Stevo Todorčević, Mathematician

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Stevo Todorčević is reputed to be one of the world’s greatest authorities in some areas of mathematics

Stevo Todorčević, Mathematician

Stevo Todorčević, born in 1955 in Mrkonjić Grad, graduated in mathematics at the University of Belgrade back in 1977, earning his master’s degree in 1978 and his doctorate in 1979.

He became a member of the Serbian Academy of Science and the Arts in 1991, and a member of the Royal Society of Canada in 2016. He has led themed mathematics programmes at the Mittag-Leffler Institute in Stockholm (2000), Toronto’s Fields Institute (2002 and 2012) and the Centre de Recerca Matematica in Barcelona (2003-2004), as well as leading scientific projects in the United States, Canada and France.

Todorčević has had, or still has, installations at prestigious world institutions, such as, for example, the University of California in Berkeley, the Princeton Institute for Advanced Study, CNRS in Paris and the University of Toronto.

When he was inducted into the Serbian house of immortals at the age of 36, the explanation wrote: “He solved several key mathematics problems that had waited several decades for a solution and provided procedures for work in several fields of mathematics, which bear his name. We can conclude with certainty that Stevo Todorčević is our mathematician who has been appreciated and valued the most by world mathematics.”

Paul Erdős, one of the most highly-rated mathematicians of the 20th century, wrote about one result of Todorčević’s work (published in journal Acta Mathematica in 1987): “This is certainly one of the unexpected and sensational results”.

On the basis of the citation for an award received by Todorčević in 2012, we see that his contributions include solutions to a number of longstanding open problems in various fields of mathematics, such as the problem of S-spaces and Katetov’s problem in topology, the von Neumann and Maharam Problems regarding measure algebra, the Davis-Johnson’s problem from functional analysis etc. He has also been attributed with research work methods in mathematics, such as the Method of Minimal Walks or the side condition method in forcing design. He is also responsible for the correspondence (KPT correspondence) between topological dynamics and the structural Ramsey theory.

Todorčević and his associate Dilip Raghavan have just solved another problem in mathematics that had remained open for more than 50 years and will be published very soon.

“And it was solved here in Serbia while walking with my son Marko around Kalemegdan” – says Todorčević with a smile and unconcealed delight, turning his look towards his wife, Vesna, who is also a mathematician and professor. Thanks to this turnaround at the beginning of our conversation, it became quite clear that Stevo Todorčević is a happy man. Our story of a great scientist – a story of love, passion, enormous sacrifice, and unselfish giving – could begin…

Stevo Todorčević, Mathematician
Vesna and Stevo Todorčević

There’s no fascination with money or some other rewards that can be compared to that experience of joy, that spark

Responding to comparisons with Mihajlo Pupin, Milutin Milanković and Mihailo Petrović Alas, he said briefly, “please don’t”. And we left it at that. We didn’t talk about recognition, awards or titles either, because Todorčević is “just a mathematician”.

“I think that of crucial importance to my scientific career was actually my open approach to mathematics. Given that back in the then Yugoslavia of 1979 I was unable to find an appropriate mentor, at the recommendation of English mathematician Keith Devlin (who was a member of my doctoral commission, at the suggestion of Professor Đuro Kurepa) I decided to go to Jerusalem and attend the lectures of Professor Saharon Shelah on proper forcing.

“I didn’t collaborate personally with Shelah, but I sat through his lectures. However, he knew that I was among the listeners, so I presume that it was because of me that he lectured in English, instead of in Hebrew. And, what is most important, he wrote everything on the board. That was very rare at that time and is even more so today. And that was of great help to me because I didn’t know English well enough. It was then that I learned a method that would later prove useful to me.

“Another important mathematician in my life was James Baumgartner, who worked at Dartmouth College in New England. He noticed that I had some talent, whatever that meant at the time, so he invited me to spend six very productive months there, where I solved two major open problems. That progress was initiated with a letter from Frederick William Galvin, sent from Budapest on 2nd November 1980, in which the most important problems of one area of mathematics were described on 22 pages. However, it took me four years to solve the most important problem on that list, with a solution about which Paul Erdős himself even expressed great jubilation.

“Richard Laver is another mathematician who was very important for me. It was from him that I learned something which is incalculably important – persistence. How not to give up and to extend further when there is no success after a year or two or three when you are passing through a tormenting crisis.”

How does one find inspiration in mathematics; what’s the feeling like when you realise you’ve solved an enduring problem, that you’ve discovered something for which science will be grateful?

“That’s something that almost nobody knows. If mathematicians knew how and what to do to gain inspiration, then there wouldn’t even be any mathematicians. It seems to me that it’s about a great desire to know something, to learn, to search for the unknown.

