## GeoMeG: Description

The general goal of the project is to study Lie groups equipped with arbitrary distance functions and to develop an adapted geometric measure theory in the non-Riemannian settings.

The main focus is subRiemannian geometry, together with implications to control systems and nilpotent groups. SubRiemannian spaces, and especially Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, subelliptic PDE's and geometric group theory.

The members of the team master methods coming from geometric measure theory, geometric analysis, and group theory; in particular, analysis on metric spaces and Lie group theory.

| Grant Agreement | University of Jyvskäylä Press Release |

## People

### Principal Investigator

Enrico Le Donne

| Personal website |

### Post-Docs

Thibaut Dumont
Francesca Tripaldi

### Ph.D. Students

Fares Essebei
Valerio Gianella
Eero Hakavuori
Ville Kivioja
Terhi Moisala
Sebastiano Nicolussi Golo

## Jyväskylä Geometry Seminar

The Jyväskylä Geometry seminar is on Mondays afternoons
in the Department of Mathematics and Statistics.
The meeting is usually from 14:15 to 16:00 in room MaD 380.

### Fall 2017

 23/October/2017 Asymptotic nilpotency of cellular automata (MaD 380) Ville Salo (University of Turku)   Part 1: Asymptotic nilpotency of classical cellular automata A cellular automaton (CA) is a continuous shift-commuting self-map of a subshift. By a classical CA we refer to a CA defined on a full shift over a free abelian group, namely on $S^{\mathbb{Z}^d}$ for a finite alphabet $S$ under the shift action of $\mathbb{Z}^d$. For $d = 1$, it was shown in [Guillon-Richard, 2008] that if such a CA is asymptotically nilpotent (every configuration converges to the all-zero configuration), then actually every point maps to the all-zero point after a bounded number of steps. This was generalized to CA on free abelian groups by the author in 2012. We explain some of the geometric ideas behind the proofs, and discuss the importance of nilpotency in the decidability theory of CA. $$\,$$ Part 2: Asymptotic nilpotency of non-classical cellular automata We call a (topological dynamical) $\mathbb{N}$-system asymptotically nilpotent (AN) if there is a special point 0 such that every point converges to 0 in the iteration of the dynamics, and uniformly asymptotically nilpotent (UAN) if this convergence is uniform over the space. A non-uniformly asymptotically nilpotent (NUAN) $\mathbb{N}$-system is one that is AN but not UAN. We say a family of $\mathbb{N}$-systems is nilrigid if it does not contain a NUAN map, and if $G$ is a group, we say a $G$-system is nilrigid if its family of endomorphisms ($G$-commuting continuous self-maps) is nilrigid. The results above state essentially that full shifts over abelian groups are nilrigid. We discuss how robust/fragile this result is, by studying nilrigidity when the "full shift" assumption is replaced by dynamical assumptions, e.g. by SFTness or soficness, when the alphabet is replaced by a manifold, or when we step into the measurable category, finding plenty of non-nilrigid situations. We then discuss the case when the acting group is non-abelian, e.g. the Heisenberg group or a free group.
 15/January/2018 TBA (MaD 380) TBA (TBA)
 22/January/2018 TBA (MaD 380) TBA (TBA)

## Recent Papers

Soon preprints related to ERC GeoMeG will be available in this space.

## Future Events

19 -23 February 2018 | Jyväskylä (Finland)
Sub-Riemannian Geometry and Beyond
| Website |

## Open Positions

Plaese, contact the PI if you are interested to do a Ph.D. or a postdoc
in the framework of GeoMeG ERC project.

New positions will be open in the next years.

Prof. Enrico Le Donne