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M. Khalkhli
Do you agree for example that Hilbert was not prophetic in his choice of problems and the course of mathematics in the 20th century was not predicted in Hilbert’s questions?
Alain Connes
Some of them played a role, but mathematics does not work like this. Nobody works on problems because they are well-known but mostly because they are interesting and relevant. The goal of the Millenium problems was to attract public attention on mathematics and in that respect it worked wonderfully.
G. B. Khosrovshahi
Which one of those 7 problems is the most outstanding problem?
Alain Connes
There is always an element of arbitrariness in choosing such problems. There are some problems that everybody agrees upon like the Riemann hypotheses. But Navier-Stokes? It is a typical nonlinear equation about which we would love to understand a lot more, but it’s very hard to say that deciding for that equation between existence of smooth solutions or breakdown, which is a very difficult mathematical problem in analysis, will be truly relevant for the concrete instances where the equation applies. There is some element of arbitrariness. So nobody can be sure that these are the most important problems and it is very difficult for some of them even to formulate the question, while it would be clear to everybody what a breakthrough would mean. This is clear for the Yang-Mills problem for instance whose “mathematical” formulation is quite difficult.
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Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society. Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik.