# Vibrating reality … so what?! (Ask Victor Burenkov)

The renowned Russian professor talks about his passion for mathematics – and the real-world applications of functional analysis and differential equations

Mathematics is interesting – mathematics is everywhere! Can you imagine, with only a piece of paper and a pen, I can conduct full-fledged scientific research?! I sometimes see formulae and solutions in my dreams.

Professor Victor Burenkov takes my Samsung mobile phone and says: “You can’t even imagine how much math has been put into it. There is a famous saying that any particular science could be considered science only to the extent of how much mathematics is in it.”

Mathematics is everywhere, he affirms, from space and aviation to gadgets in our day-to-day lives.

Victor, an internationally recognized professor, Director of the SM Nikol'skii Institute of Mathematics and Head of the Department of Mathematical Analysis and Theory of Functions at the Peoples’ Friendship University of Russia (RUDN), is an expert in functional analysis and differential equations – which he says can be used to explain virtually all vital processes, described mathematically with the help of differential equations.

Differential equations are very diverse, he explains; there are thousands and thousands of them describing processes in physics, astronomy, chemistry, biology, economics and other sciences. They play an important role in a variety of industries, from aviation, space research, engineering and medicine to agricultural technology.

One of the challenges of a modern mathematician is to solve differential equations explicitly or approximately by using various mathematical methods, and to analyze properties of the solutions. Note that it may happen that no solution exists for a given equation.

“So what?!” Victor proclaims. “If you prove that no solution to an equation exists, you solve the equation. You can make far-reaching conclusions from this observation.”

“So what?” is one of his favorite phrases and one he repeats frequently, along with this one: “Only problems yet to be solved are better than the solved ones.”

## “Good” and “bad” vibrations

A scientific field that has fascinated Victor since 2000 is spectral analysis, particularly the spectral theory of differential equations.

Spectral analysis focuses on the analysis of the spectrum, which, physically, is a collection of an object’s vibration frequencies. From a mathematics point of view, frequencies are the eigenvalues of a certain operator.

“Everything in the world is vibrating,” Victor explains.

For example, in music, the notes A and C are the purest vibrations, vibrating with one fixed frequency.

Most objects have many frequencies, and knowing the frequencies, their features and properties is essential. To emphasize the importance of these frequencies, Victor gives some practical examples:

You may have heard that marching soldiers are cautioned to break stride on a bridge. Do you know why? Because the frequency of their steps could match the bridge’s frequency of vibration, which could damage and even destroy the bridge.

In the aircraft industry and in mechanical engineering, special attention is paid to the selection of materials and structures whose frequencies would not match those of the environment when a plane or a vehicle is in use. Otherwise, serious problems may arise.

Frequency coalescence (or resonance) can be very dangerous. However, at times resonance could be beneficial. For example, in radio technology when you tune your radio to a station; and in medicine, in the magnetic resonance method.

Thus one must learn how to find eigenvalues (frequencies), Victor says:

It is very difficult to precisely calculate them; it is rarely achievable for very “good” simple objects, but we need to calculate them for every object, including “bad” ones. In order to do so, one may use a variety of methods yielding approximate estimations. All of the methods are based on the fact that a “bad” object (e.g., a very crooked acute-angled area, or an area with a very rough surface) could be replaced by a close “good” object (an area with no acute angles, or an area with smooth surface), and all of the calculations are performed for the selected “good” object. It is reasonable to ask in which cases this replacement is justified, and what is the difference between the estimated eigenvalues (frequencies) and their true values (keep in mind that there are cases in which such replacement is not justified). This is called the spectral stability problem.

## Research on the method of transition operators

Professor Burenkov and his colleague Professor Pier Domenico Lamberti of the University of Padova in Italy have developed a method called *method of transition operators*, which is used to solve spectral stability problems for a wide range of objects. The method allows one to assess deviation of eigenvalues (frequencies) of “bad” objects from those of “good” objects via geometric characteristics (which show to what extent “good” objects differ from “bad” ones).

Professor Burenkov and his colleagues have published many articles on the subject in leading mathematics journals. Here is a sample of his works published in Elsevier journals. We have made the articles freely available until December 31, 2016:

V.I. Burenkov, P. D. Lamberti: “Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators ,” *Journal of Differential Equations* (2007)

V.I. Burenkov and Pier Domenico Lamberti: “Spectral stability of the pp-Laplacian,” *Nonlinear Analysis: Theory, Methods & Applications* (September 2009)

V.I. Burenkov, E. B. Davies: “Spectral stability of the Neumann Laplacian,” *Journal of Differential Equations* (2002)

V.I. Burenkov, P. D. Lamberti: “Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators,” *Journal of Differential Equations* (2007)

## Can mathematics be commercialized?

Victor says commercialization is not something he focuses on, but he wishes it could play more of a role in theoretical mathematics and fundamental research in general.

The notion of intellectual property does not apply to theorems and formulae, which also cannot be patented and cannot be considered inventions or discoveries (which is totally unfair). So the intellectual property in mathematics is something vague and not widespread.

As I mentioned before, your mobile phone is based on mathematics (together, of course, with physics and technology). Ideally, it would help if mathematicians would be receiving at least a small portion of cellphone manufacturers’ profits. If this were the case, all mathematicians would be wealthy people, and they would not have to “beg” governments to provide funds for carrying out fundamental research in mathematics. Unfortunately, it is yet not clear how to organize this process.

## On Russian mathematicians

Victor speaks with pride about Russia’s leadership in the field:

Russian mathematics is still highly regarded in the world. Mathematicians from Russia are working in many universities around the world. Any university considers it a privilege to employ a mathematician from Russia. In some universities, there are several Russian mathematicians. This was the case in Cardiff University (in Wales), where I worked for 15 years. There was even a joke there: ‘You have a higher chance of being understood in Cardiff School of Mathematics if you speak Russian than if you speak Welsh.’

## On the philosophy and psychology of math

For Victor, philosophy and psychology play a key role in mathematics – and who will be good at it:

You know, mathematics is interesting. It is based on a specified set of axioms and an elegant logical system of reasoning. There are much fewer extraneous factors than in any other science. Mathematics is the “purest” object for philosophical and psychological study of the process of scientific knowledge.

I think that mathematics is a science for people who have more abstract thinking (they are more willing to do purely theoretical things just for fun).

As for his own work, he sums up success succinctly: “When you do something which is being valued by others and used in their work, this is what I call recognition.”

## Victor Ivanovich Burenkov

Dr. Victor Ivanovich Burenkov is a Doctor of Physical and Mathematical Sciences, an internationally recognized professor, Director of the SM Nikol'skii Institute of Mathematics and Head of the Department of Mathematical Analysis and Theory of Functions at the Peoples’ Friendship University of Russia (RUDN University).

In 1963, he graduated from the Moscow Institute of Physics and Technology. He has been working at RUDN University since 1981. From 1994 to 2014, he worked abroad as mathematics professor in Great Britain (Cardiff University), Italy (Padova University) and Kazakhstan (LN Gumilyov Eurasian National University). Starting from 2014, he has been a professor at RUDN University.

Professor Burenkov is an Honorary Professor at Cardiff University in Wales, Padova University in Italy, LN Gumilyov Eurasian National University in Kazakhstan, Russian-Armenian (Slavonic) University in Armenia, and Aktobe Regional State University in Kazakhstan.

From 2003 to 2013, Professor Burenkov served as the Vice-President of the International Society for Analysis, its Applications and Calculation Computation (ISAAC).

He is Chief Editor of the *Eurasian Mathematical Journal* together with the Rector of the MV Lomonosov Moscow State University and RAS academician V.A. Sadovnichiy and Kazakhstan’s NAS academician M. Otelbayev.

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