A New Million Dollar Prize?



Possible new problems to add to the Clay list



Grigori Perelman, the solver of the three-dimension Poincaré Conjecture, has declined to accept the first awarded Clay Prize. I may not understand him, but I think he is one of the most colorful mathematicians ever. Perhaps the award should have been offered to both him and Richard Hamilton—recall Hamilton created the Ricci-flow approach used by Perelman. Perhaps together they would have accepted. Oh well.

Today I will talk about what the Clay Mathematics Institute should add to their list to get it back to the magical number seven. I do not know if they plan to do this, I doubt they will ask me for advice even if they do, but let’s discuss what they should add anyway.



Seven is a magical number. One example is the work of George Miller. When I was at Princeton I was honored to meet George—one of the world’s great psychologists. George has done many things, but he is famous for the rule of seven. He discovered that most people can remember in their “working memory” only things. I probably am on the lower end of this bound these days, but his paper is one of the most cited in all of science. Perhaps his rule is one reason the Clay Institute should get back to seven problems.

I once worked with George on a large project, although we never wrote a joint paper together. The large project was called the Massive Memory Project: our thesis was that large computer memory was as important as having many processors, sometimes even more important. It was under consideration for funding by NSF, but first we needed to “pass” a site visit. It went well until I was asked a certain hard question by a visitor X. I tried to answer X’s question, but was unable to convince them. Then, George Miller answered why memory was the answer to X’s question:

The brain uses memory to do this, so I think we should use memory too.

This convinced X. We were soon funded by both NSF and DARPA.

Problems

What problem should Clay add to their list if they decide to get back to seven problems? Here are seven they may wish to consider.

Jacobian Conjecture: I just discussed this here. I think it is one of the great open problems.

Hilbert’s Tenth Over Rationals: Suppose is an integer polynomial. The question is do there exist rational numbers so that

Of course if have to be integers, then this is in general an undecidable problem. This follows from the beautiful work of Martin Davis, Yuri Matiyasevich, Hilary Putnam, and Julia Robinson on Hilbert’s Tenth Problem. However, when the can be rational numbers the problem is open.

Graph Isomorphism: The current Clay list does not have any graph theory problems. One that I might suggest is the famous—infamous?—problem of deciding in polynomial-time whether or not two graphs are isomorphic. This problem has some beautiful partial results, but the question is unresolved.

Quantum Computing: The question I would suggest is: Can quantum computers be simulated in polynomial time by classical ones? This seems to be one of the outstanding open questions in this area. An answer would have many consequences, and would shed light on both physics and computational complexity.

Goldbach Conjecture: Recall this simple to state conjecture asks: can every even number greater than be written as the sum of two primes? I think this is one of those—“how can it be open questions”—I have called them mathematical embarrassments. Paul Homer just posted a comment here about this conjecture.

An Explicit Quadratic Boolean Circuit Lower Bound: The problem is to give an example of a boolean function family so that two things are true:

The family can be computed in polynomial time by an explicit program. The general boolean circuit complexity of grows as .

I selected quadratic growth, but of course even would be a major result.

Invariant Subspace Problem: The invariant subspace problem is the question: “Does every bounded operator on a separable Hilbert space over the complex numbers have a non-trivial closed invariant sub-space?” See this for some history. See Terence Tao for a recent discussion of this famous problem.

Open Problems

Do you like any of my suggestions? What problem do you think should be added to the Clay list?