“That’s a psychological thing and is set largely in the vocation of mentoring. A mentor (or oneself) must recognise when a student (or oneself) should stop, to cease with attempts, and where to direct themselves. During a certain period, our chances of solving a problem grow, but the moment comes when those chances fall away and it is then that a risky phase emerges.

“I also had such moments, when a problem started to torment me when nothing went my way. And then, in one instant, I would grasp where my mistake was, but I couldn’t formulate it in the right way. That’s because in order to reach the solution to a problem you need to create a system (a trap when it comes to hunting) – why something should coincide.

“I believe that the majority of mathematicians deal in maths precisely because of that – because of the spark that they experience when they reach the desired result. There’s no fascination with money or some other rewards that can be compared to that experience of joy, that spark.

“Joy is also experienced when you are introduced to an open problem and the connections that solving it could bring. That rapture is one of the most important factors of success because, without it, chances are small. It gives us strength in the process of solving that open problem. Children have that. I saw and realised that while watching my son Marko. He would play and become interested in something, and when noticing something more interesting he would suddenly leap, going to a completely different place and involving himself in some new thing with absolute attention as if he hadn’t previously been in some other game!

Stevo Todorčević, Mathematician

Studies and exams aren’t tailored to those seeking deeper knowledge. And that’s one of the major shortcomings of our education

The satisfaction shown by Todorčević through observations of his son returns our interlocutor to his own childhood, his youth and student days…

That part of my life, from a primary school in Ubovića Brdo and Banatsko Novo Selo, to high school in Pančevo, was very important for my upbringing, because it was there that I met people from whom I learned.

Those were my teachers of physics, music and fine art in Banatsko Novo Selo, and professors of physics and art in Pančevo. They taught me that schooling is not just for the sake of learning known facts, but also because of revealing that something new exists, something that needs to be discovered. It’s important to have someone like that during your schooling, especially at the beginning; someone who conveys their knowledge with such selflessness and joy or, if you want, love for some field.

One such professor at the university was Professor Kurepa, an excellent pedagogue who delighted in the successes of his students. During my studies, I engaged less in exams, because I mainly read mathematical literature that I could access and scientific works that had nothing to do with exams. The college couldn’t satisfy all my interests. Studies and exams aren’t tailored to those seeking deeper knowledge. I would say that one of the major shortcomings of our education also lies in that. Our colleges aren’t sufficiently adapted to those who want to deal with science.”

Todorčević headed out into the world at a time when that it wasn’t a trend. He went in search of knowledge and was very courageous in doing so. He today attributes that to youth that has no hesitation towards confronting the fears of the unknown.

“I went to Jerusalem at the suggestion of Keith Devlin, at my own expense. My master’s thesis that solved Blumberg’s problem (Andrew Blumberg) interested Canadians from the University of Toronto. I had great difficulty obtaining a visa for Canada. It seems to me that this was a crucial period in my life. If I hadn’t succeeded in going to Canada, I would have had to return. However, by some miracle, good fortune, I don’t know myself how I happened to cross several European borders and even the American border without visas or tickets for return flights. That sounds unbelievable, and it really was unbelievable. I only had a Canadian visa and nothing else.

“That workshop in Toronto gave me the opportunity and enough time to solve a new problem, to get into the problems of some other areas.

According to his wife, Vesna, “Stevo was a scientist-nomad. He always had to prove himself throughout his life, in order for others to realise how much they actually needed him. He received calls from one university to another and that’s why he had to be quick and successful because he otherwise would have had to return…”

“That was the very beginning,” adds Stevo, quite brave and typical for young people… “and I transfer that experience of mine to my students – to be ambitious and persistent, without calculations. That represents a breakthrough in life.”

Asked what is implied by contemporary mathematics, whether mathematics consists of different fields or is a unified science, whether its areas are strictly divided or overlapping, Todorčević says: “Both one and the other must exist – fields must be developed but must also be connected. In order to solve some problem, a mathematician must find ideas somewhere, so that the repository of ideas must exist in order for mathematics to advance. This also applies when it comes to the application of mathematics. For instance, today we have a lot of mathematical applications in informatics (Computer Science), and we know that the theoretical part of computer science separated from mathematics about 50, 60 years ago. So there is no surprise today when some mathematician returns to the roots of mathematics in an attempt to solve some open problem from Computer Science. This actually happens often.

“There are times when you’re in the mode of solving open problems and times when you’re in the mode of overviewing mathematics from a broader perspective. It seems to me that that’s also an issue of energy. For each solution to a difficult problem you invest huge energy that gives you a new picture of mathematics, which you are the only one who is able to see at that moment, so little else interests you then, not even publishing your work on that solution. That’s usually how it happens.

