They say variety is the spice of life and to us at Chalkdust, maths *is* life so it makes sense that maths is made better by variety. A variety of topics, a variety of people, a variety of poorly constructed maths puns. Ulrike Tillmann embodies this ethos with her work bridging the gap between pure and applied maths. Despite spending most of her academic career in the UK, Ulrike has lived in several other countries. She was born in Germany and then went on to study in the US. She is now a professor of pure mathematics at the University of Oxford and a fellow of the Royal Society, balancing her time between research, teaching, and outreach. She sat down with us to chat about her career and what the future holds, both for her and maths in general.

## Taking the reigns

If you’ve been following maths news in the past few months, the name ‘Ulrike Tillmann’ may be particularly familiar to you. It was announced recently that she will be the next president of the London Mathematical Society, one of the UK’s five ‘learned societies’ for mathematics. She will also take up the mantle as director of the Isaac Newton Institute, a research institute at the University of Cambridge, in autumn of this year. Research institutes are perhaps the least well-known entities in the academic world (as viewed from the outside), often only visited by some of the most senior academics in a field. We asked Ulrike to explain what they are all about. “The Isaac Newton Institute runs mathematical programmes in quite a broad range of areas. These programmes typically run between four and six months and researchers come from all over the world to concentrate on their research.” The programmes are beneficial not only to individual mathematicians, but to the community as a whole. “Being together with your colleagues who are also experts in your area, and who are often completely spread all over the world, is a fantastic thing. It brings the field forward and it can make a big difference to that research area.” On paper, the role of director will involve overseeing the organisation of these programmes, but she sees it going beyond this, including “making sure that things like equality and diversity are not just observed, but also incorporated.”

Diversity matters a lot to Ulrike and she has spent a lot of time thinking about what can be done, so she turns our conversation towards representation in mathematics more generally. “The most important part seems to me to be ensuring that women, and also other minorities, are welcome; and fostering a very open society.”

Ulrike is involved with many events for women in mathematics, both as a speaker and organiser. Indeed, encouraging more women into mathematics was part of her motivation for taking on her new positions. “I think as women we have to occasionally come forward and do these roles, even though sometimes we shy away from them. Being a presence is important.” She hopes that by increasing the visibility of women in mathematics, women will be encouraged to study maths and stay in academia. “We’re always drawn to groups where we see people who are similar to us, where we can identify, where we are obviously welcome. So I think we need to make that part of our culture: to just be open.” Unfortunately we know all too well that any change like this will take time, and she acknowledges the difficulties. “If I had some solution, I would have implemented it by now. But go back 30 years and there has been a big change. I think that’s encouraging and we just need to make sure that it is pushed in the right direction.”

The most important part seems to me to be ensuring that women, and also other minorities, are welcome; and fostering a very open society.

Ulrike, of course, knows personally the importance of diversity as a woman in mathematics, but she is keen to impress that diversity goes beyond gender, and the other underrepresented groups we often talk about. Really she sees the need for diversity of experience, thinking, and background. “In terms of excellence you need a mixture of people—not just the stars—you need a whole mixture of people all striving for excellence in their own ways. This cannot be measured on a simple one-dimensional scale. I think geographic diversity is also another aspect of this which is really important. And we will all be better off if we spread things around a little bit in a sensible way.”

## Seeing the shape of data

Some of Ulrike’s most recent research has been in the fledgling field of *topological data analysis* (TDA). “It’s really trying to capture the shape of data. You can imagine data as a point cloud in some Euclidean space, and when you have such point clouds, what does it mean to be a shape?”

The idea of studying the basic shapes in data is nothing new. “There is clustering, for example: people already understand clustering relatively well—there’s a bunch of points here, a bunch of points there—they seem to be separated and maybe that separation is meaningful for the data. Or in linear regression, you are trying to fit a line to your data, and then that gives you some understanding of the data.”

Topological data analysis seeks to use advanced topological techniques to detect more complicated structures hidden in complex data. “You are looking for holes in dimension one or two and then you can use different techniques to approach the same data from different directions and try to understand a little bit more about the shape. The idea is that, especially for complex data, the shape should be meaningful.”

It’s always a two way dialogue between the mathematician and those people who want to apply it.

