Ideas do not have any birthday. A good idea never dies.
Simple ideas may fit into anybody’s brain and be realized by a single person. A complex idea needs several brilliant minds to be hosted properly and people of good will to be put forward.
Poor ideas are easily shared. A rich idea demands deep efforts to be communicated from one person to another.
Herewith a virtual shelter is offered to this idea. This website is the reunion point for all the human and material resources cooperating in keeping alive and breeding this idea.
As a 0-th order approximation it might be said that:
“The idea of Diffiety School is to continuously produce experts in a new area of Mathematics called Diffeotopy, who can cooperate in overcoming the long-resisting difficulties aroused by the crisis of the traditional approaches in facing problems linked with non-linear PDEs and the precise mathematical description of quantum phenomena.”
Subsequent higher-order comprehension of this idea takes much more time and effort. And it can go in different directions, depending on people’s attitudes, interests and capabilities. There are several ways and different levels for becoming an active part of Diffiety School:
The crisis of the traditional approaches in facing problems linked with the non-linear PDEs and the precise mathematical description of quantum phenomena, produces this historically unique situation in which mathematics has to win a challenge whose dimensions fairly exceed the capabilities of any single mind, even a genial one. Therefore to overcome the difficulties that have been aroused and that still resisted for a long time, a coherent and well programmed work of a team of mathematicians trained to face the challenges of diffeotopy is needed. And the aim of Diffiety School is the continuos productions of experts in this area.
Napoleon used to say that there cannot be a great nation without great mathematics. He was right. Following this idea, he brought prosperity and glory to France by founding the famous Ecole Normale Supérieure and other Universities of great renown in which teaching mathematics was held at unmatched levels for that age. Thanks to this culture of mathematics, France since then until now has always been a great power in the world. For it has the capability of building spaceships to explore the cosmos. Nuclear industry is fairly developed there, much as everything which requires both advanced mathematics and technologies based on it. On the contrary, the southern neighbourhood of France, Spain, basically the same concerning human and natural resources, does not build airplanes neither goes out to cosmos, and does not produce higher technologies. Even though Spain appears among the seven countries considered to be the most industrially advanced, she is just a mere province of Europe, incapable of affecting what happens in the rest of the world. Mostly, this is due to the fact that Spain is scarcely inspired by mathematics. Countries belonging to the so-called Third World, like India, China, Brazil, that steadily and dynamically make their way to the leadership of world development, not only already posses the mathematics needed for such an effort, but put adequate resources to its improvement. This is brightly confirmed by mathematical international olympiads, where the young Chinese mathematicians constantly beat the European and American ones leaving behind the mathematical teams from old-culture countries like Germany, Italy, etc. Also one may notice that China itself, and India and Brazil too now are capable of producing airplanes, fly to cosmos, and do many other things far beyond Spain’s and Italy’s powers.
Health of mathematics is an indicator of a nation’s scientific and technological state, and of its own development possibilities. Looking back to history, it is easy to understand why mathematics - a weird thing poorly comprehensible for the wide public, seldom mentioned by mass-media - indeed is the chief and most powerful impulse to any country’s development. Recalling Archimedes - legend has that he managed to set Roman ships on fire by using parabolic shields - it is obvious to conclude that he was capable of such a trick only because he was the greatest mathematic of his time and knew about geometrical properties of parabola. He also sank Roman ships by using mechanical devices for launching heavy projectiles, some of which he built and operated by himself, relying on the mathematical equations describing them. Such equations were already known at the time, or were discovered by Archimedes. This is one of the first times in history when mathematics took part to a weaponed conflict and turned out to be the best army of that period.
Napoleon had huge respect for mathematics for he was an artillery officer. He knew well that not only theory was the God of the war in that time, but also that the most effective schemes for fire management can be obtained exclusively on the base of mathematical computations. To such examples thousands more might be easily added.
Laws of Nature are written in a mathematical language. For this very reason, any kind of exact knowledge is mathematical in its essence. The dawn of the new era began with Newton, who found the mathematical formalization of the chief equations of mechanics and on this basis he managed to explain mathematically why all massive bodies and in particular the formerly divine planets move precisely accordingly to such laws. To achieve all of this, he was forced to create a new mathematical language, the differential calculus, to deduce and then to solve the first instances of differential equations in celestial mechanics. It was an epochal achievement of the human thought not only because he profoundly changed our perception of the world but also because it made evident how far human mind can reach if reinforced by mathematics. Figuring out how planets fly being incapable of touching them from such a big distance, is undoubtedly not a trifle.
We live in the electromagnetic era: we listen to the radio, we watch TV, we use mobile phones and we spend even more time in Internet. Inside of all our house devices there his one kind or another of electric component. Humankind probes the deepest places of the Universe by using electromagnetic waves from all the frequency spectrum. But few know that the electromagnetic waves were discovered mathematically. This was accomplished by James Clerk Maxwell whom tradition holds as a physicist, but he was much of a mathematician as he was of a physicist. Indeed he was able to collect numerous empirical facts regarding electricity and magnetism due to his predecessors Coulomb, Oersted, Ohm, Faraday and others, developing for this purpose that mathematical theory which nowadays goes like “tensorial analysis”. Only thanks to this mathematics Maxwell succeeded to write down the differential equations bearing his name that exhaustively describe all electromagnetic phenomena. Then studying equations he discovered on his sheet of paper the same electromagnetic waves that are the foundations of the modern civilization. Only a long lime later the existence of the electromagnetic waves was confirmed by experiments, after which it began the epoch of their practical applications, that have reached impressive proportions in the present.
