Much of modern pure mathematics research, as it is practiced, is highly specialised, highly technical and abstract – and as such is frequently challenging to communicate to anyone outside a fairly narrow specialist audience. As a result, much of the general public is completely mystified by academic pure mathematics and has no idea what the daily routine of a mathematical researcher even looks like. How do we change this, and why do we have to?

Of course, most research at universities and research institutes, in particular fundamental research like pure mathematics, is publicly funded – meaning we have an obligation to present our work to the public and make it accessible. We owe it to people to let them know what our common money is being spent on. This is in part also in our self-interest – we need to advertise our field in order to convince those paying for it of its usefulness to society as a whole. Pure mathematics has already landed on the financial chopping block in multiple European countries in recent years. When austerity measures where introduced and research funding cut, this has usually impacted fundamental research with no obvious application, so pure mathematics, even though cheap to research, is an obvious target. (In response to this, mathematicians in the UK have founded a dedicated campaign, Protect Pure Maths.)

I am no expert in science communication; far from it! I have only recently started taking a serious interest in this subject, however, the little experience I have has convinced me that there is a lot of interest in learning about a research area and profession frequently perceived as arcane. In this article, I want to share some of my ideas.

Almost nobody outside has seen "real mathematics".

The first hurdle encountered by most mathematicians when trying to communicate what they do to someone whose mathematics education stopped at secondary school is that most people who are not mathematicians have never or only rarely encountered mathematics in its "real form", that is to say, as a creative form of logical exploration and puzzle solving. This is because mathematics education at school tends to relegate this aspect of mathematics to a side activity and – forced by the curriculum, not because teachers don't want their students to have fun – instead chiefly focuses on technical manipulation of symbols. This complaint is of course nothing new and in no way an original observation – an intense exposition of this argument is the famous essay A Mathematician's Lament by Paul Lockhart, which eviscerates the US-American high school mathematics curriculum (a recommended read, even though I am personally far more positive about my own mathematics instruction at a German secondary school).

When I started as an undergraduate student of physics at Heidelberg University in 2010, the student organisation of the faculties of Mathematics & Computer Science and Physics & Astronomy provided all new students in these fields with a guide to university studies and the subjects we would be taking (often together). My favourite statement about the first-semester (abstract) Linear Algebra course was (translated from German and paraphrased):

"The mathematics lectures at university will have about as much in common with your maths lessons at school as with your PE lessons."

This is of course an exaggeration, but it is true in essence. (Of course, lectures at university, while following a rigorous, proof-based approach of mathematics, sadly also often fail to convey the wonder of original mathematical discovery, just like school lessons.)

As a result of the current predominant way of teaching mathematics in schools, a large proportion of the population leaves school with the impression that mathematics is at its heart about complicated calculation and symbol manipulation (and of course it can sometimes involve that, but never for its own sake), and that even basic concepts go way above their head – what mathematician hasn't heard the reply "Oh, I am bad at maths" or "Oh, I hated maths in school" when stating their occupation?

But it is important to stress that this does not mean that the public is mentally disengaged and has no interest in mathematical research. Many people simply lack confidence in their own mathematical ability, but I have found that they are usually still interested in learning more about what mathematical research and mathematics at university involves, as well as the applications in industry, and will find the subject fascinating when presented with it in a different manner than they saw it at school.

Talking about model problems that illustrate the beauty of mathematical research and discovery.

Thus, the first task of any mathematician doing public outreach is almost always to set the record straight on what our subject is and how mathematical thinking works: Mathematics is about discovery and logical construction of ideas, used to solve more or less elaborate questions and riddles – whether one views maths more as a process of discovery or building being a philosophical question. It is also about abstraction – not for its own sake, but for the sake of clarity and a more general understanding of the problem at hand, or at first distinct-looking problems at once.

Since mathematics has been made to look unappealing or at least entirely unapproachable to so many people, mathematics outreach centres around turning the public perception around: There are already many initiatives bringing a fun kind of mathematics based on these fundamental principles – mathematical problems turned into puzzles, logic games, geometric arts & crafts – to a general public or an audience of school children, in the form of university science festivals, science museums or books. Universities are frequently looking for volunteers to help run outreach events based around such activities – try it out at an event near you!

Some of us are also invited to give public lectures for a general audience: The best, most impressive such event that I have attended so far was the public lecture at Manjul Bhargava's Fields Medal Symposium in October 2016. Among many other things, he spoke about learning about the Fibonacci sequence from his grandfather, a linguist and expert of Sanskrit – before this event, I was entirely unaware that the Indian Sanskrit scholar Virahanka had already described and studied this sequence centuries before Fibonacci and that the sequence was used in the metric patterns of Sanskrit poetry. To me, this is a beautiful example of the universality of mathematics, illustrated with a simple, yet powerful piece of number theory.

But what about my specific research project?

Now, all mathematics outreach in which I have participated had one thing in common: The mathematics being presented was quite old, mostly from the 19th century or earlier. Truly modern pure mathematics research is rarely showcased in such forums, except as underlying some modern physics, which is generally where the focus lies (which is of course a great way of doing it). The obvious explanation for this seems to be that (with exceptions) modern mathematical problems are frequently difficult even to state without a lengthy introduction and quite specific preliminary knowledge.

The challenge is thus to cut back on technical detail and to pick simple examples, to the point that the subject becomes generally understandable, while still leaving the essence of the project intact. Any such attempt will of course involve compromising on one of these goals, and the presentation of certain research projects may never be able to get all that close to both goals at once. To be clear, I have not succeeded yet when it comes to this research project, FuSeGC. When I do, I will certainly post the result on this very web page!

As an intermediate goal, I have instead chosen to present my research area to mathematics students: not experts, but equipped with a good amount of background and preliminary knowledge. That is what this Blog is primarily for.

An exercise for the reader:

Have you done it, or do you know good examples of modern pure mathematics research presented to a general audience in a concise and accessible way? Let me know using the e-mail address at the bottom of the page.

(Last edited: 11.10.21)