Morse theory for plane algebraic curves

Knot Theory and Plane Algebraic Curves

Knot theory has been known for a long time to be a powerful tool for the study of the topology of local isolated singular points of a plane algebraic curve. However it is rather recently that knot theory has been used to study plane algebraic curves in the large. Given a reduced plane algebraic curve C2 passing through the origin, let Lr = \ @B4 r be the intersection of with a round ball in C2 ...

Plane Algebraic Curves

We go over some of the basics of plane algebraic curves, which are planar curves described as the set of solutions of a polynomial in two variables. We study many basic notions, such as projective space, parametrization, and the intersection of two curves. We end with the group law on the cubic and search for torsion points.

Real Plane Algebraic Curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...

Topology of Plane Algebraic Curves

Morse Theory from an Algebraic Viewpoint

Forman’s discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.