## Algebraic topology: a black box turning shapes into numbers

Read an interview with mathematician Kathryn Hess Bellwald on the occasion of the MUNI Seminar Series.

**Kathryn Hess Bellwald’s profound fascination with mathematics has led her to a distinguished career as a professor at the École Polytechnique Fédérale de Lausanne (EPFL). Renowned for her research in algebraic topology, prof. Hess has bridged the gap between complex mathematical theories and tangible applications. Through her involvement in the Blue Brain project, she applies topological insights to the complexities of brain networks, advancing our understanding of neuroscience. As the founder of the extra-curricular Euler course for children, she inspires young minds, fostering a new generation of mathematicians in Switzerland. Moreover, while actively supporting her peers through initiatives like the Women in Topology network, she demonstrates her commitment to creating a more inclusive and supportive scientific community. **

*Photo: Professor Kathryn Hess Bellwald*

Beyond her research, she is an exceptional educator and communicator. She was voted by EPFL students to receive the Polysphere d’Or Teaching Award in 2013, she gave a speech for 2017 TEDx Lugano and a public lecture at the European Congress of Mathematicians in 2021. She is also the latest addition to the MUNI Seminar Series, which introduced the audience with topological perspectives on networks, and also made this interview possible.

**Mathematics changes the rules**

*You have mentioned, in your past interviews, that as a child, you wanted to be an astrophysicist. It was not until your undergraduate studies that you switched to mathematics. What do you think is the most fascinating thing about math?*

The fact that you can change the rules. We are always working with some system of axioms (i.e. rules or statements that are generally accepted to be true without proof - author’s note). You can explore, what happens if you don't include an axiom or change something about another axiom? What is the impact of these sorts of games that one can play?

I was in a fast-paced math class when I was 12 years old or so. There was a woman mathematician who once gave us a lecture on Euclidean geometry, by talking about non-Euclidean geometry - like what happens if you change the parallel axiom, if parallel lines could meet for instance. Things like this just blew my mind, as well as the fact that it came from a single female lecturer at that time.

*Has this experience inspired your decision to create the Euler course at EPFL for young mathematic talents in Switzerland?*

Indeed. The idea is that the children should start when they are about 11 or 12. And then they follow it through till the end of their high school studies. As a part of the programme, we invite advanced mathematicians or computer scientists to talk about their own fields, to give the kids a direct contact with real research mathematics.

*How do you select participants for the course?*

We have drawn on the model created at Johns Hopkins University. They select kids by giving them the SAT tests, otherwise used in the US for high school leaving exams. It may seem like a crazy thing to do, to give such a test to a bunch of 10 and 11 year olds. But it turns out that kids, who have a certain mathematical ability, tend to be very good at reasoning backwards. So they look at the question, see the possible answers from the multiple choice, and they have a good intuition for what is probably going to be the right answer. For our purposes, we created something similar, more appropriate for the Swiss population. The students with top results are then interviewed with their parents to make sure that it's not just the parents pushing, but the kids are actually interested. The last application round saw about 270 applications, we choose the top 30 or so.

*What is the success rate?*

About half of the kids make it all the way through. Others, when they reach a certain point in adolescence, spending time on math is not what they want to be doing. But even those, who leave early, say that it has been a great experience to be in contact with other like-minded kids.

And do you follow up on those who successfully complete the course. What path do they follow?

The very first group started in 2008, and we followed them quite closely. Out of the 13 kids, who completed the course, at least half of the class got their PhDs in math, physics or related fields.

**Mathematics of shapes**

*Your areas of expertise include homotopy theory, category theory and applications of algebraic topology. Please tell us what you do in particular?*

Topology is the mathematics of shape. To demonstrate what it means, we can use the famous joke about topologists not knowing the difference between a coffee cup and a donut. The essential common point for both is an existence of a hole - be it in the middle of the donut or in the mug handle. If a coffee cup and a donut were made of playdough, you could just squeeze and deform them, making one look like the other in continuous deformation. Topologists are then interested in what properties of shapes remain the same under such continuous deformation, where you don't allow yourself to cut or to glue or anything like that.

*Photo: topology explained*

The situation then gets more complex, when you are interested in multidimensional objects, where visualization is more difficult. We need some sort of a black box that will ingurgitate these different shapes, and then spit out something you can actually calculate or recognize more easily - like a number, a polynomial, a matrix and so on. We are looking for a means to compare. At the same time, we are collapsing away some of the information, but you can at least determine something. The idea of algebraic topology is in taking a topological problem about shapes and translating it into a problem about algebra, which is somehow easier to work with.

*So like in a map, you are creating a simplified model of reality, but it allows you to see how to get from point A to point B? *

Exactly.

