CH 22: Do I Have a Question for you!
A realistic attempt to isolate the most difficult problems in mathematics and then offer a bona fide $1 million dollar prize is certainly worth mentioning. Fame and fortune thus await those super clever individuals who can crack these most challenging and relevant of mathematical puzzles, dubbed the Millennium Prizes. It appears that in many respects, and these problems attest to this fact, the field of mathematics is still wide open and unexplored. Here are the seven Millennium Prize Problems in abbreviated and lay terms.
1. P vs NP (exponential time). This is perhaps the most important question in theoretical computer science. The riddle is whether a computer that can verify a solution in a certain time frame can also find a solution in a certain time frame. Are there questions that would take infinite time to solve with infinite computational resources?
2. Hodge Conjecture. This problem deals with investigating the shapes of complicated objects. This process of cataloging different shapes has become a powerful tool for mathematicians over the years. However, some of the underlying geometry has become obscured and the Hodge conjecture could fill in the missing geometric pieces. It currently remains a major unsolved problem in the field of Algebraic Geometry.
3. Wavier-Stokes Equations. Fluid mechanics, which is applied math that deals with the motion of liquids, is immensely useful and effective. The challenge with this problem is to fill in the gaps on these insights, which still remain elusive. This would go a long way to better understanding turbulence, for example.
4. Birch Conjecture. Consider the equation x2 + y2 = z2. Then consider the whole number solutions that might exist for this equation. Now think of more complex equations and finding solutions can become near impossible. The Birch conjecture asserts that in certain complex cases, there is information that we can glean about the nature of these solutions.
5. Riemann Hypothesis. Perhaps the most technically difficult challenge, it is a deep problem related to number theory, the math that is concerned with the properties of numbers. It would yield answers regarding the distribution of prime numbers, which has profound implications in cryptography.
6. Yang-Mills Theory. This deals with physics, and proving that quantum field theory (you probably have heard the term quantum mechanics) is provable in the context of modern mathematical physics.
10. Poincare Conjecture. Interestingly, this problem was recently solved. On March 18th, 2010, the Clay Mathematics Institute awarded Grigori Perelman of St. Petersburg, Russia for his work on the Poincare Conjecture. The CMI writes, “It is a major advance in the history of mathematics that will long be remembered.” It should be noted though that Mr. Perelman issued a statement a few months later indicating his displeasure with the mathematics community and his belief that the mathematician, Richard Hamilton, is as deserved of credit, he walks the walk and talks the talk too, he also turned down the $1 million dollar prize.
The Poincare Conjecture deals with Topology, which is the math that is concerned with spatial properties and the solution, which made very adept use of differential equations and geometry, answers a very fundamental question about the shapes that form our cosmos.
Michael Phelps is impressed with this Medal