How do math problems arise

New concepts of space in mathematics

What is theoretical mathematics good for? Little is heard in public about the results and successes of mathematics. This sometimes creates the absurd impression that there is little development in mathematics, that everything essential is already known, or that perhaps all that matters is to solve some long-known problems (such as Fermat's famous problem, the recently resolved). Paradoxically, one reason why so little is made public is precisely the rapid progress and great success of mathematical methods. Joachim Cuntz from the Mathematical Institute explains what it consists of.

The mathematical techniques are so sophisticated that it is extremely difficult to even begin to understand them to a non-specialist. For similar reasons, the question of the usefulness of mathematics and especially of "pure" mathematics is difficult to answer. There is therefore a lack of clarity about the role of modern mathematics even in the informed public - and this at a time when all areas of science and everyday life are becoming more and more inexorably pervaded by methods and ways of thinking that come from mathematics. Let us therefore first devote a few sections to an attempt to explain the nature and goals of modern mathematics.

The peculiar way of doing mathematics is to do thought experiments. The starting point are abstractions of operations that can actually be carried out in reality. This is a human ability to which we have become so used that we hardly notice it anymore. A child who wants to build a tower from toy blocks has to learn how to place the blocks so that the tower does not collapse immediately. After some practice, this can be anticipated in the imagination without actually moving the stones. At a slightly higher (or sometimes lower) level, this technique is the basis for most of our decisions. We play through the consequences of an action in our minds and then decide on those whose results are likely to come closest to our wishes. The concept of number is also an abstraction from such mental operations.

The experimental laboratory of the mind

The most interesting aspects of mathematics, however, are those that transcend these "everyday" applications and investigate operations that in reality can only be carried out with difficulty or in principle not at all. To give just one example, the infinite is consistently built into the methods of mathematics in the most varied of forms. Areas can be examined in the mind that are inaccessible to daily experience.

The Platonic Reality of Mathematics and Spiritual Adventure: The mathematician examines a Platonic reality that exists outside of himself. We do not want to make any philosophical claims here, but only to reflect an experience that every active mathematician has. Indeed, the world of mathematical objects and structures shows many characteristics of objective reality (repeatability, verifiability). The mathematician explores this world just as a naturalist or geographer explores the real world. The only arbitrariness consists in the names he gives to the objects he finds. A proof is a path in this landscape from the area that we know to the desired goal (issue). On the way there we get to know other parts of the area and will soon find shorter and more convenient ways or new ways to results that we already knew. The development does not only take place in small individual steps, but at important points in global visions, which mathematicians of all times from Archimedes to Galois to Poincaré have reported (as well as the great philosophers Rousseau, Descartes, Pascal ...).

Human possibilities are notoriously limited. The certainty of the mathematical results, however, makes it possible to add up the contributions of countless individual scientists and combine them into a large puzzle. Because nothing is lost in mathematics - what has been recognized as correct remains correct. Over time, therefore, a building of unheard-of complexity has emerged. What are the relationships with objective reality and what are the practical benefits? The applications of mathematics are by no means just a useful side effect. To stay with the picture of the researcher of mathematical reality described above: There could be a risk that this researcher only stays in his ivory tower in order to examine it more and more closely. In order not to wither, mathematics needs the fresh air of new, unexplored areas. It lives from examining new structures in constant contact with objective reality, which then find their place in the large building. Relationships with existing theories then arise almost automatically. This classification makes methods directly available for dealing with the new phenomena. On the other hand, the new structures can complement the already existing ones in often very unforeseen ways and thus revolutionize entire areas and lead to new applications (including practical ones) in completely different places. For example, ideas that come from theoretical physics can lead to applications in number theory or vice versa.

Mathematics overcomes the physical limits of our existence

For centuries, mathematics has developed in a fruitful symbiosis with physics. In many places it is not clear whether a special result is to be assigned to mathematics or theoretical physics. Often physical discoveries were anticipated through purely mathematical considerations. Well-known examples are the electromagnetic waves or the general theory of relativity. Theories such as theoretical mechanics, electrodynamics, general relativity, quantum mechanics or chromodynamics are basically completely mathematical theories, which incidentally are just as difficult to make understandable for the layman as are all mathematical structures. Many physical facts can only be adequately formulated in very sophisticated mathematical language. When physicists describe these structures in popular science presentations, they are generally working with rather vague analogies. We shall try below to explain a new part of modern mathematics in a similar way with the help of analogies.

