Parameter spaces in holomorphic dynamics. – LAMBDA

LAMBDA is a project in pure mathematics, in the field of dynamical systems. Its purpose is the study of parameter spaces of
holomorphic dynamical systems in one and several complex variables.

An emblematic and classic example in this topic is the family of quadratic polynomials
z^2+c, with c in C, which gives birth to the celebrated Mandelbrot set.
Nonetheless the theory of spaces of rational mappings on the Riemann sphere has diversified
considerably in the last few years, with the introduction of modern tools from higher dimensional complex analysis and
geometry, as well as non-Archimedean techniques. At the same time, the exploration of parameter spaces of higher dimensional holomorphic dynamical systems has begun.

The ambition of this project is to consider these problems as a whole, by initiating a systematic study of
higher dimensional parameter space phenomena. By this we mean higher dimensional parameter space of one-dimensional mappings, as well as parameter spaces of higher dimensional mappings.
For this, we have gathered a large number of experts in one and several complex variables holomorphic dynamics,
in order to make significative progress on a wide range of topics, including:
-techniques of pluripotential theory (bifurcation currents, etc);
-complex and non-Archimedean geometry of the moduli space of rational maps on P^1 (special subvarieties, compactifications, etc.);
-measure theoretic issues (variation of Hausdorff dimension of Julia sets, entropy, etc.);
-bifurcation theory in higher dimensional holomorphic dynamics.

Another main objective of the project is to break the cultural barriers between the
communities of dynamicists in one and several complex variables.

In addition to these theoretical aspects, we will develop numerical tools for computer assisted proofs of parameter space theorems.

We will also undertake actions promoting this field of research among a general audience (in particular high school and university students),
in association with the IREM of Toulouse.

The team is divided in two partners respectively centered in Paris and Toulouse. It comprises 23 members, among them 17 with permanent
positions, with a dense array of active collaborations. Many of its members are widely recognized experts in the topic.
We will organize two internal meetings every year to help homogenize mathematical knowledge inside the group, stay informed about the state of the art,
and report about work in progress by the members.
An international conference will be organized in the second half of the project to review the most recent advances in the topic.

The acronym LAMBDA was chosen to allude to a well-established notational tradition for parameter spaces in this area of mathematics.