David Ayotte, PhD

David Ayotte, PhD

Data scientist

Google Summer of Code 2021 program

1 minute read

Published:

I will explain in this post a brief summary of what is the Google Summer of Code (abreviated GSoC). In short, the GSoC 2021 program is a 10-weeks program by Google where students all around the world are paired with mentors and work as developpers for an open source organization.

The organization that I will be working for this summer is SageMath together with the mentor Vincent Delecroix. SageMath is a mathematical software that can perform multiple computing task and is very useful for a wide variety of students and researchers. I personally used a lot this software in the past during my study, so I am very enthusiastic of being a part of this for the summer 2021.

My main goal will be to implement a new feature into this software, which is the “ring of quasimodular forms”. A more precise description of my project can be found here.

The 10-weeks coding period of GSoC will begin in June 7. Hence, every two weeks during the program, I will document my journey here on my blog by explaining my work and sharing my ideas. Before this coding period, I will be in a preparatory phase where I will study the mathematical side of quasimodular forms and explain what is the current state of SageMath regarding modular forms.

If you read this blog and find any mistakes, or have any question, do not hesitate to contact me via email. My email address can be found below my picture at the left of this page.

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Google Summer of Code 2021 summary

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In this post, I will give an overview of my work for the SageMath open-source mathematical software. If you are new to this blog, I suggest that you start reading from the first post.

Partitions and the Bloch-Okounkov Theorem

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In this post I will briefly explain what is a partition of a positive integer and how it is possible to relate this theory to quasimodular forms via the Bloch-Okounkov theorem.

How to Write a Modular Form In Terms of a Set of Generators?

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In this post, I will explain an algorithmic way to write any modular form \(f \in \mathcal{M}_*(\Gamma)\) as a polynomial in the generators of the graded ring \(\mathcal{M}_*(\Gamma)\) (\(\Gamma = \Gamma_0(N), \Gamma_1(N)\) or \(\mathrm{SL}_2(\mathbb{Z})\)).

The Pushout of Two Modular Forms Spaces

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In the previous post, I discussed about some changes made in the code for the graded ring of modular forms in SageMath. However, there was one lacking feature and it was the pushout of two modular forms space. I will explain this feature in this post.