*And where can we see the results of your research in real-life applications?*

To be honest, when I first started, I did not care about whether there were any applications, because it was a kind of mathematics that fit my brain, and I really enjoyed just thinking about it. But little by little, I realized its usefulness in fields like neuroscience, material science and computer science.

Building on principles of knot theory, you can think of taking a string, tying a knot and then gluing the ends together. When you look at your knot, you start thinking, whether there is some way you can carefully pull it apart, so that it just gives you a circle. Or if you have two knots, can you by moving some of the strings around, actually change one of the knots into the other? If you take an example of polymers, the long chains of monomers, as they float in solutions, they can start knotting themselves up. You are then looking for a link between the structural properties of such polymers and their other properties - e.g. stiffness.

*That can be of a particular interest to industries using plastic materials for instance, which are derived from polymers. So you basically help other fields to build a model where they can test their hypotheses?*

Exactly, and sometimes they may have not even considered such hypotheses before. I have once invited a polymer scientist to be on the thesis defense of my PhD student. And the scientist was so impressed by the ideas she presented, that he actually gave her a job immediately.

It is just interesting to have a conversation with specialists in their own fields, to see what their problems are, and learn to speak their language. In my opinion, mathematicians are very good at getting to the heart of a problem. This doesn't mean you understand everything about the field, but enough to work on this particular relevant problem.

*Please tell us about your involvement in the Blue Brain Project.*

Since about 2014, I've been working closely with the people in the Blue Brain project of EPFL. They started to build a computer-based reconstruction of a rat brain. By studying individual parts of the brain we obtain dense maps of the underlying network. Using the tools of topology, we can say that certain parts of the network are more complex than others. We then study, why do we have this diversity of the local structure in the network? What is the influence? What is the impact on the brain's function?

We see that the more complex parts of the structure are there to ensure that it's really robust and reliable. And you want your brain to be reliable, right? On the other hand, it is expensive to have such complex wiring. It is slower, it uses more energy, and sometimes you would like to be able to encode information more efficiently. And that is what the less complex parts of the network are for, that efficient coding. Topology helps us to see whether this is the only thing special about the network compared to just any other network, and to study the biological implications.

*While creating these kinds of maps of a rat brain, can we expect that healthy individuals would all have comparable maps, and any differences would signal abnormality or a disease?*

Something like that. For example, if you are under a lot of stress, then your dendrites, the part of the neuron that is gathering information from elsewhere, shrink. And you can see that as the dendrites shrink, the complexity of the network really falls off.

Then you could say, well, that does not look very good for the neurons, but what is the impact for the brain as a whole? There are various neurodegenerative diseases, where the branchiness of the neurons goes down, you have fewer dendrites. Thus you have a decrease in complexity. So this gives you some insight into what may be going on in pathological conditions.

*So your model is able to capture the current state of the network and make predictions?*

Exactly. Algebraic topology also gives hopes for future research to be less dependent on animal testing at certain stages and rely on computer-based models, such as in testing of new pharmaceuticals.

**Women in Topology**

*In your public appearances, you openly advocate for a more diverse research environment. How do you feel as a woman in science today?*

Things have gotten a lot better since the beginning of my career. At that time, there were maybe two women in algebraic topology, who were more senior than me. That has gotten a lot better. I remember last year, I spoke at a retirement conference of a colleague and, as I looked out at the audience, a third of them, at least, were women. The diversity of the public was all the colors of the rainbow, basically. It was really nice to see.

Many more things are in place to support women at conferences, such as child care. And there is a lot more awareness of the fact that it is just not acceptable to organize a conference with only male speakers.

*Do you think the academic world is ready or even able to recognize different needs of women, who take care of their children? You have mentioned at various occasions that as a mother of four, you had to slow down your career considerably, for some time. But the academic hierarchy and requirements are set up in a way that expects a certain pace throughout the whole working life. *

I think it is still not entirely ready, no. There is still not a deep understanding of the specific needs. There are some women who are absolutely remarkable in keeping the pace. I have a young colleague at EPFL, who is a highly productive mathematician, and she is currently expecting her fourth child. I don't know how she does it. I really did slow down for a good ten years with my four sons. It was a choice and I don't regret it, but it definitely happened.

*You support fellow female mathematicians through the Women in Topology initiative. Please tell us more about how it works.*

The idea was based on something that the Women in Number Theory set up originally, not just to get together but to do mathematics together, to create really strong bonds. And so with three colleagues, we decided in 2011 to write to almost all the colleagues we could think of to say, do you have some female PhD students or postdocs who would be interested in such collaboration in topology?

We then recruited every tenured woman or more senior woman that we knew to act as a team leader for these projects. The idea is to have a couple of senior women working with three or four junior women on a project together. They work remotely for some months and then they meet for a workshop, where they finish the project and then write a paper together.

