Geometric Structures seminar - Igor Zelenko - Thursday 20 Nov - 14:00 - Room 005

[Tommaso Rossi]

Dear All,

This is to announce the next seminar from the series "Geometric Structures".

*Date and venue*: Thursday 20 Nov - 14:00 - Room 005

*Speaker*: Igor Zelenko (Texas A&M University)

*Title*: Local Geometry of Distributions: Symplectification, Cartan 
Prolongation, and Maximality of Class

*Abstract*: In 1970, N. Tanaka introduced a method for constructing a 
canonical frame for distributions with constant Tanaka symbol. In 2009, 
B. Doubrov and I, building on earlier works of A. Agrachev, R. 
Gamkrelidze, and myself, employed a symplectification procedure to 
obtain a canonical frame for distributions independent of their Tanaka 
symbol. However, this required an additional assumption - the maximality 
of class of the distribution.

In a recent joint work with N. Day, we proved that all bracket 
generating rank-2 distributions with 5-dimensional cube are of maximal 
class at a generic point. This result allows one to assign a canonical 
frame at a generic point to every rank-2 distribution that is not of 
Goursat type. On the optimal control side, this result implies that for 
bracket-generating rank-2 distributions with 5-dimensional cube, there 
exist plenty of abnormal extremal trajectories starting from a generic 
point.

Further, in the rank-2 case, I will give an interpretation of the 
symplectification procedure in terms of a classical construction known 
as Cartan prolongation and discuss the question of the minimal number of 
iterative Cartan prolongations needed for the Tanaka symbols to become 
unified or finitely unified.

In contrast with the rank-2 situation, we found examples of rank-3 
distributions with 6-dimensional square that are not of maximal class. 
In particular, I will present a (3, 8) distribution of non-maximal class 
whose symmetry algebra has dimension 29 and contains a semidirect sum of 
the exceptional Lie algebra $\mathfrak g_2$ with a copy of its adjoint 
module.

Finally, if time permits, I will discuss the analogous results for 
normal geodesics in sub-Riemannian geometry and more general geometries, 
yielding algebraic proofs and extensions of several results of Agrachev 
that were originally obtained by analytic methods.


More information can be found here: 
https://researchseminars.org/seminar/Geometric_Structures_SISSA