Another mathematician, Nicolai Zhukovsky discovered more than one hundred years ago, the law for the lift force acting on wings just by finding necessary solutions to the differential equations that describe the behaviour of the gaseous media. From that moment, the study of the equations of hydro- and gas-dynamics became essential for naval and aeronautical industries. In such a business the job of the mathematical engineers is the most important. And Zhukovsky was for many years the president of the mathematical society of Moscow.
People do not know that the utilization of mathematics by almost all the manufactured devices we daily use has huge proportions. The more technologically advanced the devices, the richer is the mathematics driving them. Mathematics is hidden in all these products in an immaterial way, so that his presence cannot strike the superficial eyes. Mathematics is indispensable for the project of any product, for computing its parameters and properties, and for programming the industrial machines dedicated to its production. To build up a system of industrial machines a lot of mathematics is needed and often such a system cannot even be projected without having at hand a mathematical idea. To reach another planet without missing the target it is necessary to compute the flight trajectory, perform the right manoeuvre at the right time, and so on so forth. This is just an example of a non-trivial problem for Mathematics which can be solved by the methods of the control theory based on the theory of differential equations.
Only the modern differential geometry can tell us about what is going on inside the black holes. Direct inspections are impossible since not even a ray of light can escape the black hole. If a medical doctor needs to figure out what is inside a sick man without cutting him to pieces, this is possible thanks to the tomography machine, a device that can compute what resides inside a body or object. And this is possible thanks to a mathematical theory called “inverse Radon transformation”.
Also simple geometrical constructions and algebraic computations are involved in the productions of almost all objects of everyday life. It is impossible to build up an house without simple mathematical computations that become even more complicated when one needs to create a bridge, a tower, a ship, and aircraft, a submarine, and so on. When we see on our PC screen virtual reality and other similar miracles we must take into account that all of this is based on one kind or another of mathematics and in particular on “projective geometry”. Therefore Mathematics is present everywhere in the modern civilization and the efficiency of the industry of scientific research is even more based on the mathematical background of the engineers and scientists working in any scientific or technological environment. For this very reason without advanced mathematics and well organized mathematical education it is impossible to stay on the top of the scientific and technological progress. Arab Emirates is now a very rich country tough it is not very extended. When all the oil resources from its ground will be burned away, it will be just a dead sandy land without mathematics, if those who rule do not make any present effort to this end.
Need for mathematics in our dynamical world is doomed to raise ever after. For before developing any noteworthy large enough engineering project-be it social economic or scientific-it is necessary to create a mathematical model for it first. In many cases this is the only way to proceed. Shortage of natural and economical resources puts even more tough obstacles against the direct experimentations and therefore the theoretical modelling is the unique path for the progress. Therefore the closest future of humankind lays in the mathematical industry. A great challenge awaits for 21st century Mathematics so that it is natural to ask whether contemporary mathematics is ready for it.
Mathematics of the past century, unlike that of the 800’s and 900’s, was mostly employed to solve its internal problems. That was physiological since Mathematics had to face-like any other living being-those current problems due to its own growing up. As an obvious consequence, the links of this Mathematics-focused to solve its internal issues-with the totality of all natural and concrete sciences, economical and social, were remarkably weakened. Expectedly inside of the mathematical community several scholastic sects were formed-they were born and died and their only effect was to slow down the advancing of mathematics. This change of interests in the mathematical society lead to a situation in which the most important sectors of mathematics-for the comprehension of the new discoveries in Physics, Chemistry and other natural sciences and for the construction of mathematical models of complex social and economical phenomena-are not suitably perfected and developed. In other words, modern Mathematics has got no ability to face the fundamental problems of our days at all. For example, the difficulties of understanding the nature of the elementary particles is due to large extent to the lack of the adapted mathematics. Actually this problem concerns everything that goes under the name of “quantum physics”. As a result, humankind is forced to waste unbelievable amounts of resources to build up experimental facilities like for instance the modern accelerators for elementary particles. We waste billions to cover the damages caused by hurricanes and many other natural cataclysms-economic and ecological catastrophes. To large extent all of this could be avoided if we were able to create and to interpret the mathematical models for all of these phenomena. It would be thousands time cheaper if the necessary resources were devoted to the development of the corresponding areas of the fundamental mathematics.
