Mathematicians absolutely love data! A common misconception by people who are not involved with the natural sciences is that mathematics is already "finished". That we already know everything we need and that no more active research is being done. It may therefore be hard to imagine that research data plays a fundamental role in today's mathematical research. But just like physicists perform experiments to analyze properties, mathematicians also gather data about the objects they study. But what does mathematical research data look like? What is it used for? And why is it important? What kind of challenges does one encounter when handling mathematical data?

The reason mathematics was originally developed was to try and understand our universe better. People observed real-life phenomena and developed abstract concepts like fractions, angles, coordinates, etc. in order to be able to formulate problems and think about them.

But as mathematics has grown, so has the complexity of the objects that are being studied. It is therefore quite common that mathematicians try to classify and list properties of the objects they're working with. Think of something like the periodic table in chemistry, but then for mathematical objects!

In the following, we're going to discuss a few mathematical objects (some simple, some more complicated) and discuss how these objects are stored as data.

Platonic Solids

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Link to open on an external website
Credits: Polyhedra interactive app by Jürgen Richter-Gebert (Science to Touch / TU Munich), part of the Math to Touch collection.

Elliptic Curves

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Credits: Simple interactive prototype by IMAGINARY.

Algebraic Surfaces

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Link to open on an external website: love.imaginary.org
Credits: Raytracer by Aaron Montag, created using CindyJS by Jürgen Richter-Gebert.

x^6+y^6+z^6-1
y*z*(x^2+y-z)
x^2+y^2-z^2+a-0.5
x^2+y^2+z^2+2*x*y*z-1
x^2-x^3+y^2+y^4+z^3-z^4
1.2*x^2+1.2*z^2-5*(y+0.5)^3*(0.5-y)^3
(x^2+y^2+z^2-1)*((x-3*a)^2+y^2+z^2-1)
4*((a*(1+sqrt(5))/2)^2*x^2-y^2)*((a*(1+sqrt(5))/2)^2*y^2-z^2)*((a*(1+sqrt(5))/2)^2*z^2-x^2)-1*(1+2*(a*(1+sqrt(5))/2))*(x^2+y^2+z^2-1)^2

With the rise of computers, we have seen a large increase in the size and the complexity of the data mathematicians use. In the past, we only had books like the atlas of finite groups in which some data was written down, but nowadays a database like the LMFDB, for example, contains multiple Terabytes of data.

And more data gets produced every day! Papers with new results get published at a very high speed and we currently simply do not have the infrastructure available for supporting the huge influx of data that comes with it. There are several aspects to consider here:

• If we make a database of objects of a certain type available, other researchers will use this data to produce new results. But we want to make sure that this classification is complete, i.e. that we have all of the objects in our database (like we hopefully convinced you there are only 5 platonic solids) and that the data is correct. But how can we ensure this? Sometimes computations can take months or years to complete. And it's very easy to make a small mistake in the code written to perform the computations. Especially if this code is very complicated.

• And what if the data was produced using closed-source software? If I can look at the code I can convince myself that the way the data was produced makes sense mathematically. But if I can't... anything could have happened behind the scenes. The authors could've written the code in such a way that it outputs exactly the results they want to see as output in the cases they describe in their paper and simply spit out nonsense any other time.

• We want other researchers to use (or reuse) the data that has been computed. But will they actually be able to find it? With the huge influx of papers, it is sometimes difficult to get an overview of everything that is out there. Additionally, authors of papers are not always aware of best practices concerning the publication of data. A common scenario is the following: An author puts the data on their own website. They move to a different university after a few years. The website gets deleted. The data is gone forever and other people will need to redo the computations.

• Who is going to pay for storing computed data? If the data is just a couple of MB it's probably not too hard to find a place to put it, but what if it is several Gigabytes?

• In other areas of sciences, most data will consist of lists of numbers. Mathematical objects however can get very complicated very quickly and one therefore has to think carefully about how to store it. Especially if you want other people to be able to load the objects again in their favorite software package (which might not be the one you used for your computations). Currently, there is no universally agreed-upon standard for this. Think about polynomials, for example. A polynomial in one variable x with rational coefficients is simple enough. But what if the polynomial was defined over a field extension L of QQ? Then we would also need to store the field L. But there is no canonical way to define the field L. We need to choose a specific polynomial defining the field extension L over QQ and store this with the data. But maybe L was defined as a field extension over a field that was not QQ. And then you need to deal with it differently again. And how do you store a polynomial with multiple variables? Do you consider x^2 +x*y to be a polynomial in y over the ring k[x]? Or is it an element of k[x,y]? There are many more questions like this that could be asked and it is not easy to find a file format that is universally accepted by everyone.

Within MaRDI we work on possible solutions for these problems with a special focus on mathematical research data. Please subscribe to our newsletter to keep up to date on our findings!