Mathematician Dalimil Peša: “I've Always Enjoyed Getting to the Bottom of Things”
Dr. Dalimil Peša, who claims that calculation isn't his strong suit, most enjoys the part of his work where he examines a problem from all angles and waits for inspiration to strike: “This phase is the biggest fun, and what’s great about it is that you can do it practically anywhere – even while taking a walk,” comments smilingly the graduate of Matfyz and the newly awarded recipient of the International Stefan Banach Prize. He is the first Czech ever to achieve this prestigious honour.
Dr. Lenka Slavíková, a colleague from your department and the 2023 Neuron Award laureate for promising scientists, says she considers mathematics a form of art, one that never reveals in advance where it will lead her. How do you perceive mathematics? What fascinates you about it?
I see mathematics as an effort to understand things in their purest form. I’ve always enjoyed getting to the bottom of things, no matter what they were, and that’s precisely why I was captivated by the abstract nature of mathematics. Unlike the problems of many other sciences, mathematical problems are well-defined and typically not overly complex. By that, I don’t mean they aren’t challenging, but the number of factors involved is usually relatively small, and, most importantly, you know all of them in advance.
This allows you to lay hold of problems as a whole, to fully understand them in detail, and to find solutions that are completely precise. Once you have a proof and you thoroughly understand it, then, in a sense, you know everything there is to know about the problem. And once this proof is correct, it remains correct forever – unless, of course, people change their minds about what it means for something to be “correct”.
As a result, mathematics can often be incredibly elegant and beautiful. Of course, this isn’t always the case, but there is a remarkable amount of beauty in it.
You specialize in mathematical analysis. How would you explain this discipline to a layperson? What is it used for?
Mathematical analysis fundamentally deals with functions and naturally related concepts (such as sequences, derivatives, integrals, differential equations etc.). Its key motivation comes from physics, as many physical phenomena can be described using functions and differential equations.
Analysis is well-known to all students of Matfyz, as it is one of the mandatory and most redoubtable subjects in the first study year. How did you cope with Analysis at the beginning of your studies? I assume it wasn’t too difficult for you, given that you later chose it as your major…
Analysis is indeed one of the most challenging subjects, not only in the first year but also in the second one. The greatest difficulty traditionally comes from the written exams, where students must solve several complex problems within a limited time. That was also the case for me – calculation isn’t my strong suit, so I struggled a lot. In fact, the only exam I ever failed was Mathematical Analysis 3, where I didn’t pass the written part on my first attempt. On the other hand, the oral exams, which involved proving theorems, were usually less problematic for me.
So, I definitely didn’t choose this field because I would find it easy. In truth, the subjects I considered the most difficult during my undergraduate studies are often the ones closest to my current specialization. Apart from Mathematical Analysis 1–4, the list also includes Measure and Integral Theory and, above all, Introduction to Functional Analysis, which I believe was the most challenging course in the entire undergraduate program.
So, what attracted you to analysis after all?
There were several factors, but the most important one is that I find analysis beautiful – not only the results and concepts it deals with but especially the methods used in proofs. In my view, analysis is abstract in just the right way. On the one hand, it’s abstract enough for the theory to be clear and elegant, but on the other hand, it’s still tangible enough for me to develop some intuition about how the objects I work with behave.
This suits me because I don’t enjoy problems that are too concrete, as they often require spending a lot of time on “dawdling over” details, which can obscure the essence of the problem. On the other hand, some fields are so abstract that, even if I understand every step of a proof, I still have no idea what’s going on. That’s why I chose a field that balances between these two extremes.
How did you find your specialization?
My supervisor, Professor Luboš Pick, played a key role. I was fortunate to meet him in an informal setting early in my studies, and I felt he was someone I would enjoy collaborating with. I also knew he encouraged students to engage in original research, even during their bachelor’s theses, which was something I wanted to try at that time. So, I approached him to supervise my bachelor’s thesis. At that point, I didn’t have any concrete idea about a specific topic – I just knew it would be analysis and hoped I would enjoy it. I was assigned to research function spaces, which have largely remained my focus up to now.
Function spaces are also the subject of your dissertation. What is it about?
Unfortunately, a function space is quite an abstract concept that is difficult to explain without referring to more advanced mathematical knowledge, so any explanation will inevitably be somewhat clumsy and not entirely precise. In principle, however, it is a space where each individual point is, in fact, a function.