Mathematical modelling of the most important natural phenomena, of the complex social phenomena and of the technological processes, and the control of complex systems, etc., is based on the non linear PDEs. For instance the entire set of knowledge of electromagnetic phenomena is encoded into the Maxwell differential equations and therefore anything else which is due to such phenomena can be derived from Maxwell equations in a purely mathematical way. This is far from easy. Moreover in the most common cases the modern mathematics is incapable to do that in real time. If we want to understand the nature of hurricanes and on the basis of such a knowledge to formulate forecasts, it is necessary to solve some mathematical problems concerning the differential equations that describe the motions of the masses of air. Solutions to some mathematical problems concerning the differential equations of magnetohydrodynamics would lead us to remarkable progresses towards the realization of the controlled thermonuclear fusion, freeing once and for all the humankind from the menace of the energy shortage. And many cases like this. Regardless to this, 20th century mathematicians busied themselves with studies in all branches but non linear PDEs. Moreover, it has been said that an unified theory for such equations could not even be formulated. Actual state of things is not so desperate though, as it seemed for a long time, and the first foundations for this theory have already been set. It has been found that this theory is extremely complicated and that in it almost all the branches of the modern mathematics are joined in perfect harmony, many of which nobody ever believed to be related with differential equations. This germs of the new theory need to be developed in several directions and many details must be perfected in order to lead it to effective applications. To reach this goal huge human resources are needed. Therefore is of a strategic importance the formation of experts in this new area of mathematics. Diffiety School is the first step in this direction.
In modern mathematics a sector called the Theory of Algebraic Equations does not exist. Yet algebraic equations are studied by Algebraic Geometry, which constitutes the modern form of the theory of algebraic equations. This is the outcome of a fairly long evolutionary process showing that the right object to study are not the sets of concrete equations but the ideals generated by them and the corresponding algebraic varieties. In other words the concept of an algebraic variety—more exactly the spectrum of a commutative algebra—is the conceptually right answer to the question of what a system of algebraic equations is. Thinking about Groetendieck’s schemes it is easy to understand that the answer to this and many other similar questions are not so obvious and naive as one may expect at first sight.
Taking this into account, it is natural to expect that the conceptually right answer to the question of what a system of non-linear PDEs is will be even less obvious than the previous question. And so it is. In particular this is due to the fact that when building the ideal corresponding to a set of differential relations one must take into account not only the algebraic consequences of such relations but also their differential consequences. It is of fundamental importance the possibility of working with all the consequences—both algebraic and differential—of the original relations, also from the viewpoint of the logic completeness of the theoretical scheme. Following this path, it is possible to discover the concept of a diffiety, a new geometrical object playing the same role in the modern theory of PDEs as the algebraic varieties play for algebraic equations. By their very nature diffieties are carrier of many geometrical structures, the most important of which is the infinite-order contact structure—the so-called Cartan distribution—containing all the information about the system of original PDEs. Form the viewpoint of the geometrical theory of non-linear PDEs the study of a concrete system of such equations is nothing but the study of the geometrical structures on the corresponding diffiety. This way, the theory of non-linear PDEs takes the rigorous and concrete form of the study of general properties of diffieties and natural operations on them.
The history of Babylon’s tower told in the Bible explains the fate of the greatest scientific project of that time which failed after a direct divine intervention. More precisely, when God realized that the construction of the tower was too much advanced, He mixed the languages of the builders and consequently the work could not be completed. But such a story has not ended yet. Indeed some stubborn people who was curious about the structure of the Universe invented a new common language which now goes under the name of “Mathematics”. Such a language helped us a lot in understanding many details of the divine creation. But meanwhile God was not idle and now an then He applied the old trick of mixing languages within Mathematics itself. Such a process can be explained more scientifically by invoking the laws of Thermodynamics, namely the second one, which inevitably affects the scientific progress in general and shows itself in the most bright way in the mathematical branches of Natural Sciences. The modern mixed state of the mathematical language and the deriving form it chaos is perfectly shown by the current A.M.S subject classification. Not to speak of the almost total impotence before the non-linear problems and the screaming mathematical ugliness of the current theory of quanta.
Diffeotopy is a new area of Mathematics generated by the attempts to find some path in the multitude of thousands concrete facts growing around the non-linear PDEs and in all those areas of Mathematics itself, Mechanics, Physics, etc., where they are applied and are indispensable. Diffeotopy began to take a concrete shape 20-25 years ago, and now it has its own precise methodology, paradigm and philosophy. In a schematic way diffeotopy can be defined as the study of the natural structures of Differential Calculus - primary and secondary ones - and their applications to problems of Mathematics itself and mathematical models from Natural Sciences. In this sense Differential Geometry and Topology, Mechanics of Continua Media, Field Theory, Control Theory, etc., constitute objects for diffeotopy - obviously not to speak of the non-linear PDEs themselves. Undoubtedly such a list is not complete and as time goes by it will be enlarged. For instance, it can be sharply foreseen that in due time Algebraic Geometry will take part of it - but understood as Differential Geometry over algebras of algebraic functions. And this will happen only after sheaves, schemes and other palliative means have been selected for extinction by the natural Darwinian process for concepts.
From a materialistic viewpoint diffeotopy represents a natural synthesis of many sectors of modern Mathematics among which one can find PDEs themselves, Commutative and Homological Algebra, Algebraic and Differential Topology, Differential Geometry, Lie Group and Algebras, and other things. Therefore the pathway to diffeotopy is not easy. But on the other hand a steady and safe progress in the key areas of Natural Sciences is not possible without a suitable development of diffeotopy.