Ring theory and algebraic geometry (Summer 2020, Session I) will study geometric objects such as curves from an algebraic point of view. In the abstract algebra class, we skipped ring theory, usually a central part of an abstract algebra class, because we were optimizing for interestingness, and ring theory on its own is not the most exciting of subjects. But the subject comes to life when used to study geometry (or number theory, but that’s a completely different class).
In algebraic geometry, we learn that there is a beautiful bijection between certain types of geometric objects known as varieties and certain types of rings. This bijection carries with it a lot of structure, and we will see that the ring provides us with lots of information about the varieties, and vice versa. For example, we can detect singular points of curves (such as nodes, where the curve has self-intersections) directly from the ring.
One of the most mysterious theorems from classical geometry is Pascal’s hexagon theorem, which says that if ABCDEF is a hexagon inscribed in a conic section so that no two sides are parallel, then the intersections of AB and DE, BC and EF, CD and FA, are collinear. While this theorem can be proven using only classical geometry, such a proof is not particularly enlightening. The reason is that Pascal’s Theorem is really trying to say something about the intersections of algebraic curves and as such follows more naturally from a key theorem in algebraic geometry known as Bezout’s Theorem.
All this is rather classical and has been known since the 19th century. But algebraic geometry was also the subject of perhaps the most important mathematical advance in the 20th century, when the subject was completely reshaped by Alexander Grothendieck and others. While much of modern algebraic geometry requires considerable technical background, we will at least study the basics of this new approach and see how it subsumes the older formulations and behaves better in many cases.