It might be difficult to imagine how complex topological features can be interpreted meaningfully in the real world, but the approach has many success stories. “There has been a famous study by Gunnar Carlsson and collaborators which looks at different types of breast cancers. The data was effectively Y-shaped, rather than just a line. Understanding that there was a third branch, a new branch, meant they could see that not all cancers were the same. There was actually a ‘good’ version that you didn’t have to treat.” Data scientists rely on TDA as without it “sometimes you just can’t predict what shapes the data has.” This last point is essential—the topological techniques can help you find patterns in your data that you would not even think to look for.

One key topological tool used in TDA is *persistent homology*. Homology is a technique which uses algebra to count the topological features of a space, for example, the homology of a torus can be summarised by three numbers counting the features in dimensions 0, 1, and 2:

$\beta_0=1 $ | It has one connected component, so every 0-dimensional point is connected to every other |

$\beta_1=2$ | It contains two ‘independent’ 1-dimensional circles (red) |

$\beta_2=1$ | Its 2-dimensional surface encompasses one interior cavity |

Persistent homology studies those topological features which can be found *persistently* in the homology of the data as you vary the scale on which you look at the data points’ interactions. In this context, clustering means that your data has several connected components, and so just corresponds to the 0-dimensional homology. TDA focuses on finding higher dimensional structures by looking at the higher-dimensional homology.

“Especially looking at biological data, it is generally not so important where exactly the points are—they are just samples anyway. So topology in particular is quite useful because it tries to study the shape in a ‘fuzzy’ way. Topology is just a poor relative of geometry where you forget about angles and distances, so that it can focus on the most important features.”

## An evolving field

Ulrike’s work in this area is formalised through her role as co-director of the Centre of Topological Data Analysis at the University of Oxford. Despite a research background in very pure mathematics, she doesn’t limit herself to the theoretical side of TDA. “Our pitch to the EPSRC [Engineering and Physical Sciences Research Council] was that we would go all the way to the applications. The application should tell us what we want to understand theoretically, and then we work backwards and forwards between pure and applied. I have been involved in a study where immune cells are analysed. How quickly can they infiltrate a cancer? That is a real life study where maybe these topological methods can be used. So you see the whole pipeline going through.”

Since TDA is a new and exciting field, it is tempting to try and speculate about what developments we can expect in the next few years. Ulrike is cautiously optimistic: “I think evolution rather than revolution is probably what we are going to see. A certain amount of new thinking has to be cultivated because topology is not one of your typical areas of applied mathematics: you tend to see more analysis, numerics, and linear algebra.” It takes time for new mathematical ideas from pure topics to infiltrate applied research groups.

“I think it needs to be popularised a little bit first because it’s always a two-way dialogue between the mathematician and those people who want to apply it and we just need to fill in that space more. But it is this interaction between topological data analysis and other techniques that will really be important.” These other techniques are by no means limited to data science—applied mathematics is about pulling together any and every tool which might be helpful. “We are also trying to mix TDA with machine learning methods to make more meaningful and also more interpretable machine learning algorithms.”

## Igniting a love of maths

You’d be forgiven for thinking Ulrike never had doubts she’d be a mathematician, but this was not the case. “It was somewhat gradual. I went to Stanford for my graduate degree and during my first year I was playing with the idea of doing something in computer science.” Although Ulrike did eventually settle on maths, she does worry that a more rigid degree course would have prevented this. “I did about a third of my undergraduate courses in mathematics. If I had come to Britain at that point I would have completely missed the train.” In fact, Ulrike believes this is a significant flaw in the British system. “I think we force our students into decisions too early. If you like mathematics you shouldn’t have to rely on your decision as a 16-year-old to pick further mathematics A-level.”

I think it’s a really deep satisfaction that comes out of being able to solve a problem. That you see connections between things that you haven’t been able to see before…

Ulrike grew up in a small town in Germany and partially credits this for sparking her joy of mathematics. “There was no kindergarten or anything like that, so I was a bit bored. I asked my mother for problems and she would set me some sums and I liked doing those.” Throughout school all the way to her undergraduate degree, maths was just something that came easily to her, rather than a strong interest. Eventually it was the puzzles that drew her in. “I really wanted to work on these challenging problems that mathematics provides. I think it’s a really deep satisfaction that comes out of solving a problem. That you see connections between things that you haven’t been able to see before, and that maybe nobody else has been able to see before. That is very exciting—to really try to understand something—and sometimes you bring new concepts together.”

