“It’s noteworthy that I initially became interested in physics because people often talked about Einstein, relativity, and similar topics”, he shares. “In contrast, I never really had the chance to develop an appreciation for mathematics before university, since it was rarely discussed and explained to the general public in the same way as physics, with big tantalizing unsolved mysteries. I think I even used to think that mathematics was a finished field, with nothing left to discover or research.” Professor Karlsson explains that once he began studying mathematics at a higher level, he realized how he found it more “logically pleasing” than physics. While the ultimate goal of every science is to understand how nature works, mathematics stands apart – constructed from the ground up using axioms. In other words, unlike biology, chemistry, or physics, mathematics doesn’t aim to explain something that already exists, but rather develops new structures based on self-built foundations. Because of this, mathematics remains internally precise and coherent – built on logic and axioms – whereas physics, for example, is grounded in observation and experimentation and must always contend with approximations and the limitations of measurement.

It's unfortunate that, unlike physics for example, we do not hear much about mathematics' great unresolved problems and mysteries. Many of these can be explained to the general public.

“How did you decide which area of mathematics to focus on in your research?” In response, Professor Karlsson explains that he has always observed a strong interconnectedness throughout mathematics so he wanted to focus on something “central” – a topic that is related to most other areas. Many fields in mathematics share similar issues, but these can sometimes be expressed in entirely different ways – which he finds quite fascinating. “Currently, I’m working on the applications of a new fixed point theorem for isometries to operator theory and machine learning”, explains the professor. A fixed point theorem is a mathematical result that guarantees the existence of points that remain unchanged under certain transformations; isometries are transformations that preserve distances, like rotations or reflections. This can be applied to operator theory, which studies linear transformations of function spaces, and also in machine learning, where understanding the structure of data and transformations is important for developing algorithms. “In another direction, I study spectral invariants of limits of graphs – like in mathematical physics – as an approach to analytic number theory, in particular for new structural understanding of zeros and special values of Dirichlet L-functions.” Spectral invariants are properties related to the “spectrum” or eigenvalues of graphs – like the frequencies of a vibrating string – that, here again, remain unchanged under certain transformations. Studying the limits of graphs means analyzing what happens when graphs grow infinitely large. The results provide tools for understanding problems in analytic number theory, a field that uses calculus and complex analysis to study numbers. In particular, the goal is to gain insight into Dirichlet L-functions, which are central objects in number theory connected to prime numbers. Understanding the zeros and special values of these functions better would reveal deep structures and patterns in mathematics.

Currently, I’m working on the applications of a new fixed point theorem for isometries to operator theory and machine learning. In another direction, I study spectral invariants of limits of graphs – like in mathematical physics – as an approach to analytic number theory, in particular for new structural understanding of zeros and special values of Dirichlet L-functions.

Professor Karlsson held deep admiration and respect for all researchers before becoming one himself. After all, discovering new concepts and results only using pen and paper seems like a daunting task, doesn’t it? But once he entered the field, he was surprised at how once you start working on a topic, it’s actually quite possible to make progress and find new theorems. There are many, many open problems to work on, although it is a challenge to choose the most fruitful directions. “My first real exposure to research came during my PhD at Yale University” says Professor Karlsson. “At the time I was there, several of the professors had made fundamental discoveries quite some time back and were laureates of the most prestigious awards. Once I began working at other universities, I noticed that research is perceived differently from one place to another. It made me think that people sometimes place too much value on what’s considered important at the moment, rather than from a longer perspective.” Unfortunately, this also means that having a successful career in mathematics sometimes takes more than just being good at it. Even a strong research paper on a topic that isn’t “trending” may not receive the attention it deserves. Meanwhile, papers on more popular topics tend to attract significantly more interest, even when they aren’t particularly innovative. On the other hand, intense focus on certain timely topics is important for crucial progress. “From the outside, one may have an idealized view of researchers – a community of passionate people, all working together in harmony to advance science. But the reality is more complex and imperfect” he explains. Time is also a judge – like the famous mathematician G.H. Hardy once said, beauty is the first test: there is no permanent place in the world for ugly mathematics.

Time is also a judge – like the famous mathematician G.H. Hardy once said, beauty is the first test: there is no permanent place in the world for ugly mathematics.

Indeed, many theoretical concepts in mathematics may initially seem abstract or even useless, but history has shown that theory often finds practical applications later on – sometimes decades or even centuries afterward. A classic example is Boolean algebra, developed in the mid-19th century by George Boole. At the time, his work was seen as purely theoretical, with no obvious use outside of symbolic logic. Yet today, Boolean algebra forms the foundation of digital logic and computer science; it underpins everything from circuit design to search algorithms. That’s why advancing theory is not just a philosophical exercise – it often lays the groundwork for future breakthroughs, even if their relevance isn’t immediately clear. It’s a reminder that seemingly “useless” mathematics can shape the world in ways we can’t yet predict. The use of number theory for cryptography is another example. “Sometimes a problem resists or all the approaches seem awkward, but then in good moments it suddenly happens that one sees it in the right way, and everything becomes simple and coherent. This is the aesthetic component of mathematics mentioned above, together with its remarkable coherence and power to unify. To me, a beautiful mathematical result can often be qualified by three features: simple, profound and surprising. But that’s what makes the difference between mathematics and real life – like it’s described in Dante’s books, whenever one tries to formulate a principle based on current experience, one can be sure that only a few days or cantos later, there appears something complicating that principle…”

To me, a beautiful mathematical result can often be qualified in three words: simple, profound and surprising.

It's interesting to notice the shifts of focus in mathematics over time. The 1960s, with its focus on more abstract ideas, had Grothendieck revolutionize algebraic geometry and number theory. With the rise of computers dynamical systems and chaos theory became a very active field. According to Professor Karlsson, probability theory has in recent decades become more central than ever. Most recently, there’s been a huge surge of interest in neural networks and machine learning, making today’s AI. This technology is much more mathematical than the impression given in the news and public discourse and this topic is of an unprecedented scale and urgency as compared to the trends discussed above. 

This article is part of a three-part series by volunteers Yuta Mikhalkin and Tanish Patil. Read the conversations with David Cimasoni and Michelle Bucher as well!

“On this note, if I may add one more general comment, it would be that everyone agrees that AI needs to train on the best data possible there exists”, says Karlsson. “If one reverses the usual direction in the intelligence metaphor, then humans should train on the best that has been thought, created and written – read and study the canonical works of literature, in particular Shakespeare and Dante. There seems however to be a misunderstanding about this.” Apart from literary classics, the professor also has some other recommendations for aspiring scientists. “Besides from what I already mentioned about pursuing topics out of passion and not utility, a good advice for young mathematicians is to not compare yourself to others, everyone has different strengths”, says the professor. “The fact that you know less than someone doesn’t automatically make you worse or less successful than them. Also, it’s always important to have confidence in your ideas and to develop them, even if you’re unsure if they’re going to work – if you constantly doubt yourself and assume nothing is good enough, you’ll struggle to make progress. "It’s crucial to have confidence in yourself to be creative and find the solutions, even after failed attempts!  And be guided by beauty!”.

It’s crucial to have confidence in yourself to be creative and find the solutions, even after failed attempts! And be guided by beauty!