The website of the Clay Mathematical Institute writes the following:
The focus [...] was on important classic questions that have resisted solution for many years.
Arthur M. Jaffe, once director of the Clay Mathematical Insitute, wrote on the history of conceiving the Millennium Prize Problems (emphasis mine).
As a first step, I requested that each SAB member submit a personal list of top questions. Each
of these questions should be difficult and important — a time-tested challenge on which mathematicians had worked without success. [...] We added questions to the list one by one. With each new question we asked whether the list should be expanded
or whether it might be improved by substituting a
new question. [...] While each problem on the list was central and
important, I want to stress that the SAB did not envisage making a definitive list, nor even a representative set of famous unsolved problems.
Rather, personal taste entered our choices; a different scientific advisory board undoubtedly would
have come up with a different list. The persons we
consulted were experts, but they were chosen under
pressure of time. However, the spirit of the selection transcends these decisions: the resulting list
represents an honest attempt to convey some excitement about mathematics. We do not wish to address the question, “Why is Problem A not on your
list?” Rather we say that the list highlights seven
historic, important, and difficult open questions in
mathematics.
The source of the above article also describes in more detail the actual process of creating the final list of $7$ problems.
For reference, the seven problems are:
- The Birch-Swinnerton-Dyer conjecture
(Andrew Wiles)
- The Hodge conjecture (Pierre Deligne)
- The Navier-Stokes equation has
smooth solutions (Charles Fefferman)
- P is not NP (Stephen Cook)
- The Poincaré conjecture (John Milnor)
- Quantum Yang-Mills theory exists
with a mass gap (Arthur Jaffe and Edward
Witten)
- The Riemann hypothesis (Enrico
Bombieri)
