It all started in 1889 with F. Engel's article on the $(2,3,4)$-distributions.
The language was PDE's. Engel proved that the couple of differential
equations in 4 variables having generic properties (in contemporary language, a
rank-2 distribution ${\rm span}(X,Y)$ in the tangent bundle to a 4-manifold such
that ${\rm dim\, span}(X,Y,[X,Y]) = 3$ everywhere and ${\rm dim\, span}(X,Y,[X,Y],[X,[X,Y]],[Y,
[X,Y]]) = 4$
everywhere) behaves LOCALLY in a unique way.
E. von Weber continued in 1898 putting forward a condition now called
the Goursat condition: IF the derived growth $(2,3,4, \ldots, n-1,n)$ occurs locally
everywhere, THEN a unique model couple of PDE's in $n$ variables occurs.
This Weber's theorem was grossly false, because it implied that the Goursat condition
was locally trivial and had no singularities whatsoever. Yet von Weber was a pioneer.
E. Cartan (1914) started to geometrize, Goursat (1922) widely popularized.
A true depth of the Goursat condition was discovered only in 1978
(Giaro-Kumpera-Ruiz). These authors produced a singular behaviour within
the derived growth $(2,3,4,5)$. A renewal of interest in the singularities
of Goursat structures has been taking place since 1996 until now.
Since 1999 (Kumpera-Rubin) other regular derived growths have attracted
interest, in particular $(3,5,7, \ldots, 2m-1, 2m+1)$ everywhere, i.e.,
so-called SPECIAL 2-MONSTERS.
Attempts at local classification of them will be outlined, terminating
in mentioning an ISSUE OF INTEREST to the dynamical systems' community:
local diffeomorphisms of ${\mathbb R}^3$ that are not embeddable in flows of smooth
autonomous vector fields. The latter are badly obstructing the
classifications. A 2010' result says that the derived growth
$(3,5,7,9,11)$ has exactly
34 local geometric realizations. A challenging difficult issue is
the derived growth $(3,5,7,9,11,13)$. The speaker, helped recently
by A. Weber, hopes for a discrete/finite? local classification
of those special 2-monsters of length 5.
Meeting ID: 852 4277 3200
Passcode: 103121