## Wednesday, August 13, 2014

### Abstract Fields

Maryam Mirzakhani of Stanford University has become the first woman and the first person of Iranian birth to win the prestigious Fields Medal in mathematics. The prize is awarded annually to mathematicians under 40 for outstanding and groundbreaking work.

Dr. Mirzakhani focuses on what are called "hyperbolic surfaces," a branch of geometry that would be described in layman's terms as "really really weird."

The normal space we live in features what we call "Euclidean geometry," named after good ol' Euclid, the fellow who codified some of the basic features of the way spaces, lines and points relate to each other. One of his postulates is that if you have a line, and a point not on that line, you can only draw one line through that point that is parallel to your first line. That means the two lines stay the same distance apart forever in either direction.

If you tweak the parallel postulate, you can suppose different things, such as parallel lines that eventually intersect. Or, as in the case of the area in which Dr. Mirzakhani works, you suppose there is more than one line through the point that parallels your first line. Usually when you take a basic postulate and assume some different condition for it, eventually you run into a contradiction that proves your altered postulate is false.

Unfortunately for the headache-free existence of non-mathematicians, starting in the 19th century it became clear that there were several geometries that did just fine with an altered parallel postulate. The "hyperbolic geometry" that Dr. Mirzakhani studies is one such vision. This may seem like not such a big deal until you try to make your mind accept the picture of two distinct lines through a single point, both of which are parallel to a third line. There is no real-world situation analogous to this idea, which means people exploring it have to use both sight and imagination to understand what they're doing. In a manner inexplicable to those of us who feel like running an algorithmical victory lap when our checkbooks balance, mathematicians such as Dr. Mirzakhani must combine their math with art in order to understand the fantastic "space" in which they work.

That's worth a medal.