Could we summarize the most difficult math problems

Michael Eisermann

Lately one has even read about the Poincaré conjecture in the press, from serious (“centuries-old mystery”) to sensational (“picture explains the millennium problem”). Often the sensation is conjured up but the actual message is only hinted at.

Another trigger in March 2010 was the lavish Millennium Prize. The following article was written in response to the question I was asked occasionally: What does the Poincaré conjecture say? I hope my (not very short) answer is useful for one or the other.

I am always grateful for.

Overview in key words

Non-math introduction:

Illustrative examples in two dimensions:

Generalization in three dimensions:

Links on the subject

Package insert: Who is this text intended for?

The following text contains a light dose of math and may cause dizziness or headaches. If in doubt, please ask your supervising mathematician. If there is no allergy, the article should be easy to understand. A bit of school mathematics is required (Cartesian coordinates and the Pythagorean theorem).

Anyone who wants to understand the Poincaré conjecture and shine with it at the next party also needs a little patience. An MP3 player or a bank card are unfortunately not easy to understand either, and so it is with many advances, including mathematical ones. Nevertheless, it is worth going on a journey of discovery and trying it!

Experience has shown that large parts of the sciences - and mathematics is no exception - can only be accessed through longer studies and turn away any fleeting visitor. Nevertheless, there are outstanding individual results that can at least be explained in general terms. This is what is to be attempted here.

In any case, I do not want to be satisfied with the - honest but resigned - summary that the assumption is "so complicated that it is not only incomprehensible to laypeople, but also exceeds the skills of many experts" (Stern, Focus, according to DPA). Before I talk about the content of the Poincaré Conjecture, however, I will begin with its eventful history and the fascination it exudes.

Perelman's spectacular proof of the Poincaré conjecture

The Poincaré conjecture was formulated in 1904 by the French mathematician and physicist Henri Poincaré and was considered one of the most difficult problems of geometry for a hundred years, more precisely the mathematical branch of topology (more on this in a moment).

In 2000, the Clay Mathematics Institute in Cambridge / Massachusetts selected the Poincaré conjecture as one of the seven most important unsolved problems in mathematics, for the solution of which it offered a price of one million US dollars.

The Poincaré conjecture has since been proven by the Russian mathematician Grigori Perelman. In 2006 he was awarded the Fields Medal of the International Mathematical Union for his revolutionary work. This is one of the highest awards in mathematics, comparable to the Nobel Prize in other disciplines. To the amazement of many colleagues, Perelman turned down the Fields Medal. He did not appear for the award ceremony at the International Congress of Mathematicians, which takes place every four years, and otherwise lives very withdrawn.

Perelman published his proof on the internet in 2002/2003 (arXiv) and presented it at a lecture series in the USA. He has refused the usual publication in a specialist journal. The evidence has been examined and worked out by several groups of experts in recent years and is now recognized. On March 18, 2010, Perelman was awarded the Millennium Prize of one million US dollars. It appears that Perelman will reject this award too.

Where does the media interest come from?

Technical progress is impossible without mathematics. Yet mathematical achievements rarely make the headlines. Mathematics shapes a growing number of technical applications in our everyday life, but mostly remains discreet behind the scenes. Basic mathematical research is hardly visible in everyday life, is considered difficult to convey and is therefore rarely discussed in the media.

So it happens that only the most spectacular individual results penetrate the public discussion. The fact that a prominent conjecture defied all attempted solutions for a century is remarkable, but not uncommon in mathematics. The fact that the lucky man who has finally solved a problem of the century rejects all honors and job offers is unique in this extreme, even in mathematics.

The unusually high prize money of one million dollars in particular contributed to the huge media coverage: This value is arbitrary but understandable for everyone. A good story can be made out of this. Much ado was therefore made about the external circumstances of the Poincaré Conjecture.

However, it was seldom clear from the media reports what the presumption actually meant. A hundred years of mathematical development is difficult to sum up in one sentence - and such an impossible task is the horror of every science journalist. (I don't even want to try that, but rather take the necessary time in the following and go a little further.)

The popular interest in Poincaré is roughly comparable to Fermat's conjecture, which was proven by Andrew Wiles in 1993. The latter has the media advantage that its statement can be explained in a few simple words, no matter how difficult the corresponding proof may be.

From a media perspective, the Poincaré conjecture is more cumbersome. It can be formulated in a few - but not very simple - words like this:

If a three-dimensional closed space is simply connected,
then this space is topologically identical with the three-dimensional sphere.

The terms used for this purpose will now be explained step by step.

What is the Poincaré Conjecture about?

In our everyday experience, we live in a three-dimensional world. If we are honest, we only know a small part of the universe: In our immediate surroundings, each point can be clearly represented by three coordinates, as we learned in school. Such a local map describes our surroundings - but not the entire universe.

