### Research group:

Problems related to Monster tower

#### 16 - 21 September, 2011, Warsaw

Over the last two years, after an exhaustive contribution, coauthored by Montgomery and Zhitomirskii was written, the Monster tower has attracted a considerable amount of attention. The main directions of research are as follows:

1. The classical (1, 2) Monster tower as a universal space for the classical Goursat distributions. In particular, a number of conjectures related to the nilpotent approximations of locally universal Goursat structures living on the stages of that tower.

2. The (1, *n*) Monster tower, *n* ≥ 3, and related classification of curves in R*^n*. A certain amount of results were obtained by P. Mormul, J. Adachi, F. Pelletier, R. Montgomery, A. Castro and some students of R. Montgomery. The main open question: is it possible to reduce the classification of points of the (1, *n*) Monster tower to the classification of jets of curves in R^*n*, for any *n *≥ 3 (for *n *= 2 it is possible).

3. The (*k, n*) Monster tower,* k* ≥ 2: construction and properties of points and curves [tangent to distributions pertinent to that tower]. Montgomery and Zhitomirskii started this reserach and discovered many surprising facts in the very construction of the relevant tower.

The planned reserach group would firstly focus on the first direction - a direct continuation of the activity of Section No 17 in the joint Israeli-Polish PTM meeting scheduled for September 11 - 15 in Łódź. Several standing conjectures bearing on the nilpotent approximations of Goursat distributions are to be discussed. We also plan to discuss the second and third directions, and related problems, which are closer to the participants J. Adachi and W. Domitrz. Yet within these directions we will concentrate on fixing open interesting problems rather than attacking them.

The main regular fully-fledged lectures will be given by P. Mormul and M. Zhitomirskii. The other participants will give "seminar-style" lectures, while all four of us would take part in the discussions following each lecture.

### Organizer

Piotr Mormul (Polish Academy of Sciences)

### Sources

- P. Mormul; Multi-dimensional Cartan prolongations and special
*k*-flags - P. Mormul; Singularity classes of special 2-flags
- P. Mormul; Do moduli of Goursat distributions appear on the level of nilpotent approximations?
- P. Mormul; Small growth vectors of the compactifications of the contact systems on J
^{r}(1,1) - K. Shibuya, K. Yamaguchi; Drapeau theorem for differential systems
- J. Adachi; Global stability of distributions of higher corank of derived length one
- J. Adachi; Global stability of special multi-flags
- A. Kumpera; Flag systems and ordinary differential equations
- A. Kumpera, J. L. Rubin; Multi-flag systems and ordinary differential equations
- A. L. Castro, R. Montgomery; Curve singularities and Monster/Semple tower
- P. Mormul, F. Pelletier; Special 2-flags in lengths not exceeding four: a study in strong nilpotency of distributions
- W. Domitrz, S. Janeczko, M. Zhitomirskii; Relative Poincare lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety
- W. Domitrz; Local symplectic algebra of quasi-homogeneous curves
- W. Domitrz, J. H. Rieger; Volume preserving subgroups of
*A*and*K*and singularities in unimodular geometry - A. Bellaïche, The tangent space in sub-Riemannian geometry, In: Sub-Riemannian Geometry, Progress in Mathematics 144, Birkhäuser 1996, pp.1-78.
- S. You. Ignatovich, Realizable growth vectors of affine control systems. J. Dynam. Control Systems 15 (2009), 557-585
- F.Jean, The car with N trailers: characterization of the singular configurations. ESAIM: COCV 1 (1996), 241-266 (electronic)
- R. Montgomery, M. Zhitomirskii, Geometric approach to Goursat flags. Ann. Inst. H. Poincaré- AN 18 (2001, 459-493
- R. Montgomery, M. Zhitomirskii, Points and Curves in the Monster Tower. Memoirs of the AMS 956 (2010).
- R. Montgomery, M. Zhitomirskii, V. Swaminathan, Resolving singularities with Cartan's prolongation. J. Fixed Point Theory Appl. 3 (2008), 353-378.
- P. Mormul, Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors. Central Europ. J. Math. 2 (2004), 859-883.

### Participants

- Jiro Adachi (Hokkaido University, Sapporo, Japan)
- Wojciech Domitrz (Technical University of Warsaw, Poland)
- Piotr Mormul (IM PAN)
- Mikhail Zhitomirskii (Technion, Haifa, Israel)