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."