Singularities of vector distributions and foliations
| 1. Equations A(x,y)dx+B(x,y)dy=0 as a starting point for several modern math topics. |
| 2. Vector fields and differential forms on the plane. Phase portraits. Integrability. Center-focus problem and other open problems. |
| 3. Foliations. Frobenius theorem. Malgrange theorem (Frobenius through singularities) and Kupka phenomenon. Other results and open problems on singular foliations. |
| 4. Generic vector distributions. Non-holonomic dynamical systems. Darboux theorem. Engel theorem. |
| 5. Singularities of distributions. Contact geometry.
Darboux-Givental' theorem. Geometric point of view on PDE's. Recent works by V. Arnol'd on local contact algebra. |
| 6. Cartan's prolongation. Cartan-Goursat flags and curves in contact space. |
| 7. Control systems - invariant point of view. Darboux and Martinet theorems in terms of control theory. |
| 8. Basic methods and techniques. |