Poincaré then asked himself: How can three-dimensional spaces look globally?

For the sake of simplicity, imagine you are an ant living on a spherical surface. At first glance, your world appears to be one great level. You can describe your immediate surroundings using two coordinates and thus create a map. This is a local description of your world.

The next thing you might ask is whether the area is limited or whether it extends into infinity. To find out, set off on an expedition that will bring you back to your starting point after circumnavigating the world - with the knowledge that the area on which you live is finally borderless.

It follows that you live on a sphere, doesn't it? Or maybe not: As we will see in a moment, there are also other surfaces that are just as finite and borderless, but differ significantly from the spherical surface. So you need even more information to be able to infer the spherical shape of your world. It is the same in three dimensions, and that is what the Poincaré Conjecture is about.

Like the fictional ant in two dimensions, Poincaré went on a journey of thought in three dimensions: What shape can three-dimensional spaces have? And what information do you need to be able to differentiate between these forms? The Poincaré conjecture characterizes the simplest three-dimensional space, the three-dimensional sphere. This assumption is outlined below.

Analogy in two dimensions: what are areas?

As an analogy, it is best to start in two dimensions, since two-dimensional spaces (i.e. surfaces) can be visualized more easily than three-dimensional spaces. Instead of an abstract definition, let's start with a few fundamental examples:

The surfaces include the two-dimensional sphere (i.e. the spherical surface), as well as the torus (the surface of a full ring), the double torus (with two holes), the surface of a pretzel (with three holes), and so on ...

Such objects are known as Surfaces. On a small scale, they all look the same: they can be described locally by two coordinates, which is why an area is also called one two-dimensional space.

Seen globally, however, these areas are very different. Topology, a branch of geometry, deals with such questions. First of all, it is a matter of recognizing such spaces as different or as the same.

How can you classify areas?

Things as different as a car tire and a coffee cup with a handle are considered topologically the same: if they were made of clay, one object could be shaped into the shape of the other without having to drill a hole or glue anything on. A ball or a pretzel, on the other hand, is topologically different from a car tire: the ball has no hole and the pretzel has three. Therefore, they cannot be transformed into one another by constant deformation.

In order to be able to tell the different types of surfaces apart, we need the right vocabulary.

An area is called closedwhen it is finite and borderless. (Instead of “finally”, topologists prefer to say “compact”). Sphere, torus, double torus, etc. are examples of closed surfaces. The Euclidean plane known from school has no edge, but is infinite. A circular disk is finite, but it has an edge.

An area is called orientablewhen it has two sides. The above examples are all orientable. There are also non-orientable surfaces, such as the popular Möbius strip. I do not want to go into these wondrous creatures here.

There are an infinite number of different areas, but luckily you can get a good overview. The classification of surfaces has been known since the beginning of the 20th century: Each closed, orientable surface corresponds exactly to one of the above models, with a certain number of holes. This is not quite as banal as it sounds: Can you identify the area shown on the left with one from our list? (How many holes does it have? One, two, three, four, ...?)

What exactly is a sphere?

Now that we've got an overview of areas, let's take a closer look at the spheres. Spheres are some of the simplest mathematical objects. They exist in every dimension and they can be described with a little bit of school mathematics: the one-dimensional sphere is nothing more than the circular line, the two-dimensional sphere is the spherical surface, and the three-dimensional sphere is ... well, we still want to find out !

In the plane we can identify each point by two coordinates(x, y) describe. The distance r this point to the origin can be calculated with the Pythagorean theorem, because it is true r² = x² + y². All points at a distance r = 1 to the origin form the unit circle. This therefore consists of all points (x, y) With x² + y² = 1. It is the one-dimensional edge of the two-dimensional circular disk. The latter consists of all points (x, y) With x² + y² ≤ 1 (marked in yellow in the sketch).

Why do we call the circle one-dimensional? Each point can be uniquely described locally by a coordinate, for example the angle. Our description by two coordinates (x, y) is created by embedding the circular line in the plane. However, only one of the remaining coordinates is on the circular line x, y freely movable, the other is calculated from it: for example we can around the point (x, y) = (0.1) around x move freely and this is calculated y = √ (1-x²).

Let's go one dimension higher. In Euclidean space we can refer to any point by three coordinates (x, y, z) clearly describe. The distance r this point to the origin can be calculated again with the Pythagorean theorem: here applies r² = x² + y² + z².

All points at a distance r = 1 to the origin form the unitary sphere. This therefore consists of all points (x, y, z) With x² + y² + z² = 1.

The two-dimensional sphere is the edge of the three-dimensional sphere, consisting of all points (x, y, z) With x² + y² + z² ≤ 1. The sphere is two-dimensional because of the three coordinates x, y, z only two can be moved freely on the sphere: around the point (x, y, z) = (0,0,1) around, for example, we can x, y move freely and this inevitably determines the third value z = √ (1-x²-y²).

