I have talked about some of the major open problems in mathematics with my MTH 101 students (either in class or outside) and these include the Riemann Hypothesis, the Goldbach Conjecture and the Collatz Conjecture. One important problem which was resolved recently is the Poincare Conjecture . As you may know, Grigory Perelman provided a sketch of the proof in his arxiv papers which was subsequently verified by various prominent mathematicians and at the 2006 International Congress of Mathematicians held in Madrid, Spain, he was awarded the Fields medal (which he declined; more recently, he also declined the million dollar prize from Clay Mathematics Institute). I have been wanting to mention this for some time but it was not easy to find a context to talk about this problem in a Calculus-1 class…..until now.

I read the OU Math Club blog regularly and over there I found a news item which I am sure my MTH 101 class will find interesting (particularly since many of them recently participated in a Fashion Show at the annual fest EnthuZia 2010). Now, Perelman actually proved a more general result, called Thurston’s Geometrization Conjecture which implies the Poincare conjecture. Cornell math professor William Thurston proposed this conjecture in 1982 and it says that, roughly speaking, a three dimensional geometric object (more precisely, a closed, oriented 3-manifold) can be cut into geometric pieces, each of which has one of the eight geometries in dimension three. These consists of spherical, hyperbolic and flat geometries and five other kinds which are somewhat difficult to explain here. For his seminal contributions, Thurston was awarded the Fields Medal in 1982.

Inspired by Thurston’s work, Issey Miyake fashion designer Dai Fujiwara created his fall-winter 2010-2011 collection which attempts to illustrate the eight geometries that are sufficient to describe a three-dimensional form. The collection was displayed in Paris in March this year and Thurston, wearing a designer jacket, attended the show. Here’s an ABC news story on the event. You can watch the interview with Fujiwara and Thurston here:

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