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A Beautiful Mind

August 13, 2008

The most important school to me was the Ludwik Zamenhoff High School, No. 31, in ŁódĽ. Back in those days, I was interested in physics much more than mathematics. I had a great physics teacher. Physics was her true passion, and you could talk to her at any time of day and night. Looking back, I can see she was very patient with me.

But my interest in physics began with my conversations with a cousin who studied electronics at the Wrocław University of Technology. He told me about lectures by Richard Feynman. The Feynman lectures were one of the first books I ever bought for my own money. I was 12 when my aunt, who was a chemical engineer, gave me a scientific book about the theory of relativity. I can still remember how to derive formulas for the shortening of a moving body or the alteration of body weight in motion. It was all written in a very straightforward language.

I also understood that you should not get too attached to what you see, but search for simple assumptions that you can use to explain phenomena you cannot understand. One such general assumption, the axiom of the theory of relativity, is the constant speed of light, measured in relation to any system of coordinates. This is essentially a very mathematical kind of reasoning where you take a system of axioms and see what comes out of it. That was how physics led me to mathematics. There was a lot of chance in it. There was a good teacher too, and my own curiosity about the world.

As a result, I enrolled at the ŁódĽ University of Technology to study at the Department of Fundamental Technology Problems, which was later converted into the Department of Applied Mathematics. My major was in mathematical statistics.

The economy of thinking. Theorems are true and succinct messages. The esthetic aspect plays a considerable role because a good theorem is a beautiful one too. A mathematical proof is a sequence of real implications. A beautiful proof is surprising and usually short. On the other hand, mathematical terms are frequently interpretable in the language of the real world, hence the enormous variety of applications of mathematics. Nowadays, practically all sciences are undergoing rapid "mathematization," which is even true of medicine, as exemplified by epidemiology, genetics and so on.

Math is not difficult as long as you like to think about abstract terms with surgeon-like precision. The problem is that few people do. Colloquial language is not very precise. Sentences usually lack any logical value, and the truth in them depends on the context. Common speech is more about communicating emotions, maintaining relations, demonstrating one's position in a group, and so on. Mathematics is completely devoid of that; it also lacks the time element.

You could think the following: by the time the dog reaches the place where the rabbit was a moment ago, the rabbit will have run to a different place. When the dog reaches that new place, the rabbit will run another positive distance away and so on, ad infinitum. This way, the dog will never catch up with the rabbit! The mistake in this reasoning is an assumption than an infinite number of repetitions of one operation takes an infinite amount of time and that it all has something to do with the chase.

Meanwhile, mathematical operations are beyond time. The sum of a sequence consisting of 100, 50, 25 meters and so on is 200 meters, regardless of how fast I can prove it and if I can prove it at all. In order to catch the rabbit, then, the dog needs to run 200 meters, which takes it 10 seconds.

Mathematics is beyond time and space. It is constant, mysterious and majestic. It is good for contemplation and it teaches humility. Humanity needed several centuries to prove what is known as Fermat's Last Theorem. Why is that? Because we are pretty frail in our minds. Apparently, people use only a few percent of their brain capacity in the thinking process.

In a way, mathematics is a science of the structure of the obvious. The proposition of a theorem is contained, or should I say, hidden in its assumptions. What we have is information overload, and it takes a lot of effort to discard what is redundant. The most pleasant thing in practicing mathematics is the moment when, after a long concentration, you suddenly see everything in the right order and discover the truth.

The subject of my Ph.D. dissertation concerned mathematical inequalities, that is the maximum amount of information you could squeeze out of data in order to estimate a parameter in a model. And I proved that "minimax" inequality of information. The proof was rather complicated, but the inequality was simple and, in my opinion, beautiful.

In 1988, Larry D. Brown, a member of the American Academy of Sciences who currently works at the Wharton School of the University of Pennsylvania [in Philadelphia, PA, USA], mentioned my work in his annual Abraham Wald Memorial Lecture at the International Statistical Institute. He devoted a part of his lecture to another proof of my inequality, which I felt enormously honored by. Later, we wrote a paper on information inequalities together.

It concerned the properties of a certain type of nonparametric estimators of probability distribution density. While doing my research, I also managed to solve two problems identified in research reports. One, raised by Vladimir Vapnik, concerned the question of whether or not estimators of the shortest distance could be effective. R.-D. Reiss had given affirmative answers to this question for some special situations, while I managed to expand that to a very general situation. The other question, raised by Luc Devroy, dealt with the possibility of obtaining, in a space with infinitely many dimensions, the same rate of convergence of the density estimator which was possible in a space with a finite number of dimensions. The answer was yes this time as well.

After that I spent several months on a Krupp Foundation scholarship at the University of Dortmund, Germany, and I also obtained a scholarship for a shorter period of time from the American National Academy of Science. I spent the latter period at the University of Maryland in Baltimore County. For over a decade, I have been going on one- or two-week visits to research centers in various European countries and the United States.

I became a professor in 1999. Apart from the schools I have mentioned, I work intensely with the Illinois State University. I have written many papers with Prof. Krzysztof Ostaszewski, who heads the university's actuarial program. We also wrote two monographs on pension plans, one for the Wydawnictwo Naukowo-Techniczne publishers in Poland and the other one, entitled Financial Risk Management for Pension Plans, for the international publishers Elsevier.

