## Laurent Bartholdi

Directeur de Recherche CDD

Équipe d'Algèbre et Géométrie
DMA - Ecole Normale Supérieure
45, rue d'Ulm
75230 Paris Cedex 05 - France
E-mail: Laurent.Bartholdi@ens.fr
Bureau: T5, Toits du DMA
Téléphone: +33 1 44 32 20 60
Fax: +33 1 44 32 20 80

### Séminaires et visiteurs:

• 06.12.2016: Après-midi de théorie des systèmes dynamiques.
14:00-14:45: Kevin Pilgrim, "Affine endomorphisms of spheres, push-point homeomorphisms, and complex dynamics"
Recent work of Selinger-Yampolsky and Bartholdi-Dudko suggests that there may be a practical algorithmic classification theory for critically finite rational functions and the countable collection of topological cousin maps in which they naturally lie. I will discuss what form this might take when restricted to the class of so-called Nearly Euclidean Thurston (NET) maps. This is a rich class of examples which are slight combinatorial perturbations of affine endomorphisms of the two-sphere to itself. Given such a map, many fundamental invariants are practically computable, and the natural dynamical correspondence on moduli space is a classical modular curve. W. Floyd and W. Parry have tabulated extensive data at http://www.math.vt.edu/netmaps/index.php. This is joint work with W. Floyd, G. Kelsey, S. Koch, R. Lodge, W. Parry, and A. Saenz-Maldonado.
15:00-15:45: Thomas Gauthier, "Classification of special curves of cubic polynomials"
Post-critically finite polynomials play a particular role in studying parameter spaces of complex dynamical systems for several reasons: their dynamics is simpler to understand and they are equidistributed with respect to a certain dynamically relevant probability measure in the parameter space. In this talk, I will describe their distribution from the point of view of the Zariski topology. This amounts to describe all irreducible algebraic curves in the space of cubic polynomials which contain infinitely many post-critically finite ones. This is joint work with C. Favre.
15:45-16:15: pause café
16:15-17:00: Dzmitry Dudko, "Pacman Renormalization and scaling of the Mandelbrot set at Siegel points"
In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of “Pacman Renormalization Theory” that combines features of quadratic-like and Siegel renormalizations. Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. We prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters. This is joint work with Mikhail Lyubich and Nikita Selinger.
• 08.11.2016: Après-midi de théorie des groupes et probabilités.
14:00-14:45: Tianyi Zheng, "Speed of random walks and minimal growth of harmonic functions on groups"
We discuss a construction of groups where the speed (rate of escape) of simple random walk can follow any sufficiently regular function between diffusive and linear. When the speed of the $\mu$-random walk is sub-linear, all bounded $\mu$-harmonic functions are constant. We investigate the minimal growth of non-constant harmonic functions on these groups and show it is tightly related to the speed of the random walk. Based on joint works with Jeremie Brieussel, Gidi Amir and Gady Kozma.
15:00-15:45: Antoine Gournay, "Mixing, malnormal subgroups and cohomology in degree one"
Given a representation of a group $G$ (e.g. the Koopman representation associated to action on a probability measure space), under certain mixing conditions, the cohomology of the representation vanishes if the cohomology of the representation restricted to a subgroup $H$ vanishes. Harmonic functions play an important rôle in the proofs and results on ${\ell }^{p}$-cohomology are obtained as an intermediate step.
15:45-16:15: pause café
16:15-17:00: Sébastien Gouëzel, "Numerical estimates for the spectral radius in surface groups"
Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups), and discuss in particular the case of surface groups.
• 02.12.2016-14.12.2016: Kevin Pilgrim.
• 07.11.2016-21.11.2016: Tianyi Zheng.
• 20.06.2016-01.07.2016: Igor Lysionok.
Exposé le mardi 21.6.2016 à 14h00, IHP (rue Pierre-et-Marie Curie) salle 01: Burnside groups and small cancellation theory
Résumé: The Novikov-Adian theorem states that a non-cyclic Burnside group B(m,n) of odd exponent n greater or equal 665 is infinite. Starting from the original approach, all known proofs of infiniteness of B(m,n) utilize the idea that the group can be described in terms of some iterated small cancellation condition. In the last decade, several explicit implementations of small cancellation conditions of this type were introduced which can be applied also in a more general setup to groups acting on hyperbolic metric spaces. I will give a brief overview of the small cancellation approach to Burnside groups and describe yet another implementation providing a reasonably accessible proof that B(m,n) is infinite with rather moderate bound n > 2000 on the odd exponent n.
• 02.05.2016-13.05.2016: Alexei Kanel-Belov.
Exposé le mardi 3.5.2016 à 14h00, salle U ou V: On geometric ring theory
Résumé: Quite recently Ilia Rips and Arieh Juhasz constructed an Engel but not locally nilpotent group, i.e. a group with identities $[\dots[x,y],y,\dots,y]=e$. This group has non-positive curvature and big commutative parts: some parts have small cancellation and some commute. --- This group looks somehow like a ring and group multiplication sometimes behaves like multiplication and sometimes like addition. The theory of canonic forms of this group is applicable for rings in particular in the skew field construction.

Note also that rings are close to semigroups too. There is a hope nowdays to develop geometric ring theory.

• 02.11.2015-06.11.2015: Yutaka Ishii
• 20.10.2015-27.10.2015: Ross Goeghegan
• 05.10.2015-11.10.2015: Dylan Thurston.
Exposé le jeudi 8.10.2015 à 15h30, salle W: Discrete quadratic differentials
Résumé: Given a filling graph embedded on a surface, we propose a model for discrete quadratic differentials. Any Whitehead equivalence class of measured foliations has a unique representative as a harmonic quadratic differential. (For instance, every simple multi-curve has a unique such representative.) This gives a new, effective way to, for instance, compute the action of the mapping class group on a closed surface.