Computational Algebraic Geometry

The main lectures will be uploaded in video format to YouTube, see the links below or the lecture playlist.

We will hold tutorial sessions in person, on Thursdays at 10:15–11:45 in Room F107, Building 1101.

Grading

I will not collect or grade homework. They are for your entertainment and growth only. This also frees me to give useful homework assignments without any regard to the ease of marking them.
If you want a grade from this lecture, then you can take an oral exam at the end of the semester. The homework questions are a good indication of what I might ask.

Lecture videos

Homework for Lecture 1

  1. Visit the Magma and SageMath calculators. Make sure these things run on your computer.
  2. If you are feeling adventurous, try the tutorial for SageMath. If you like it, try the “first steps” pdf file for Magma.
  3. For the technically minded, I recommend that you attempt installing SageMath and Julia to your own computer. Good luck!
  4. Read about the text editors Vim, Emacs, Atom, and Sublime. Play around with the ones that appeal to you. Decide which one(s) you want to use in the next weeks.

Homework for Lecture 2

Consider an ellipse of the form ax^2 + by^2 = c with a,b,c complex constants.

  1. For which a,b,c will the geometry of the set of complex solutions change?
  2. If a,b,c are rational, can I still consider finite reductions for each prime? What can go wrong? How can you go around it?
  3. (Challenge) If a,b,c are possibly transcendental, what would be the special reductions that would be loosely analogous to the finite reductions we studied?

Homework for Lecture 3

  1. Display primes up to 1000 that have 3 as a quadratic residue but not -1.
  2. Read about the boolean operators in the Magma documentation.
    • Start by searching for eq and ne. What happens if you write something silly like (3 eq true)? Does Magma have a more flexible equality operator?
    • Read about the “comparison” operators for integers: lt, gt, le, ge.
    • Fiddle around to get a feel for the syntax. Write simple “if” clauses, then make them more and more complicated. Extra points if you manage to get confused.
  3. Read about “while” and “repeat … until” constructions. Study the examples there. Again, fiddle until you are confused for maximum results.
  4. How many twin primes are there less than 10000? Display all pairs.

Homework for Lecture 4

  1. Prove that the complement of the affine n-space in projective n-space is a projective (n-1)-space.
  2. Prove that the projective real line is a circle and the projective complex line is a sphere.
  3. Fill in the details in the argument showing that the projective closure of the complex circle is a sphere.
  4. Read about the commands Homogenization and IsHomogeneous in the Magma handbook. Fiddle around.

Homework for Lecture 5

  1. Implement your own forest fire algorithm, as in Section 1.2.8 of Solving Problems with Magma.
  2. Implement the (2 dimensional) Game of Life in the same spirit as the forest fire algorithm.

Resources