What does the “subspace” of Star Trek look like?

It is part of the essence of mathematics to recognize functions once they have been understood. This leads to the universality of mathematics and the applicability of the methods developed by it to the most diverse mechanisms and processes in our environment. Recently, mathematical methods have also found their way into many other individual sciences. Examples of applications of newer mathematical methods (which already existed before these applications) are somewhat random: computed tomography, Fourier analysis and decomposition into wave packets ("wavelets") in oil drilling, brain wave measurements or echo sounders, applications of game theory in economics, logic and algebra in formal linguistics, statistical methods, image processing, "fuzzy" logic, search methods, coding and encryption. Simpler applications of mathematical terms have become so commonplace that you don't even notice them anymore.

Conversely, the problems from these areas also provide impetus for new mathematical developments. For this reason, too, the development of mathematics is faster today than ever before. An important goal of basic mathematics is to be able to master and understand extremely complex structures and relationships. Most of the time, the methods developed are used in completely different areas after a short time. Because in mathematics it almost never depends on the individual problem, but rather on the methods that enable its solution. All the different parts of mathematics are interrelated and mutually beneficial. Even if someone wanted to take the position that only mathematics should be practiced that can be directly applied, for example, to technical or engineering problems, he would soon have to recognize that for these purposes too, concepts and results are astonishingly quick from other areas of mathematics (such as geometry or topology or the theory of analytical functions) are of decisive relevance.

Spaces whose dimension is not an integer

Finally, a few words about the role of the computer in modern mathematics. First of all, it plays the role that it also plays in other individual sciences. It is used in mathematics to simulate complex dynamic systems, to convert mathematical knowledge into practical calculations and to test hypotheses. In addition, however, the investigation and consideration of its functioning also gives rise to new mathematical investigations and structures which, just like those that come from the other sciences, find their place in the building of mathematics and give new impulses to the rest of mathematics. In basic research, however, the computer does not play the central role that the public often suspects (and cannot play it for reasons of principle). How big the misunderstandings are here is illustrated by the following episode: In an interview, the former French President Giscard d'Estaing cited the solution to Fermat's problem as an example of the success of the ever increasing performance of computers. In reality, this solution relies on extremely refined and complicated theoretical methods that have only recently been developed. The computer's only contribution was word processing while typing the article. Mathematics will always remain man's tools to understand and structure his environment and also to deal with the increasingly complex mechanisms that are emerging in technology and society like computers, for example, can still be mastered and processed.

We now come to the more specific content of this article, namely the presentation of new mathematical ideas that form the basis of the research of our working group at the Mathematical Institute of the University of Heidelberg. Here, too, we want to go back a little further and trace the origins of these ideas. John von Neumann (born 1903 in Budapest, after a short period in Hamburg and Berlin professor at Princeton) was one of the most important mathematicians of this century, who was involved in almost all areas of mathematics, from the basics of mathematics and logic to functional analysis, contributed important ideas and results. He is the founder of game theory and developed the concept of the computer used today; he has analyzed the theoretical foundation of quantum mechanics and the measurement process in quantum mechanics; Incidentally, he also belonged to the group of scientists who worked on the development of the atomic bomb. In a long-term collaboration with Francis Murray, in which he invested an extraordinary amount of energy (two out of six volumes in his collected works are dedicated to this area), he examined rooms whose dimensions are not necessarily an integer (1, 2, 3, ... .), but any positive number (such as 5.761). That is, in such a room there are, so to speak, 5,761 different "directions". This is not to be confused with the non-integer “metric” dimension of fractal spaces, which is a much less deep-seated phenomenon. Von Neumann came across these spaces in connection with his investigation of the phenomenon observed in quantum mechanics that the result of several successive measurements may depend on the order of the measurements (so you may get different results if you first look at the position and then the Momentum or measuring first the momentum and then the position of a particle). The two measuring processes are not interchangeable or not commutative, as one says in mathematics. Value ranges of non-commutative quantities are typically discrete or fractal. What does the room look like on a microscopic scale? We don't know - but there are many indications that the structure of space on this level cannot be described with the continuous geometry we know. For example, there could be such a thing as a smallest length, that is, the distance between two “points” should be at least this length. Nevertheless, objects in this room should still be able to move continuously. The concept of a point itself would then have to lose its meaning. For mathematicians, all of this is already a reality. Such spaces exist in mathematics and can be examined and described. They are based on a far-reaching further development of the Neumann ideas mentioned above and can have non-integer dimensions. If you multiply “functions” on these spaces, the result depends on the order of the factors; they are therefore also called non-commutative spaces. Such spaces can, for example, be shifted or rotated in directions that do not exist in our usual Euclidean space and then show completely new structures or relationships between sub-areas (this is how one might imagine the “subspace” in Star Trek). In keeping with the essence of mathematics, these objects, which are described here as flowery and apparently speculative, are examined with logical accuracy, and the statements about them are of course fully demonstrable.

