JUST ABOUT ANYBODY who read a book titled Prime Obsession

will know all about

David Hilbert’s 23 problems for the twentieth century (Chapter 12). Well, a

22-year-old Swedish mathematician has just cracked part of the 16th. Here

is a report. This

mathematician, by the way, is female–so much for the idea that women can’t

do math.

You can find all 23 of Hilbert’s problems, in the form he originally

presented them, here.

The sixteenth, to

put it at its simplest, starts with inquiries into the topology of algebraic

curves in the projective plane. Um, look: the graph of x-squared plus

y-squared equals a-squared comes out as a circle of radius a if you draw it

on a regular plane, right? And the circle divides the plane into two parts,

an “inside” and an “outside,” right? Well, Hilbert’s sixteenth concerns

questions that arise when you generalize this notion to curves of higher

degree (higher than “squared,” that is — cubed, fourth powers, etc.) and

replace the regular plane by a projective plane (which you can do by adding

a line at infinity), and then proceed to higher dimensions (so you have

surfaces in space instead of curves in a plane). Please don’t ask me to

explain further, it’s not my area… though I cannot forbear noting that

one of the classic texts on the topic was written by a Harvard prof. named

Coolidge.