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