And this turns out to work really, really well in creating networks. It gives everybody more publications, which is not a bad thing. For many of the junior women, it's their first publication and it also gives them someone who can write letters of recommendation for them. And we observe that the number of women with permanent positions has just skyrocketed lately.

*What do you think works as a motivation for women to enter math?*

Role models help a lot. Few years ago, EPFL was celebrating its 50th anniversary. They had portraits made of all the women professors and a little biographical text. And then some of the faculties kept the portraits up at the entrance to their buildings. At the Basic Sciences faculty, where mathematics is, we were discussing if we should do this too. Some of my colleagues were worried that we could make our male counterparts feel left out or excluded.

But in my opinion, this was not about us saying “look at us”. This was for the young women who are maybe 20 % of the student body, to say look, we have eight women professors, as a means of encouragement.

*Can you identify someone that you are being a role model to?*

I have had a number of younger women say to me, thank you for being there, for showing me that it is possible to have children and a career and a good life.

*How does it feel?*

It is extremely rewarding and highly motivating for me to continue. I think that when you have been lucky enough to be given a lot, to receive a lot, you have an obligation to give back. It is not enough just to be in a corner by yourself.

**The Power of Serendipity**

*You came to Masaryk University on the occasion of the MUNI Seminar Series, where you are giving a presentation this afternoon. How did you come into contact with our university and faculty? *

That is an illustration of pure serendipity. Last summer, there was the Nordic Congress of Mathematicians in Aalborg, Denmark. I was invited to be a plenary speaker, and so was Dan Král’ from the Faculty of Informatics MU.

I gave the first plenary lecture on a similar topic to what I'm going to talk about today for MUNI. And Dan gave his talk the second day. And he mentioned a number of things from my talk that resonated with him. These kinds of connections. And even if he hadn't, I would have seen them with what he was talking about. And afterwards we spoke at lunch, he suggested that I come to MUNI with my lecture. So it was definitely unexpected.

*Photo: Prof. Hess’ lecture at MUNI Seminar Series*

*Do you foresee any other collaboration in the future?*

It's not impossible. I had the opportunity to meet colleagues from his lab this morning. It is really important as a researcher to be open to that kind of contact, and being aware that you cannot predict when these new sorts of connections are going to take you. You have to be ready to run with it, and say “Let's talk more!”.

**What do you consider your biggest success in your career? **

That is a hard question. One thing was setting up the Euler program for the students. Then the Woman in Topology initiative. As the Associate Vice President for Student Affairs and Outreach at EPFL, I have also put into place a mental health task force EPFL and laid the foundations for developing a strategy for public engagement at EPFL.

I feel like if I write another paper, maybe 50 people will read it. Whereas, if I put something into place to improve the mental health of 12,000 students on campus, that means a lot to me. We are part of a society, and we really need to give back. That is the only way for society to continue moving forward.

*I could not miss that you work in Lausanne, the city of Olympic spirit, lake, mountains, chocolate… Do you ever find time to enjoy your surroundings?*

Absolutely. Hiking in the mountains is my favorite leisure time activity. I live about 40 kilometers from Lausanne in the Rhône valley, so it is very easy to get up into the Alps and go hiking. To me, that is the best way to relax. Spending time in nature like that is magical. The lake is so beautiful. I try at least a couple of times a year to take one of the boat cruises. And yeah, chocolate, hmm.

*Thank you very much and good luck with your speech today. *

**Professor Dan Král' from FI MU appreciated Prof. Hess' lecture and the subsequent discussion with the following words:**

"It was a great pleasure to welcome Prof. Hess to Masaryk University, where she presented her research to our staff and students in a lecture organized as part of the MUNI Seminar Series. She focused on the use of topological methods to analyse networks, which she presented directly on specific applications within the Blue Brain Project, which aims to create a realistic model of the mammal brain. During her lecture, which filled the prepared lecture hall in the building on Komenský náměstí to the last seat, Professor Hess demonstrated, for example, the difference in topological parameters of networks of neurons in the cerebral cortex of mice and humans and their distinct difference from chaotic networks with similar density of connections.

She also used the occasion of her visit for an intensive discussion with staff and students of our Discrete Methods and Algorithms Laboratory (DIMEA) at the Faculty of Informatics of MU. We discussed combinatorial approaches to network analysis that explain some of the observed properties of the networks studied in the Blue Brain Project. The combinatorial parameters addressed by the lab, like the topological parameters presented in Professor Hess' talk, can distinguish between chaos and structure in networks, leading to questions about whether they capture the same or different structural properties of the networks studied by the Blue Brain Project."

Author: Marta Vrlová, Office of External Relations and Partnerships at FI MU

Photo credits: Irina Matusevich (MU)

The image of donut/coffee mug deformation is adapted from a video by Henry Segerman & Keenan Crane: https://youtu.be/9NlqYr6-TpA?si=Gn6BKGOZgq9eSBXt- Attachments