **A**lgebra and the **M**andelbrot set) funded by the European Research Council (ERC) under the Advanced Grant scheme.

The project is directed by Prof. Dierk Schleicher, and is aimed, by exploiting various mathematical disciplines, to advance the theory of holomorphic dynamics in several important directions from polynomials to rational and transcendental maps. The research project started in **October, 2016**, and will run for a period of **five years (2016-2021)**, with the total budget over **2.3 million Euro**.

Research of our dynamics group in this project is focused into the following four major directions:

**Combinatorics and Rigidity of Rational Maps**

The goal is to develop classification of rational maps, in particular, to provide good combinatorial models for their dynamics. Moreover, we aim to establish a Rigidity Principle asserting that the additional challenges of non-polynomial rational maps are encoded in the simpler polynomial setting.

**Thurston Theory, Laminations, Entropy, and Transcendental Maps**

We work on advancing Thurston’s fundamental characterization theorem of rational maps and his lamination theory to the world of transcendental maps, thus developing a novel way of understanding of spaces of iterated polynomials and transcendental maps.

**Newton’s Method and Root Finding**

Here we target at developing an extremely efficient polynomial root finder based on Newton’s method, which allows factorizing polynomials of degree several million in a matter of minutes rather than months. Our considerations will provide a family of rational maps that are highly susceptible to combinatorial analysis, leading the way for an understanding of more general maps.

**Iterated Monodromy Groups (IMGs)**

IMG is an innovative concept that helps to 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”. One of our goals in this theme is to advance the theory of IMGs of rational and transcendental maps.