In 80th, V. Arnold conjectured that generic finite-parameter families of vector fields on the two-sphere behave almost as regularly as generic vector fields. In particular, he conjectured that a generic finite-parameter family is structurally stable.

A few years ago, Yu. Ilyashenko, I. Schurov and me provided an example of a locally generic structurally unstable 3-parameter family of vector fields on the two-sphere, see http://rdcu.be/IqCp for details.

During the last year, we (N. Solodovnikov, N. Goncharuk, and myself) found several even more surprising examples. Some of them were mentioned by Yu. Ilyashenko in his talk at Oliver Club last spring, while others were found in July.

I plan to give 2 talks on this subject. During the talks, I'm going to show many nice pictures, and formulate some open problems that I belive to be doable by graduate (or even strong undergraduate) students.