For this class, I want students to complete a smallish project, which should involve some mathematics and some computation.
Ideally, this is done in pairs.
Each group will make a presentation to the rest of us of their project DATE TBD.
This will be about 30 minutes and would include a brief discussion of the mathematics, as
well as a brief demonstration of some software that was written for the project.
You would also hand in a short typed (LaTeX) description of your work.
It is not intended to be onerous, but to stimulate each of you to go a little further than
the lectures.
Different topics may have different emphasis on how much to write up and how much to code.
Possible Topics:
Appendix D of Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea, has some suggestions of projects, some of which may be suitible for this
class.
SAGBI Bases. This algorithm is for Subalgebra Analog of Gröbner Basis for Ideals.
These are also called Canonical Subalgebra Bases.
One Introduction is in Chapter 11 of the very readable book Gröbner bases and convex polytopes by Bernd Sturmfels.
You could also consider other topics from Gröbner bases and convex polytopes.
Exploring the use of resultants for implicitization of surfaces, with software.
Study and present Newton polytopes + Mixed volume + Bernstein's theorem.
Exploring the Grassmannian further than was done in class. Such as the Plücker ideal, or Schubert varieties and Schubert calculus.
Tropical geometry, specifically looking at what + and * are for.
This pertains to both the previous topic and the next one. Using the software GFan to compute Gröbner fans and tropical varieties.
Gröbner fan and Newton polytopes. (This would include Universal Gröbner bases.)
There are many topics in numerical computation that we have not yet gotten to.
There are questions in arithmetic geometry that are amenable to computataion, including Galois groups.
Projects that were done in a previous classes
Free Resolutions and Regular Sequences
Smale's 17th problem for binomial and trinomial systems
Special Concise Tensors
Numerically Computing Newton Polytopes
Viro's Construction
Equations of polynomials of power form
The Plücker ideal for the Grassmannian
Equations for Schubert varieties
Degenerations of toric varieties and regular subdivsions
Certified path-tracking of Newton Homotopies
Computing Amoebas
Syzygies, free resolutions, and Hilbert Polynomials
