This semester, Assistant Professor of Mathematics Benjamin Allen is challenging the students in his Senior Seminar to grapple with some of the field's most difficult problems-the solutions to which have stumped mathematicians for centuries.
The Clay Mathematics Institute (CMI) established the Millennium Prize Problems in 2000 to celebrate mathematics in the new millennium. CMI's Scientific Advisory Board selected the seven problems, considered, at the time, to be the greatest unsolved problems in the field, and the Institute's Board of Directors allocated $1 million for the solution of each problem. Though there are millions of dollars on the line, Allen hopes his students take away lessons that are much more valuable.
"I wanted to give students a sense of math as an active, ongoing process," Allen said. "The material that students learn in undergraduate courses has been settled a long time ago. I want them to understand that there are many questions that are not yet settled, and I want to give them the sense of the processes and techniques mathematicians are using to investigate them."
Allen and his class spend a week on each problem, first reviewing the history and mathematical background to get a better understanding of the problem's origins. For most of the problems, Allen said, the class is able to build up to the actual statement of the problem, but in a few cases the problem itself is so deep that he can only hint at what it says. For all of them, the students practice techniques that mathematicians have used to solve similar, but much simpler, problems.
Nearly 14 years later, six of the problems still remain unsolved. The Russian mathematician Grigori Perelman solved the Poincaré Conjecture in 2003, but turned down the prize money.
"The Poincaré Conjecture has to do with how we can tell mathematically whether two shapes are different," said Assistant Professor of Mathematics and department chair Yulia Dementieva. "For instance, a coffee mug may look different than a doughnut, but if you made a coffee mug from flexible clay, you could deform it into a doughnut shape by 'squishing' the cup part into the handle. On the other hand, there is no way to squish a doughnut into a solid ball without collapsing the hole in the center."
Allen's students attempted to mathematically distinguish between the doughnut and sphere shapes by drawing triangles on regular spherical balloons and special doughnut-shaped balloons and then counted the number of vertices (V), edges (E) and faces (F) that result. With the spherical balloons, the students discovered that the equation V+F=E+2 was true no matter how many triangles were drawn. For the doughnut-shaped balloons, the triangles satisfied a different equation: V+F=E.
These relationships, which were first noted by 18th-century Swiss mathematician and physicist Leonhard Euler, form the basis of the modern research field of algebraic topology.
"I want to deepen their awe and wonder of mathematics," Allen said, "and also give them some understanding of the landscape of contemporary research mathematics. These problems are far from comprehensive, but they are a good starting point as to what mathematical questions are important right now."
Even before the Millennium Prize Problems were established, mathematicians have devoted their careers to each of these, Allen noted. The Riemann hypothesis, in particular, has puzzled mathematicians for centuries, and quite possibly, for centuries to come.
"The nature of mathematics is such that it is possible to spend one's career just making progress on one of these problems, leaving the actual solution to future generations."
Millennium Prize Problems
Yang-Mills and Mass Gap
Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.
P vs NP Problem
If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.
This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.
The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.
In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture. Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.
Birch and Swinnerton-Dyer Conjecture
Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.