Excursus: What shape does the earth have?

Is the earth's surface a disk or a spherical surface? This question has an eventful history. Every educated person today thinks of course that they know that the earth is a sphere. But how did you get that? How do assumptions become reliable knowledge? The main difficulty lies in the fact that it is not possible to differentiate locally between disk and spherical surface, because both look the same in a small environment. It is a global question. What information do you need to clarify this question?

The curvature of the earth's surface shows that the earth certainly does not surface Disc is: of approaching ships, the sail appears first on the horizon, then the hull. This observation has been an indication of the spherical shape of the earth since ancient times, as taught by Aristotle (384–322 BC). This argument is plausible, but by no means compelling: other curved surfaces are conceivable, and the spherical surface is just the simplest of them.

Eratosthenes (276–194 BC) first calculated the circumference of the earth from the measured curvature and the distance between the Egyptian cities of Alexandria and Aswan. However, this was only possible because he was already convinced of the spherical shape of the earth. Viewed critically, his calculation is based next to the local Measurement (within Egypt) mainly on the global Assumption that the earth is a sphere and the curvature is the same everywhere. That too is initially only plausible but not mandatory.

The first circumnavigation of the world refutes the disc theory, but also does not yet prove the spherical shape of the earth! Other shapes are conceivable, for example one can also circle the torus. This fictional geography is very popular with computer games: if you leave the rectangular playing field at the edge, you enter again at the opposite edge. Pac-Man lives on the torus!

What distinguishes the two-dimensional sphere?

From the excursus on the spherical shape of the earth, we abstract a mathematical question: How can an inhabitant of a closed surface be sure that he lives on a spherical surface? In other words: what distinguishes the sphere among all surfaces?

The following picture shows a simple but fundamental observation: If a resident of the spherical surface stretches an infinitely elastic rubber band around his world, he can always let it purr to one point. Other surfaces, such as the torus, do not have this property: Here the rubber band can be stretched so that it encircles a hole and cannot be pulled together.

On the sphere you can each Continuously contract the closed curve to one point. Mathematicians call a room with this property simply connected. In our list of surfaces above, only the sphere is singly connected. Since our list is complete, this property characterizes the sphere below all closed surfaces. In other words:

If a two-dimensional closed space is simply-connected,
then this space is topologically identical with the two-dimensional sphere.

With this we have found a topological characterization of the two-dimensional sphere. Poincaré suspected that it was the same in three dimensions. In order to be able to formulate this assumption a little more precisely, we still have to clarify what a three-dimensional sphere should be.

What is a three-dimensional space?

Let us now take the bold step from two to three dimensions.

As a reminder: a surface is a two-dimensional space, because it can be described locally by two coordinates. (To determine the position of a vehicle on the earth's surface, one uses, for example, latitude and longitude.) We saw above that this local description still leaves many possibilities for the global shape of an area.

Similarly, a three-dimensional space can be described locally by three coordinates. (To determine the position of a satellite, one uses, for example, latitude and longitude as well as the height above earth.) Here too, this description is initially only local and still leaves many possibilities for the global shape.

A three-dimensional space is very simple on a small scale. Unfortunately, as a whole it cannot generally be represented by simple images as was the case with surfaces. But this is not an objective-mathematical problem but a subjective-psychological one. As happened above, we can embed surfaces in the three-dimensional visual space; this is also possible in higher dimensions, only here our view fails.

However, mathematically speaking, three-dimensional spaces are just as real as their lower-dimensional siblings. The fact that they are difficult to draw is irrelevant: it is enough to calculate with their coordinates. “Because what you have in black and white, you can confidently carry it home.” It is time to underpin this with a concrete example: the three-dimensional sphere.

What is the three-dimensional sphere?

You already guessed it: Just as there is a two-dimensional sphere that borders the three-dimensional sphere, there is also a three-dimensional sphere that borders the four-dimensional sphere. (If this gives you a headache the first time, don't worry: this is normal. If you feel dizzy, you can skip the rest of this section.)

In order to remove all doubts, we need a precise definition. In four-dimensional space we can identify each point by four coordinates (x, y, z, w) describe. Don't try too hard to visualize it, I think it's impossible. I cannot draw pictures for you either, but we can count with these coordinates as before:

For the distance r of the point (x, y, z, w) applies to the origin r² = x² + y² + z² + w². The three-dimensional sphere consists of all points (x, y, z, w)who have the equation x² + y² + z² + w² = 1 fulfill. It is the three-dimensional edge of the four-dimensional sphere. The latter consists of all points (x, y, z, w) With x² + y² + z² + w² ≤ 1.

Why do we call this sphere three-dimensional? Of the four coordinates (x, y, z, w) only three can be moved freely on the sphere: around the point (x, y, z, w) = (0,0,0,1) around, for example, we can x, y, z move freely and this inevitably determines the fourth value w = √ (1-x²-y²-z²).

