**Igor Klebanov** sees the world through a “mathematical microscope”. The SUSU scientist creates caricatures in a form of equations. Creative approach helps finding unexpected solutions and obtaining beautiful results.

Igor Klebanov was born in 1969 in Chelyabinsk. He graduated with honors from the Faculty of Physics of the CSPI (1991) and completed his postgraduate studies at the CSU’s Department of Theoretical Physics (1994). Today he is Associate Professor and Senior Research Fellow at the Department of System Programming of the SUSU School of Electrical Engineering and Computer Science, Candidate of Sciences (Physics and Mathematics). His hobbies are philosophy, literature, theatre and sculpture. He is married and has a son.

*Photo of Igor Klebanov*

### Under the sign of symmetry

My interlocutor is a keen scientist. In 26 years of research activity and teaching Mathematics at the School, the objects of his research were statistical physics, physics of liquid crystals, and mathematical modeling in humanitarian research. Nowadays, his professional attention is focused on the “mathematical microscope”. Speaking in scientific terms, the group analysis of mathematical models.

“In the basis of research by the ‘mathematical microscope’ lays the symmetry,” says Igor Iosifovich. “It rams through the entire universe. From the position of mathematics, symmetry is some changes that leave an object unchanged. Let us consider a human body, for example. If not to consider its internal structure, it is clear that there is a reflexive symmetry. Or let us consider a usual equilateral triangle: there are three axes of symmetry (angle bisectors).

We can also rotate it by 0, 120 or 240 degrees regarding the center, and it will coincide with itself. We are working on more complex things. It turned out, if we know the symmetry of a system of equations that model a physical process, we practically have a microscope in our hands. It allows seeing a solution which is simply invisible without this technique, or might be an object of a genius guess. It happens that mathematicians who don’t know how to apply the symmetry analysis intuitively guessed the possibility of such solutions. Whereas the symmetry analysis allows obtaining these solutions distinctly, with the use of algorithms.

Everything had started in the 70s of the 19th century in Norway, where an outstanding European mathematician named Marius Sophus Lie invented the ‘mathematical microscope’ – an apparatus for symmetry analysis. Contemporaries, as it frequently happens, didn’t duly appreciate his invention. They considered it only to be a new language for reciting old things. Only two of his closest friends realized that the discovery has far-reaching implications. That is exactly what happened.

Sophus Lie passed away in 1899, and during the following 50 years the group analysis wasn’t bringing up much of attention. But everything had changed in the middle of the 20th century, where science started dealing with gas dynamics for military purposes. Specialists in the sphere of gas dynamics have to solve very difficult equations, which require a lot of efforts even using modern computers. And so, two brilliant minds – the American mathematician G. Birkhoff and the Soviet scientist Lev Ovsyannikov – realized that group analysis of mathematical models can play a huge role here. Like this, the technique started getting implemented into applied mathematics. The metaphor of ‘mathematical microscope’ was introduced by an outstanding contemporary mathematician, Nail Ibragimov.”

### Microscope for the Universe

**– What did you manage to see using the “mathematical microscope”?**

– Now I keep working on things that were being done at the School named after Academician Ovsyannikov in Novosibirsk, but with a little modification. They had been studying equations of classical gas dynamics, but I also added gravity. In other words, I study the equations that model gas taking into account the force of its own gravitational field. The sphere of using this study is Astrophysics: the theory of formation and evolution of stars, and formation of the large-scale structure of the Universe.

These equations have never been considered under the “mathematical microscope” until now. For the last three years, I’ve been researching this more actively, and published several papers. I managed to discover symmetries of these equations, as well as find some new solutions that can be of interest for astrophysicists. By the way, I was beginning my career as an astrophysicist, so I understand the essence of the processes that take place. But as of today, I am an applied mathematician and physicist. I don’t keep track of events in modern astrophysics anymore.

**– Are these “new decisions” a scientific discovery?**

– The word “discovery” sounds too pretentious. This is a result, a serious, important and beautiful one. I don’t like using words lightly. Discovery is something fundamental; something that rapidly brings forward an entire sphere of knowledge. Meanwhile, I only obtained a result which in time might lead to some discoveries.

**– Are you being modest, like a true scientist?**

– You know, I can’t say that I’m a modest person. More likely, I am an honest man. After all, a scientist should be honest.

**– You started talking about origin of the Universe. What theory do you adhere to?**

– The Canonical one. The Big Bang theory has some difficult moments that make some scientists have their doubts. But if we are not going too deep into it, this basically is a quite adequate theory. You know, in order not to adhere to some common version, you need t have some valid reasons and carry out research in this sphere of knowledge. I simply have another interest – the sphere of applied mathematics.

### Scientific forecasts

**– Generally, the sphere of astrophysics has many unclear moments, even for scientists. Considering we start thinking about infinity of the Universe.**

– You know, regarding this issue, a Noble Prize Winner and respected by me Lev Landau has two good quotes: “Cosmologists are always wrong but never in doubt” and “Modern science helps us understand things that we can’t imagine”. The first quote, obviously, is just a joke on astrophysicists, but the second one is very profound.

Indeed, there are things that we can’t imagine but can only understand. Imagining means create a visual model close to our comprehension. But who are we? We are macroscopic three-dimensional creatures living on Earth. We think by space categories, available for our sensory organs. An as the Universe goes beyond our sensory organs, we can’t visually imagine it entirely. Though we can understand it, which means making some forecasts.

