How can you win a million dollars

In 2000, the Clay Foundation presented seven major math problems and donated $ 1 million each to solve them. So far only one of them has been solved, namely the Poincaré conjecture (see below). All of these so-called Millennial Problems are extremely difficult - great mathematicians have tried them without resounding success. For most of them it is difficult to understand the problem in outline without studying mathematics: Therefore we only offer short informal sketches here and refer to the English-language website of the Clay Institute for details.
Three of the "millennium problems" are particularly famous: the Riemann hypothesis, the "P not equal to NP" problem, and the Navier-Stokes equations. We want to start with these. What's the matter?

P not equal to NP?

The "P not equal to NP" problem asks whether there really are computational problems for which solutions can be checked very quickly, but the solutions themselves cannot be found quickly. If the answer is yes, then the "traveling salesman problem" ("find the shortest round trip through a list of cities that each city visits only once") is such a problem; or the rucksack problem: Can you make a selection from a given set of numbers that results in a given sum?
We can also understand the strange name of the problem: "P" denotes the class of problem types that can be solved quickly ("in polynomial time", hence the "P"); "NP" are the problems that can be checked quickly ("nondeterministic polynomial" - so guess first, then check quickly, hence "NP").

The Navier-Stokes equations

The Navier-Stokes equations describe currents with eddies and turbulence (for example in a wind tunnel or in a river). Whenever things get turbulent, the usual differential calculus tools that you learn at high school, for example, fail. The Millennium Problem asks for a solution theory to precisely these equations. This is important because Navier-Stokes equations are solved every day (for example, this results in the weather report or calculations for the virtual wind tunnel to make cars streamlined and airplanes stable in flight), but without good theory, one cannot trust the mainframe computers. We use large computers in this area, but there is no reliable theory. Still.

The Riemann hypothesis

The Göttingen mathematician Bernhard Riemann proposed the Riemann hypothesis in 1859. It is about a very precise estimate of the distribution of the prime numbers - i.e. the numbers like 2, 3, 5, 7, 11, ... which cannot be broken down into smaller factors. Exact estimation means, for example: How many prime numbers are there that have exactly 100 digits?
We will probably never know for sure. But if the Riemann hypothesis is true, then it provides a very precise answer for it. She does this (unexpectedly) with the aid of differential calculus, namely by estimating the so-called zeta function, which Riemann introduced.

The Birch and Swinnerton-Dyer conjecture

Bryan Birch and Peter Swinnerton-Dyer, two now retired professors from Cambridge University (England), made this assumption in the 1960s - another great mystery of number theory. This involves plane curves called "elliptic curves", "rational points" on these curves that have fractional numbers as coordinates, and the relationship between the divisibility properties of integer solutions and the variety of rational points. They should have a close relationship. This has been confirmed experimentally, but has not yet been proven at all.
The mathematics of elliptic curves is theoretically important (it plays an important role, for example, in the proof of the Fermat conjecture by Wiles), but it is also very practical: for example, the rational points are used for complicated encryption procedures.

The mass gap of the Yang Mills theory

The Yang-Mills equations can describe elementary particles: complicated differential equations that describe and predict many properties of real particles. But is it really true that the solutions of the quantum version of the Yang-Mills equations cannot have arbitrarily small mass?
So is there a mass gap for these equations? It looks a lot like it experimentally and in computer simulations - but the evidence is missing and would be gold plated with a million dollars.

The Hodge conjecture

W. V. D. Hodge (1903-1975) was a British mathematician who made fundamental contributions to algebraic geometry: that is, to the understanding of the solution sets of polynomial equations. Such equations can describe many basic forms of nature, such as circles, ellipses or straight lines in the plane, spheres, eggs and many more complicated and exciting figures in space - the IMAGINARY exhibition from the mathematics year 2008 shows this impressively. The theory of such figures is sophisticated, especially when it comes to calculating complex numbers, which makes the theory simpler but makes the idea of ​​it much more complicated. The Hodge conjecture is a technically difficult but important question: can the substructures of such figures be described again by polynomial equations? For low-dimensional figures (which we can imagine) this is correct, but the general form of the Hodge conjecture is open. And it may well be that Professor Hodge is wrong about that.

The Poincaré conjecture

In 1904, the French mathematician Henri Poincaré asked whether the 3-dimensional sphere is the only 3-dimensional spatial form that is simply connected, i.e. in which every closed curve can be contracted to a point. The 3-dimensional sphere is the spatial form that is obtained when the 3-dimensional space is closed by a single point "in infinity". The Poincaré conjecture is a special case of a very general “geometry conjecture” that the American William Thurston (1946-2012) put forward in the 1970s - and that of the Russian Grigori Perelman from 2002/2003, based on an approach by Richard Hamilton has been fully proven.