I am a member of the Polish Mathematical Society. For many years I was also a member of the American Mathematical Society, but I have given up my membership, because over the past 10 years I have been more preoccupied with the applications of mathematics in economics and insurance.

Mathematical modeling of social transfers. I am the grumpy perfectionist type, and I correct my papers repeatedly before I submit them for print. I would first of all like to have more time to write papers. I am interested in problems such as the transfer of wealth, the rate of corporate development, risk transfers, and the impact of risk on social and corporate development. Then there is modeling based on empirical data, which means statistics in the general sense of the word.

What I have been doing lately can be used to reduce the cost of reinsurance contracts. It can also help improve the management of interest rate risk in pension plans and life insurance companies. One can also use this research to improve the financial efficiency of insurance companies and the supervision of pension funds and insurance companies.

[The astronomer Nicolaus] Copernicus above all. He built a beautiful yet simple model that explained the complicated movements of celestial bodies and put it through empirical verification. He was the first real scientist in the contemporary meaning of the word. At the same time, he had to stand up to almost the entire academic community of the day who taught the theory of Ptolemy. Remember that classical theories were highly esteemed in those times. Copernicus had the academic community against him, and later also the Church, which became involved in the scientific dispute. Today, naturally, we know that planets do not move in perfect circles and that the Copernican model was highly imprecise. Still, its strength lay in its brilliant simplicity and the anticipation of Newton's law of universal gravitation, which determines that it is the Earth that revolves around the Sun-which is thousands of times heavier than the Earth-and not the other way round.

As for more contemporary times, I admire Marie Curie-Skłodowska the most-more than Einstein. That petite woman literally shoveled tons of radioactive ore by herself in order to find the truth in science. She had to overcome not only the resistance of matter, but also social prejudice. She was the first and, as far as I know, the only woman to ever receive the Nobel Prize twice. It all happened at a time when women were not supposed to deal with science. In fact, the common belief was they should never go to college.

I received the prize together with Prof. D. Zagrodny for a paper we published in 2004 in the Journal of Risk and Insurance in the United States. The paper was "adjudged the best contribution to the actuarial literature published in 2004," according to the citation for the prize.

To cut a long story short, the paper concerns an optimal transfer of risk from the insurer to the reinsurer, and we showed that this problem is beautifully equivalent to the problem of testing the accumulated damage distribution hypothesis. We also obtained an overt form of the optimal reinsurance arrangement.

Poland never really lagged far behind the rest of Europe in the sciences, not even under communism. This obviously is not enough any longer because European science itself is not very strong these days. A well-thought-out scientific development policy is needed.

How does one devise a policy like that? It's not my job, but as an academic I probably have the right and duty to express my view. I believe that we first need to look at what we are good at now and decide what we want to be good at in the future. We should then determine what it takes to strengthen our position and be successful in these areas. Finally, we should adopt a development plan for several years, one that defines specific goals and staging posts and secures all the necessary means. This is precisely the part of the policy that the Polish government has been unable to come up with for well over a decade.

The other part is to put the plan into action through a system of competitions, national and European grants, and consistent work to carry out the consecutive stages of the project. We could carry out this part of the project pretty well, because Poland has a well-developed system of ministerial grants, which superseded grants previously available from the Scientific Research Committee.

The first part of the plan keeps failing for a number of reasons, mainly because of the ineptitude of the politicians who make all the decisions, but also because of the attitude of the scientific community, which does not entirely believe in the transparency of procedures in this area. In general, it seems that there is no other way than to have the government make all the strategic decisions in consultation with independent experts.

The main reason is the structure of our minds. We are not very clever. We have difficulty putting several simple implications together into a chain of logical reasoning. Besides, the education system is still haunted by the ghost of J.-P. Piaget. The structuralists did a lot of good to the formalization of psychology, but a lot of harm to school practice. Educating children through teaching them structures, from the whole to the detail, is the most unnatural manner one can think of. For thousands of years before the structuralists, people had no clue about this approach. They were learning things on the basis of their geometrical intuition and quantitative relations in the world they saw, and they communicated with one another using colloquial language. A formal language should be administered in doses, after the students develop an interest in fascinating problems that can be comprehended using colloquial language.

He is a renowned expert in pension plans. When he was a minister in Jerzy Buzek's government (1997-2001), he headed Poland's Social Insurance Institution (ZUS) for two years, reorganizing its operations. He enjoys a lot of respect among academics dealing with mathematics of finance, especially actuarial science and insurance. He has had his papers published in many Polish and international magazines and has co-written a book entitled Financial Risk Management of Pension Plans, together with Prof. Krzysztof Ostaszewski of Illinois State University. The book was published by Elsevier scientific publishers.

Gajek is Poland's first recipient of the David Garrick Halmstad Prize, an award that is given out annually by the U.S.-based Actuarial Foundation (TAF) for "significant contributions to actuarial science." Gajek received the award for a work entitled Reinsurance Arrangements Maximizing Insurer's Survival Probability that he has published together with Prof. Dariusz Zagrodny of the Cardinal Stefan Wyszyński University in Warsaw.

The award is granted annually by the Board of Trustees and the Actuarial Education and Research Fund (AERF) Committee of the Actuarial Foundation (TAF) for the best paper on actuarial research published each year. Few European scientists have received the prize since it was established 30 years ago.

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