## Studying surfaces and spaces

The problems which really appeal to Ulrike come from geometry and topology, in particular the so-called *moduli spaces* of surfaces. These turn out to connect with several areas of maths and physics: “You know what a surface is, and one way to think of moduli spaces is to understand them in families.” A simpler one-dimensional example might be to think about all possible circles in the plane. A circle is determined by its centre $(x,y)$ and its radius $r>0$. Therefore choosing a circle in $\mathbb{R}^2$ is the same as choosing a point $(x,y,r)$ in $\mathbb{R}^2\times\mathbb{R}^{>0} =: \mathcal{M}$. This set is the *moduli space* for circles in the plane (see figure below). The key idea is that this moduli space is *itself* a geometric space (not merely an abstract set), and so you can study it using geometric and topological tools—and hence study all possible circles at once. For example, following a path in $\mathcal{M}$ corresponds to continuously deforming one circle into another.

Since surfaces are two-dimensional, and geometrically more complex, their moduli spaces are a lot more complicated, but that also makes them more interesting. “Surfaces are one of the foundational objects in mathematics. They appear in geometry, of course, but also dynamics and number theory; they’re all somehow connected to surfaces in one way or another, and the moduli space is of interest to a lot of these subjects.” In fact, it was through the applications of moduli spaces to physics that Ulrike first became interested in the subject, working with fellow Oxford topologist Graeme Segal. “He was interested in conformal field theories and topological quantum field theories and it is the physics story behind it that made it very interesting for me. I’m still very excited about this physics part of it, because some of the theorems that we were able to prove can be interpreted as classifications of so-called *invertible topological quantum field theory*—so the story behind it is quite important.” Again the go-to technique for studying moduli spaces for Ulrike is homology.

## A mathematician’s apology

People engaged in basic research—research which has no immediate application—are often called upon to justify their work, whether to family and friends, funding bodies, or even policymakers and the general public. Sometimes this may amount to no more than a minor inconvenience. However, in recent months the topic has risen to the fore since the University of Leicester began a consultation on proposals which, as part of broad restructuring across its faculties, include the disbanding of the pure maths research group in favour of a focus on applications of mathematics to artificial intelligence, computational modelling, and data science. Of course, a consultation is not the same as enacting a proposal, and there are likely to be many factors involving the plans which are not in the public domain; nevertheless as a pure mathematician who works on the interface with applications in TDA, Ulrike is well-placed to comment in general on the idea of doing away with pure mathematics in a research intensive institution.

Turing, for example, was a mathematician first and then the inventor of the Turing machine. It feels to me that research culture…ought to support foundational mathematics.

“Of course I don’t know the precise situation, there are often financial considerations and so on, but I find it a little bit puzzling frankly. I’m a pure mathematician who also moved into data science and it seems to me that a new subject like data science will certainly benefit from pure mathematics, where many of the new ideas are coming from. Turing, for example, was a mathematician first and then the inventor of the Turing machine. It feels to me that research culture, in a research university—especially one that hopes to do something as technical as data science—ought to support foundational mathematics.” The Centre for Topological Data Analysis stands as a prime example of how pure mathematics works in tandem with its applications, although of course immediate applicability is by no means the sole justification of pure maths.

Undergraduate degrees tend to have curricula designed to ensure a strong basis in pure mathematics. This allows students to develop a sufficient grounding and enable them to specialise in topics ranging from applied to pure. How these pure modules will be taught without pure research mathematicians is clearly a question which must be tackled. “In Leicester’s case in particular, what troubles me is that they still hope to have an undergraduate maths degree, but having teaching-only staff to do the pure modules, and I don’t think that’s a great solution. For me, teaching gets more interesting and is invigorated by research, so the best teaching is often inspired and kept relevant by research.”

Much of the criticism of Leicester’s proposals centre around legitimate concerns for the individual workers who are now facing the prospect of redundancy during a pandemic, but any argument in defence of basic research in mathematics or any other subject needs to stand independent of the present situation. It is no great secret that academia is a very insular field, and one might reasonably try to argue that mathematics needs to modernise, consider new ways of working, and what is the harm if one university opts to focus solely on technology and applications? “I think actually Britain is generally quite advanced in thinking in terms of impact. Of course it takes some effort to make these connections and they are also not necessarily done by pure mathematicians themselves. But pure mathematics brings the practice and culture of rigorous thinking. That’s really important, and it’s often our students who make the applications. Britain has a science and technology based industry and economy, and we need more people educated in Stem subjects, of which mathematics is a foundational part. I don’t think we can get away from that.”