Projects in Algorithmic Algebraic Geometry

Instructor: Frank Sottile
Lectures: Mondays 8:00–10:00 024, Arnimallee 3
Fridays 8:00–10:00 007, Arnimallee 6
Exercises: Tuesday 12:00–14:00 007, Arnimallee 6

For this class, I want everyone to complete a smallish to medium-sized project. This should involve some mathematics and some computation. Ideally, you would do this in pairs.
Each group will make a presentation to the rest of us of their project either during the week 2–6 March or on 20 March. 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, and to satisfy whatever requirements we need for official purposes. Different topics may have different emphasis on how much to write up and how much to code.

Chosen Topics:

Monday 2 March	12:20	Abbas Murrey and Amoah Brefo	Approximation theory with Bézier curves
Wednesday 4 March	12:20	Monica Soto and Enrico Bartoletto	Binomial edge ideals of graphs.
Thursday 5 March	11:40	Alexey Gonus and Natalia Konstantinova	Tropical geometry
Thursday 5 March	12:20	Niklas Livchitz and Matthias Himmelmann	Tropical geometry
Friday 6 March	9:40	Lea Fliegner and André Zepernick	Implement Buchberger's algorithm in Singular
	10:20	Andrei Comaneci and Sofia Garzon	Toric ideals and integer programming
	11:00	Zijun Li and Lingwei Ye	The group law on an elliptic curve
	11:40	Shuvadarshan Bhual, Subodh Singh Khanger, and Abinaya Jayaprakash	Computation of Free Resolutions over R-modules.
	12:20	Fawzy Naguib	Syzygies, free resolutions, and Hilbert Polynomials.

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.
Degenerations of toric varieties coming from weighted orders and regular subdivsions.
Exploring the use of resultants for implicitization of surfaces, with software.
Implementing the group law on a cubic curve.
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.)
Gröbner bases for submodules of free modules, monomial orderings on free modules
Computing free resolutions over graded rings. ?Betti Tables? Schreyer orders?
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 (graduate) class at Texas A&M

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

Last modified: Mon Mar 2 12:21:18 CET 2020