**– So you’re saying that scientists read the tea leafs as well? And they are scolding astrologists, for example, for doing the same thing.**

– The background is entirely different here. Scientists are making forecasts on the basis of a model or hypotheses that can be verified. Basically, science is possible when verification is possible, i.e. check-up. When I am constructing a mathematical model, I draw a conclusion based on calculations. If an experiment proves it, this model is good; if not – the model needs to be changed. To tell the truth, sometimes it happens that an experiment doesn’t prove a theory, but then it turns out that experimenters had missed something.

Astrologists use another technique, the esoteric, philosophical one. By the way, I am not with the people who are scolding astrologists. Of course there exist charlatans who write their forecasts for everyone at once in yellow rags. But if we address the history, the medieval astrology had nothing in common with the modern quackery. It was a serious science based on other premises rather than on contemporary Natural Science. Just like alchemy was based on very profound premises connected, first of all, with human transformation.

**– Then why is all that now condemned?**

– I think, this happened in the new time, when the positive science based on experiments got into rivalry with traditional medieval sciences. The first one won, as it appeared to be more reliable in forecasting, but as they say, the baby was thrown out with the bath water. What did alchemists really do? Roughly speaking, they were searching for the philosopher’s stone in order to turn everything into gold. But we forget about one important thing. There was a condition: before you find the philosopher’s stone, you must change yourself. Only the one who transforms oneself and become “twice born” (there is such concept in philosophy) can find the philosopher’s stone. This is exactly what was thrown away; only some strange pattern remained.

### Space brothers

**– Let us go back to your research in the sphere of astrophysics. Can one see, for example, space brothers by watching through the “mathematical microscope”? When it comes to speaking about Space, the question that arises every time is whether we are alone in the Universe…**

– The “mathematical microscope” does not search for space brothers (laughs).

**– Though astrophysicists do, obviously?**

– Yes, they do. But this isn’t the mainstream. As of today, as far as I know, none of acknowledged astrophysicists dedicate their life to detect signals from intelligent creatures from Space.

**– Then space brothers do not exist because no one is searching for them?**

– They are being searched for. But, first of all, who said that they should live far beyond the Earth? After all, there is the theory, according to which they can be in the same room but in a parallel world. There is no telling that they possess the same communicative abilities. We send a signal, but what if they can’t decipher it? They might have a completely different language system. All these plays in searching for space brothers are based on anthropomorphism. This means that we consider them to be similar to us. But let us recall the Solaris. What did the other mind look like? It was an ocean, reading minds and reacting to them. Therefore, the search for space brothers somewhere around Alpha Centauri or even further is an out-of-date idea of the 60s of the past century. By the way, back then this search indirectly led to an interesting discovery. Strictly periodic signals, at first assumed as aliens’ calling signals, were detected. But then it turned out that it was a rapidly rotating neutron star. This is how radio pulsars were discovered.

### Unfashionable science

**– And still, what is the applied relevance of your research?**

– A huge one! In any sphere of science, the most important thing is to have a good mathematical model, meaning the equations which can clear up many things. We find new solutions. We watched under the microscope a system of gas dynamics taking into account its self-gravity – and obtained interesting results. There is a law of the expanding Universe, as known as the Hubble's Law. We proved its group-theoretical nature and discovered that this is one of the possible symmetric solutions. But there might be another way, there are alternative scenarios. And then, astrophysicists should consider more subtle effects and determine whether there are indeed some essential amendments to the Hubble’s Law.

**– But doesn’t the presence of alternative disprove the Hubble’s Law?**

– If the alternative variant gets proved experimentally… But as of today, we can’t assure to have disproved the Hubble’s Law. Unfortunately, the “mathematical microscope” is a tool that only a few have mastered. This is a very useful though unfashionable sphere of mathematics, because it requires a lot of analytical work. Nowadays, the fashionable are abstract areas of knowledge, which are too far from life yet: algebraic topology, algebraic geometry, and the theory of categories and functors. If we bring together all acknowledged specialists in group analysis from all around the world, they are unlikely to fill the SUSU Activity Hall. Let us recall the story of Perelman who proved the Poincare conjecture. I tell you what, only 10 people from all over the globe could really comprehend his works. At that, it took them two years to figure everything out. In general, pursuing fashion in science is silly. A scientist should do what he or she is interested in, and what can bring results.

**– So you are lucky! Because, in your case, this is exactly what happens.**

– My work in the sphere of mathematical physics and mathematical modeling is similar to work of a caricature artist. How does one draw a caricature? You need to detect some characteristic feature of a person, enhance it, and diminish everything else. In other words, if a person is tall and skinny, I can draw him in a form of spaghetti. Like this, an applied mathematician of mathematical physicist, in fact, draws caricatures of the object under research. He chooses several of the most characteristic features, hyperbolizes them, and pushes everything else into the background, simply neglecting it, to be exact. For example, I believe that there are factors A, B and C, at that, other factors do not interest me. So I write mathematical equations as if there is nothing else except for these A, B and C. So generally, I am drawing a caricature of an object and, by doing this, I learn new things. Therefore, any mathematical model is a caricature. By drawing caricatures we learn the world.