Université Bourgogne Franche-Comté (UBFC)

PhD thesis : Random walks and surface diffeomorphisms

Université Bourgogne Franche-Comté (UBFC)

DIJON, France

Thesis subject : Random walks and surface diffeomorphisms

The subject is at the interface between dynamical systems and group theory, proposing to study groups of surface diffeomorphisms. Little is known about these groups, but a very   recent work by Bowden, Hensel, Webb promises to revolutionize the field. In fact, the group of surface diffeomorphisms naturally splits into two parts: the mapping class group is the quotient group with respect to the isotopy relation, and it is well studied by geometrical group theory and topological methods. On the other hand, it is difficult to study the subgroup of diffeomorphisms isotopic to the identity by classical methods of geometric   group theory. This is now possible because Bowden, Hensel, and Webb introduced a hyperbolic space on which this group acts in a non-elementary way. The challenge is to extract information on the dynamics on the surface from the action on this hyperbolic space.

We propose in this context a probabilistic approach, namely we want to consider a random walk on the group of diffeomorphisms isotopic to the identity and extract information from it by the random walk induced on the Bowden-Hensel-Webb space. The study of random walks on groups is a very classic subject: Furstenberg was one of the first to consider random products of matrices (which amounts to a random walk on GL (n, R)), and more generally it is a classic procedure to consider random walks on Lie groups. A random walk on a group of diffeomorphisms is the nonlinear analogue of this classical problem, and it often requires non-uniformly hyperbolic dynamical systems methods. For the case of surface diffeomorphisms these techniques were implemented in a masterful work by   Brown and Rodriguez Hertz, then recently taken up by Cantat and Dujardin in the holomorphic context. On the other hand, random walks on mapping class groups have been well studied by geometric group theory methods, and the Bowden-Hensel-Webb space could allow the use of similar techniques.

One of the objectives of the project is to make a connection between the work of Brown-Rodriguez Hertz and the random walks in space of Bowden-Hensel-Webb.

 

Bibliography:

-Bowden, Hensel, Webb – Quasi-morphisms on surface diffeomorphism groups, JAMS (to appear)

-Brown, Rodriguez Hertz – Measure rigidity for random dynamics on surfaces and related skew products, JAMS (2017)

-Cantat, Dujardin – Random dynamics on real and complex projective surfaces, arXiv:2006.04394

-Maher – Random walks on the mapping class group, Duke Math. J. (2011)

Keywords: Dynamical systems, random walks, group theory

Host Laboratory:

IMB – Institut de Mathématiques de Bourgogne

Adress Host Laboratory:

Université de Bourgogne – 9, av. Alain Savary – 21078 Dijon Cedex – France

Supervisors:

Taflin, Johan – johan.taflin@u-bourgogne.fr

Triestino, Michele – michele.triestino@u-bourgogne.fr

Contract duration: 36 months

Jobs Hours: Full time

Deadline application: May 15th, 2021

Starting job: October 1st 2021

Candidate profile:

Qualification: Master degree

It is recommended that the PhD student has knowledge in dynamical systems and ergodic theory, as well as in geometric group theory and surface topology.

Application:

Please send the following documents (all in one PDF file) by e-mail to johan.taflin@u-bourgogne.fr, michele.triestino@u-bourgogne.fr  :

1) For EU candidates: Copy of your national ID card or of your passport page where your photo is printed.

For non-EU candidates: Copy of your passport page where your photo is printed.

2) Curriculum Vitae (1 page).

3) Letter of motivation relatively to the position (1 page).

4) Copy of your Master degree and/or Engineer degree if already available.

5) Copy of your final marks and ranks.

6) Coordinates of reference persons (maximum 3, at least your master thesis supervisor): Title, Name, organization, e-mail.

If you have questions regarding the application, please contact the supervisors.

Apply with CV and Cover Letter

Must be a .doc, .docx, or .pdf file and no larger than 1MBMust be a .doc, .docx, or .pdf file and no larger than 1MB
Position

PhD thesis : Random walks and surface diffeomorphisms