Seminar talk, 18 May 2011

Speaker: Rafael Sarkisyan

Title: A remark on the Cartan-Kahler theorem

Abstract:
The talk will discuss an example of algebraic PDE system $R^{q}\subset J^{q}(X,V)$ and a semi-algebraic set $D^{q}\subset R^{q}$ (this set consists of non-involutive points) such that in $D^{q}$ there are two everywhere dense subsets $A^{q}$ and $B^{q}$ having the following property. While for each point $a\in A^{q}$ it is possible to divide the set of partial derivatives of total order greater than $q$ into parametric and main part (as in the case of involutive points of $R^{q}$), this is not possible for points of $B^{q}$. We show how to state correctly an initial value problem for any point $b\in B^{q}$. This example is related to a question of V.I.Arnold.