GSoC 2023: algebras & Gröbner bases - project scope and references?

[Jamie Kai]

Hello SageMath team,

My name is Jamie Kai, I'm at the University of British Columbia in Vancouver, Canada. I'm in year 1 of a 2-year Second Bachelor of Computer Science program, and I have a previous BA in Math from McGill University.

I have experience with Python and Cython programming for large-scale statistical calculations, and several years of MATLAB and professional full-stack experience.

I've also taken several upper-year/grad math courses: advanced linear algebra (tensor, exterior and symmetric algebras, topological vector spaces), group theory, analysis (real, complex, harmonic). I have a growing interest in abstract algebra, so SageMath's list of project ideas this year is very exciting!

I am quite interested in submitting a proposal for one of the project ideas in computational algebra under Travis Scrimshaw, in particular: Improve exterior algebra and Gröbner bases code and expand to graded commutative algebras

I have a couple of questions:

Are there any suggested references (books, papers, websites) for computational ring theory or commutative algebra that I can read to get some theoretical background?

Would the 175 hour version of the exterior algebra & Gröbner bases project include just the first goal of improving performance with Gröbner bases, or also some work on the case of general graded commutative algebras?

Cheers,

Jamie

[tcscrims]

Dear Jamie,

   Thank you for your interest. There are no specific suggested references, but it is very easy to find the necessary background information on Groebner bases for polynomial rings through a Google search. It is also easy to find information regarding those for the exterior algebra (also somethings known as a skew-commutative polynomial ring or Grassmann algebra). For example, you would quickly find starting points such as

http://www.reduce-algebra.com/reduce38-docs/xideal.pdf

For the project itself, you are free to propose whatever you want to do. My dream would be to have a faster native version of graded commutative algebras within Sage that also implements fast GBs (relative to what plural does for us currently). However, if you want to focus specifically on getting the exterior case super quick, that is also fine.

Best,

Travis

[Jamie Kai]

Hi Travis,

Thanks for the reply. I've got some good references (including the one you linked) on Gröbner bases for exterior algebras and Gröbner basis algorithms for now.

I'm currently setting this up on my machine and exploring the code mentioned in the issues. Hoping to have a project idea soon.

Best,

Jamie

[Jamie Kai]

Hi,

Just want to say that unfortunately I won't apply to GSOC this year as my course load is too high over the summer.

It's been interesting to learn about SageMath, I'll keep an eye out for similar projects next year (and have studied more ring theory, ideally!)

-Jamie

[tcscrims]

Dear Jamie,

   No problem. I hope you will consider submitting some code to SageMath when you have time, even if they are just small improvements or your own code that you have found useful. (I found that writing code is a very useful way to help understanding how certain computations are done.) I have a number of other smaller things I would like to include that I haven't had time yet to implement (i.e., smaller than a GSoC project, possibly an afternoon or two).

Best,

Travis

[Jamie Kai]

Hi Travis,

Thanks. Yes, I will definitely continue to explore the SageMath system and repo, and contribute if I can.

If there are any particular issues that you have in mind, please feel free to link them and I’d be happy to check them out.

Best,

Jamie