Related: https://manifold.markets/Thomas42/will-any-of-the-remaining-clay-mill?r=VGhvbWFzNDI

All seven questions and my comments

1. Birch and Swinnerton-Dyer conjecture (1960s --): Mostly number theory, rational points on elliptic curves. Quick literature search did not turn up any relevance to elliptic curve cryptography.

2. Hodge conjecture (1930s --): algebraic topology. Not sure who is interested other than algebraic topologists and geometers (two of the most abstract fields of modern mathematics...).

3. Navier–Stokes existence and smoothness (1850? -- ): It has strong connection to classical mechanics. Presumably physical intuitions can help. Terence Tao has constructed (2016) a singular solution for averaged NS equation that allows liquid computer that constructs a smaller and faster copy of itself, and so on, allowing finite-time blowup. He believes it's possible for this to work with NS itself. See https://www.nature.com/articles/s42254-019-0068-9

4. P versus NP problem (1971--): Extremely difficult. Implications across all computer science, mathematics, and possibly philosophy. See Why Philosophers Should Care About Computational Complexity.

5. Poincaré conjecture (1904 -- 2003): Solved by the Ricci curvature flow (PDE on manifolds).

6. Riemann hypothesis (1859--): Strong connections with number theory and complex analysis. Mathematicians typically say this is the most difficult one.

7. Yang–Mills existence and mass gap (1960s --): A problem in axiomatic Quantum Field Theory. I have no intuition for how difficult it is, but considering how many mathematical physicists there are, probably it's very difficult?

There have been 500 years of cumulative problem time, and only 1 solution, so the baseline rate is 1/500 per year. This gives a probability of 6/500 * 7 = 9% of any solution before 2030.

All seven questions and my comments

1. Birch and Swinnerton-Dyer conjecture (1960s --): Mostly number theory, rational points on elliptic curves. Quick literature search did not turn up any relevance to elliptic curve cryptography.

2. Hodge conjecture (1930s --): algebraic topology. Not sure who is interested other than algebraic topologists and geometers (two of the most abstract fields of modern mathematics...).

3. Navier–Stokes existence and smoothness (1850? -- ): It has strong connection to classical mechanics. Presumably physical intuitions can help. Terence Tao has constructed (2016) a singular solution for averaged NS equation that allows liquid computer that constructs a smaller and faster copy of itself, and so on, allowing finite-time blowup. He believes it's possible for this to work with NS itself. See https://www.nature.com/articles/s42254-019-0068-9

4. P versus NP problem (1971--): Extremely difficult. Implications across all computer science, mathematics, and possibly philosophy. See Why Philosophers Should Care About Computational Complexity.

5. Poincaré conjecture (1904 -- 2003): Solved by the Ricci curvature flow (PDE on manifolds).

6. Riemann hypothesis (1859--): Strong connections with number theory and complex analysis. Mathematicians typically say this is the most difficult one.

7. Yang–Mills existence and mass gap (1960s --): A problem in axiomatic Quantum Field Theory. I have no intuition for how difficult it is, but considering how many mathematical physicists there are, probably it's very difficult?

There have been 500 years of cumulative problem time, and only 1 solution, so the baseline rate is 1/500 per year. This gives a probability of 6/500 * 7 = 9% of any solution before 2030.

I made the same market two months ago, and it's currently out of step with this one, so I'm posting it here for anyone who wants to make a dutch book:

You've got to be either the highest tier of A.I doomer or insane to trade yes on this position.

@Phan I dunno, Navier-Stokes looks like it might fall this decade.

With the help of the exponentialy growing field of AI. I would expect at least one of the Clay Mathematics Institute Millennium Problems to verify and accept a solution by 2030.