“That’s precisely what happened to me when I was working at Berkeley and Princeton. I solved some major problem that everyone expected me to publish immediately. But I wasn’t interested in that at all. And that was possible at that time, while today the system has changed and universities force us to publish our works. If you want a job at a prestigious university today and accept such a job, you are also obliged to help improve the reputation of that university by publishing important results. From 1980 to 1986 all my works were in handwritten manuscript form and were only later published.”

The intensive expansion of mathematics; its unceasing growth through the circle of users – from natural and social sciences, via the arts, medicine, economics and finance, to social networks – precisely through mathematics unites the so-called “economics of thinking” and numerous syntheses. Where is mathematics in this common thinking process and is mathematics an essential tool in many scientific fields?

“Exactly that. And it shouldn’t only be a tool of mathematicians, but rather some kind of literacy. Mathematical literacy is needed so that a person can properly approach not only mathematical problems. The mathematical way of thinking, mathematics, isn’t just natural science, rather it naturally imposes itself in many areas of creativity.”

The last century was the golden age of modern mathematics. Many open problems that had existed for 100, 200 and 300 years received solutions.

In the 20th century, mathematics reached maturity and depth. Solutions to many of these problems were reached through the development of areas which seemingly, at first glance, share nothing with the topics to which these problems relate. This maturity is also based on the finding that there are limits to certain mathematical ideas and that new ones are needed.

In order to solve some problem, a mathematician must grab ideas, they must nurture that repository of mathematical ideas in order for mathematics to advance and be applied

The field of Set Theory launched Stevo into the world of mathematics and, as he says, enriched his way of thinking enormously.

“That’s a kind of language that’s always been useful for me… In mathematics, you initially don’t even know the area in which you will specialise. You start in one area and end up in a completely different one. In most cases today it is the mentor that determines the area that you will be interested in initially.”

And so we arrive at another of Todorčević’s great passions that he also favours and which ranks him at the very top of world science: mentoring.

“Mentoring is a highly demanding job because it requires the exertion of enormous energy. Just one student is too much, not to mention two or more. If you take mentoring seriously, it occupies you completely and can be terribly exhausting. That’s because you have a student coming to you who has been attempting, sometimes for years, to discover something, confronting enormous difficulties, and who is unable to explain those difficulties to you. On the basis of that contact with the student alone, the mentor has to recognise these obstacles, to evaluate them correctly and suggest the next step.”

Todorčević is considered one of the most beloved mentors at the universities where he lectures. As much as the students love him, he returns no less love to them. Vesna notes that although Stevo is currently in Belgrade, he actually spends all his time with his students. How is this connection so highly valued and most often crucial in the success of a scientist?

– Students probably feel like they couldn’t succeed if there was no mentor. On the other hand, the mentor is very pleased to see a new researcher arrive on the scene. From a mentor’s perspective, it’s also very important to teach a student to think in a certain way. When I see that a student has accepted a certain way of thinking, I’m then sure that he or she will be a successful researcher. The mentor must have a correct vision of the mathematical field in question, a vision that he should transfer selflessly and enthusiastically to students.”

The best professors are, as a rule, those who are surpassed by their students.

“That’s correct! That’s also how it should be because otherwise there’s no progress. One such student, for example, was Justin Tatch Moore, who accepted one of my ways of thinking and made something much better of it.”

Speaking about the place of theoretical mathematics in the IT industry, artificial intelligence and how our future will look, Todorčević says:

“Whether or not artificial intelligence will become dominant in relation to the human brain is a philosophical question. I think that most mathematicians will tell you that that’s not possible. It’s not clear to me how artificial intelligence will succeed in nurturing a repository of ideas, ordering them into a series and how a machine will, at a certain juncture, select the precise idea that will work. That seems impossible to me, but artificial intelligence will undoubtedly play an important role in the future development of mathematics.”

As an educator with vast experience, we asked what his message would be for the many talented young mathematicians that Serbia has.

– First and foremost, to be ambitious and unrestricted, and to familiarise themselves with as many different areas of mathematics as possible, in order to be able to recognise what suits them the most. It is also absolutely essential to communicate with mathematicians who are experts in different fields. Of course, this precedes the acquisition of knowledge, but also the willingness to communicate liberated from vanity. I’m always delighted by mathematicians who are able to return to childhood and go back to school benches in other fields of mathematics; when top mathematicians who are very successful in one area listen to lectures from other fields like schoolchildren, with enormous enthusiasm and joy. Those types always managed to also become successful in these new fields!