# Parallelism & Borcherds’ method

We will now cover some material that is not detailed practically and exhaustively within the PDF file. Indeed, one of the apparent challenges which had to be overcome during this thesis is that we had to temper our use of computer-based solutions and refrain from making any explicit reference to these solutions within the PDF file. We indeed had to avoid putting ourselves in an awkward position from which our work could have been perceived as non-compliant with the rules of etiquette in the field of study of pure mathematics. During the last months of our thesis and especially during the writing of the manuscript, trying to find a balance between the pure math and computer-based aspects of our study put us in an awkward and uncomfortable position. We had to find our answers to the two following questions :

What is the frontier between CS and pure maths, i.e.,
what is the frontier that should not be crossed?
Is the usal dissertation, paper or PDF, a suitable format which enables us to express ourselves in such a way that the contribution brought by our thesis to the
field of study of K3 surfaces can really be highlighted ?

To the first question, we ended up finding our own compromise. To the second question, the best answer we could find is mixed. We thus felt that setting up an online platform dedicated to our thesis was the way to obtain more leeway and highlight the contribution brought by this thesis. We approached the classical manuscript as a technical appendix and took care to make it compliant with the standards of pure mathematics. There is no doubt that the computer-based side of our study is formulated in a very general, moderate, and code-free way within the PDF dissertation. The computer-based side of our research, however, cannot be neglected.

Doing so would amount to missing out on the intent and purpose of our thesis.

This website thus came up as an online complement to the manuscript, compensating for the latters’ inherent limitations. This online platform enables us to approach the computer-based side of our study in a way that would not have been feasible within a PDF file. People may consider that our work is on the frontier between pure mathematics and a more computer-oriented field of study, right on the outer edge of the pure mathematics spectrum.

We, however, maintain that we never crossed the line. We never crossed the line that would have led us out of the spectrum of pure math. We put computer-based solutions at the service of pure mathematics, nothing more.

Trying not to cross this line is one of the two reasons behind the fact that the PDF file of our thesis does not cover advanced technical details regarding parallelism. The other reason is that providing such information clearly cannot be done in a classical thesis manuscript. We nevertheless did our best to offer an accessible vision of things, although voluntarily limited, as indicated earlier on this page. Anyways, two dissertations cannot be written into a single one :

• Bringing major innovations to Prof. Shimada’s ten years old material to push further the computer-based study of $K3$ surfaces is one thing.
• Doing so while enforcing parallelism on various levels is another…

We now develop the second point by adopting a more technical approach in an online section.

To do so, we proceed as follows :

• We will start by explaining how using process-based parallelism with Python’s multiprocessing library enabled us to address the most critical and apparent internal point of the method for which there is a lot to gain by enforcing parallelism: Congruence Testing.
• We will then give precise characterizations of the other internal points of the method, which can be improved thanks to process-based parallelism.

We will then focus on enforcing parallelism not only at the internal level of Borcherds’ method but also at the level of the method itself. To this end :

• We will show how the exploration of the chamber structure & computation of sets of walls can be parallelized and explain the inner workings of the Autonomous Poolized Functional Block, also called Borcherds’ worker APFB.
• We will then explain how to push even further the deployment of the Borcherds’ method in parallel at the network level.