Questions and ideas for graph theory and algebraic geometry projects

[Jing Guo]

Dear all and mentors,

I have background knowledge in math (graph theory, number theory, commutative algebra, and algebraic geometry) and computer science, so I am interested in the following two projects:

1. Improve Height Functionality

2. Edge connectivity and edge disjoint spanning trees in digraphs

However, I do have some questions that are not answered by the documentation and/or project descriptions. I was wondering that could you please help me clarify?

1. For the first project, it's mentioned that interested participants should know "basic algebraic geometry and number theory". So I was wondering that could you clarify more specifically what one needs to know: Does one need to know schemes and elliptic curves? Being able to read and understand Krumm's and Kutz15 [1] papers? First chapter of Hartshorne and/or "Ideals, Varieties, and Algorithms"?

2. While the second project is interesting on its own, I have a potential project idea to suggest: Odd-cycle transversal [0], which removes some vertices in a graph to make it bipartite.

After doing a simple search in the source code, I could not find anything related to this, so I think it may be a nice addition to Sage. I have experience implementing relevant approximation, heuristics, and (recent) FPT (fixed-parameter tractable) algorithms in Python (with networkx) and C++ as part of my undergraduate research project.

Thank you for your time.

Jing

[1]: https://en.wikipedia.org/wiki/Odd_cycle_transversal

[2]: https://arxiv.org/abs/1210.6246

[Jing Guo]

I just think of another idea in number theory that may be pursuing: General number field sieve.

As it's written here [0] that Sage (at that time) only supported quadratic sieve and elliptic curve factorization for factoring integers, so I think having the general number field sieve could be another helpful and nice addition for Sage.

[0]: https://github.com/sagemath/sage/blob/a89f81d7783df2d4e4a187bf209e94d128301f83/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/integer_factorization.rst