Describing the geometric shape of such structures is a problem that has only recently been addressed, as the necessary methods have only recently been developed (with the very active participation of Heidelberg scientists). It is an idea of ​​modern differential geometry and algebraic topology that geometric properties such as the curvature or the global shape of a geometric object can be described by numbers (or similar quantities), so-called "invariants". Such invariants must have the property that they do not change if the geometric object is displaced or deformed without changing its shape. (To get an idea of ​​what is meant here by shape in the simplest case, imagine the surface of a car tire and the surface of a ball. Although both are two-dimensional closed surfaces, their shape is fundamentally different.) If two objects are different Numbers are assigned as invariants, so they also have different shapes. In this way, for example, one can distinguish the clover leaf loop from its mirror image. In general it is not easy to find such invariants.

How to tell the clover leaf loop from its mirror image

It has recently been possible to find invariants which can be constructed in a very surprising way for the very general kind of non-commutative spaces described above. Our Heidelberg working group has made a significant contribution to this development. The new methods are based on so-called homology and cohomology theories. In these theories, roughly speaking, a complicated structure is broken down into simple individual pieces and a systematization of the description of the relationships between the individual pieces is used. This systematization takes place algebraically, so that geometric facts are translated into algebraic formulas. A result obtained in Heidelberg makes it possible to actually calculate the invariants with the help of special decompositions and also to compare the various known cohomology theories with one another. The fascinating thing about the theories for the non-commutative case is that they use methods from almost all areas of mathematics and, on the other hand, can also be applied to problems in many different areas. As expected, the new ideas also led to a new and better understanding of the terms previously used for "normal" geometric objects. The idea is that classical geometric structures can be embedded in non-commutative spaces. The homology and cohomology invariants can be obtained by examining how the given objects move and deform in this non-commutative environment.

Is it all just a math gimmick? The non-commutative structures are examined because, as described above, they were on our way in Platonic reality. But: Your investigation has of course, again as described above, resulted in completely new applications in the most varied areas of mathematics. There are also applications to very "everyday" problems. So the invariant that distinguishes the clover leaf loop from its mirror image, or more generally, different knots from one another, was found precisely in this way.This, in turn, was useful in molecular biology to classify the knots of DNA molecules. It is also hoped that the non-commutative spaces could indeed provide a suitable framework for describing the geometry of our world on a microscopic scale. A number of outstanding mathematicians and physicists are working on this program.

Two Fields Medals have been awarded in recent years for work in the new field of non-commutative geometry - one to the French mathematician Alain Connes, who co-founded the field and shaped the field through his contributions, and the other to the New Zealand mathematician Vaughan Jones, who is responsible for relations with the Has discovered classification of knots, quantum symmetries and statistical mechanics.

In Heidelberg, the techniques were developed that allow the above-mentioned invariants for non-commutative spaces to be defined and calculated. Research in this area has been funded by the DFG within the research group “Topology and Non-Commutative Geometry” over the past 6 years.

Author:
Prof. Dr. Joachim Cuntz, Mathematical Institute, Im Neuenheimer Feld 288, 69120 Heidelberg,
Telephone (06221) 54 56 92