curso por Dmitry Fuchs (University of California, Davis, EUA)
1ª lição: Equations of degree 3 and 4
It is broadly known that along with the formula solving a quadratic equation, there exists a formula for solutions of a cubic equation. Most people, however, never use this formula, because it is, allegedly,
too long and not convenient. We will consider (and prove) the formula and will find it short and easy to memorize. We will find, however, that it is sort of not reliable: very often it gives the solutions in a
non-satisfactory form. I will explain what the formula actually gives, and will deduce a similar formula for equations of degree 4.
2ª lição: Equations of degree 5
I will give a full proof, elementary and convincing, of Abel's theorem stating that equations of degree 5 cannot be "solved in radicals". The proof will not look like usual mathematical proofs: we will discuss obvious things, seemingly not related to our problem, and at the moment when you will feel that it is time to stop walking around and to start proving the theorem, we will find that the proof is over.
3ª lição: Geometry of equations
A simple drawing in the plane will provide a machine solving cubic equations. This can be generalized to many other, non-algebraic, equations. A generalization to equations of degree 4 requires a drawing in space, strangely enough, the same, which will be instrumental in the same day lecture of Serge Tabachnikov's course.
4ª lição: Straight lines on curved surfaces
A surface is called ruled if for every point of this surface there is a line which contains the point and is contained in the surface. A surface is called doubly ruled, if for every point of the surface there are two such lines. There are classical doubly ruled surfaces: one-sheeted hyperboloids and hyperbolic paraboloids. It turns out that there are no other doubly ruled surfaces. We will prove this and will discuss some geometric properties of the doubly ruled surfaces and lines on them.
5ª lição: Twenty-seven lines
Cubic surfaces (that is, surfaces determined in space by equations of degree 3) are not ruled. Still, they contain many straight lines; for almost all cubic surfaces the number of these lines is 27. Usually, however, many of these lines are not real (complex). Still there are surfaces with all the 27 lines being real. We will give a proof of these results (very beautiful) and will discuss the properties of the 27 lines. For example, each of these lines crosses 10 other lines and misses the remaining 16. Also, there are precisely 45 planes that contain 3 of the 27 lines each. And so on.