One commonly encounters a one-dimensional space, i.e., a line. Similarly, we often deal with a two-dimensional space, i.e., a plane, and, of course, the three-dimensional space around us. Each point in such a space can be described using coordinates, which are essentially just groups of several numbers: one in the case of a line, two in a plane, and three in space. It is the quantity of required numbers which determines the dimension of the given space, and it turns out that if one wants to work with such a space, the coordinates alone are often sufficient. Once we work solely with coordinates, i.e., groups of numbers, nothing prevents us from using larger groups than just triples. In this way, we can easily implement, for example, a seven-dimensional space simply by considering coordinates with seven components. Such a space, of course, cannot be visualized, but this does not prevent us from working with it, because as long as it is described by coordinates, it behaves in many ways similarly to two- or three-dimensional spaces.
The key aspect in this case is that points can be added up (by summing the corresponding coordinates) and multiplied by a number (by multiplying all coordinates by that number) and we can also measure distances in these spaces by applying the Pythagorean theorem to the individual coordinates. Functions, in a sense, are quite similar to these coordinates, except that there are infinitely many of them – each point in the domain of a function corresponds to a function value that acts as a coordinate. In principle, a function space is an infinite-dimensional variant of conventional, visualizable spaces. The important feature is that addition and multiplication are still possible, and that also distances can be measured within such a space.
Can you briefly summarize what your research focused on, what you discovered, and what potential benefits it may have?
In my dissertation, I focused on three areas. In the first part, I studied an abstract class of function spaces. This class is defined by axioms, meaning that you do not have an explicit specification of which functions belong to the space or how distances are measured within it; you only know that the space must satisfy certain given properties. The goal is then to determine what can be said about such a space using only these axioms. This is highly useful because when working with a specific space, it suffices to verify that it meets these axioms, and suddenly, you have access to a wide range of tools that can significantly simplify your work. In fact, I came to this topic just because, while studying specific spaces, I wanted to apply some abstract results but could not do so, as the existing theories relied on overly strong assumptions that were often not met by these spaces. This led us to develop a theory encompassing all the cases relevant to us, initially in collaboration with Associate Professor Aleš Nekvinda from the Czech Technical University and later with other colleagues.
The second part of the dissertation focuses on a specific class of function spaces called Lorentz-Karamata spaces. These function spaces have recently appeared relatively frequently in various areas of mathematics (such as interpolation theory), but no comprehensive theory existed for them, and many of their properties were not well described. The intended benefit, in this case, is that mathematicians working with these spaces will be able to easily find everything they need to solve the problems they are interested in.
The third part then dealt with applications, meaning that methods from function space theory were used to solve problems from other fields.
Describe to me what working on something like this looks like in practice. Do you just sit down at a desk with a pencil and paper and start calculating?
That is not too far from the truth, but it is only one part of the whole process – and it usually does not involve calculating in the way people typically imagine. Working on an open problem generally consists of four stages. First, it is necessary to study the available literature related to the problem, meaning reading articles and books. This provides the necessary context, the tools that can be used, and often inspiration for how to approach the problem. Then, one needs to come up with an idea, which means thinking about the problem from different angles until a way to construct the proof is found. This phase is the most enjoyable and is particularly appealing because it can be done practically anywhere, even while taking a walk. The third phase is the one that most closely resembles your idea – it requires sitting down with paper and working out the details. This often includes some form of calculation, but in an abstract sense, where the result is not a number but, for example, the validity of an estimate for a given function or something similar. Finally, everything must be written up in a form suitable for publication.
Of course, it is never this simple and straightforward. Problems can arise at any stage, requiring a return to previous steps. It is entirely common that even during the process of writing up what seems to be a completed result, it becomes apparent that some parts of the proof are not fully developed and that a proper solution still needs to be worked out.
Do you have any ritual that helps you get started before diving into work? Some people make coffee, others go for a run… How does a mathematician approach this?
Personally, I have never really been into rituals, so I don’t have anything like that. The closest thing would probably be that when I am in the phase of contemplation about a problem and I am looking for ideas, I tend to move around, to walk around my apartment, the building, or even outside in the surrounding area.
Where do you see your career heading, and what would you like to focus on? Do you plan to stay at Matfyz and continue with the topic of your dissertation?
I definitely want to continue in research. I am currently working as a postdoc at TU Chemnitz on a two-year contract. Beyond that, I don’t have specific plans yet, but in the long run, I would like to return to the Czech Republic and find a position at a university or the Academy of Sciences. As for research, I am currently trying to take advantage of the new environment to broaden my horizons so that I can work on a wider range of problems in the future. However, I have certainly not abandoned my previous topics – especially in abstract theory, there are still many interesting open problems.
What does the Banach Prize mean to you?
For me, it is primarily proof that my work so far has been meaningful, as well as a great motivation for the future to strive for further interesting results.
The International Stefan Banach Prize is awarded by the Polish Mathematical Society to authors of outstanding doctoral dissertations in the field of mathematics. The prize is named after the Polish mathematician and founder of functional analysis, Stefan Banach, and aims to support young scientific talents from Central and Eastern Europe.
OPMK, photo by Tomáš Rubín