In addition to the three-dimensional sphere, there are many other three-dimensional spaces, and mathematicians have developed tools to deal with such abstract objects. In contrast to surfaces, a complete list of all three-dimensional spaces is not yet known.

Excursus: What shape does our universe have?

The theory of three-dimensional spaces has proven itself because it can be successfully applied in a wide variety of situations, for example in physics.

Cosmologists study the properties of our universe, such as the global shape of the universe, just as our ancestors speculated and made astute observations about the shape of the earth. For cosmologists, however, there is no prospect of a “circumnavigation”: They have to observe passively and are dependent on the radiation reaching us.

For all we know, our universe has no edge. In addition, some cosmologists suspect that it is finite. It is therefore a closed three-dimensional space. The three-dimensional sphere would be the simplest model here, but many other spaces are conceivable. (In recent years, for example, the theory has been put forward that the universe is a dodecahedron space, a further invention by Poincare.) As always in the natural sciences, ultimately only observation can decide.

The mathematical occupation with spaces in three (and more) dimensions, on the other hand, is of a theoretical nature. It is by no means just a gimmick, but results from observing the world around us and trying to understand it and to describe it as precisely as possible using theoretical models.

At the end of the 19th century, based on mathematical considerations, Poincaré began to systematically examine three-dimensional spaces. However, his scientific interests were very diverse: He was not only a mathematician but also a physicist and one of the founders of the theory of relativity. The latter adds a time dimension to three spatial dimensions and works quite unabashedly with four-dimensional spaces. (How simple and familiar the three-dimensional spaces suddenly appear to us in this light ... ;-)

The Poincaré conjecture

In 1904 Poincaré suggested that, among all three-dimensional spaces, only the sphere is simply connected. That is the famous Poincaré conjecture.

With the above preparations we can now summarize the Poincaré conjecture in a few words - however, each of these words contains well over 100 years of mathematical experience, it is simply not possible without precise terms:

If a three-dimensional closed space is simply connected,
then this space is topologically identical with the three-dimensional sphere.

In memory of: three dimensional means that the considered space can be described in every small environment by three coordinates. Closed means finite and borderless. Simply connected means that every closed curve can be continuously contracted to a point. The three-dimensional sphere apparently fulfills all these conditions, and Poincaré suspected that it is characterized by these properties alone.

Unlike for areas in dimension 2, this assumption in dimension 3 has proven to be extremely difficult. Three-dimensional spaces are a great challenge to the imagination and also to the inventiveness of mathematicians. Numerous attempts at solutions have been published during the 20th century, many of them by eminent mathematicians. No attempt at proof stood up to scrutiny. It was not until a hundred years after Poincaré that Perelman was able to present solid evidence that Poincaré was correct in his assumption.

The Poincaré conjecture has thus become a mathematical proposition.

The proof idea

In his proof, Perelman ingeniously uses the idea of ​​curvature. In the case of surfaces, the curvature obviously plays a role, as noted above, and this idea has also proven to be an important aid in three dimensions.

Perelman's revolutionary results have brought the geometric classification of all three-dimensional spaces within reach, as it was undertaken by William Thurston since the 1970s. Building on the work of Richard Hamilton from the 1980s, Perelman has developed new techniques with which the curvature of these spaces can be controlled and skillfully modified.

To illustrate this, here is a sketch in dimension 2:

Put simply, extreme dents are repaired and the room is as round as possible. Finally, if it is just as uniformly curved as the sphere, then it is also a sphere topologically. The exact elaboration of this ingeniously simple idea is difficult - even for mathematicians. But the result justifies all efforts, and Perelman's groundbreaking ideas will probably still arouse admiration a hundred years from now.

Links on the subject

Some early reports from German-speaking media:

The reporting on the award of the Fields Medal reached a first high point in August 2006:

The Millennium Prize in March 2010 also met with a strong response:

Reports from foreign media that are worth reading and listening to:

Popular science books:

  • Donald O'Shea: Poincaré's conjecture. S. Fischer, Frankfurt 2007.
  • George Szpiro: The Poincaré Adventure. Piper, Munich 2008.
  • Keith Devlin: The Millennium Problems. Basic Books 2002.

Further links to mathematics:

© Michael Eisermann,, modified Friday 5/04/2019. Readers beware: No warranty! The information on this website is provided by the author and has not been reviewed or approved by the University of Stuttgart. It is distributed in the hope that it will be useful, but it is provided without any warranty whatsoever, without even the implied warranty of fitness for any particular purpose. The author is not responsible for the content of external webpages. So, as always, please remain critical of what you see and read: not everything on the internet is true and good and useful. Well, I assume that I am preaching to the converted: who on earth reads disclaimers and footnotes in the first place?