If mathematical brilliance generated electricity, there'd be enough wattage at this restaurant table – a stone's throw from the American Institute of Math – for a good 20 million homes. The conversation flows from food to hobbies to, well, zeta functions. But as dinner winds down, George Csordas, a distinguished-looking functions theorist from the University of Hawaii, confesses that he never balances his checkbook. "I hate doing the math," he says with a broad smile.

Victor Vinnikov from Israel's Ben Gurion University jumps in. "It's restaurant checks for me," he says. "I can calculate the 15 percent tip, but adding it back in – forget it."

These two are part of a group of 28 mathematicians who recently assembled to ponder some of the hardest math problems since Archimedes – part of a week-long workshop at the institute.

Now, before you flip away to read about something simpler – like, say, the Middle East – let me state up front just how much math you'll need to understand for this article: zero.

Let me also state the most important number for this group of mathematicians: zero.

Finally, let me offer a definition of math from David Farmer, director of programming at the institute: Math is about making connections, especially where they aren't obvious.

In other words, it's a dating service for numeric ideas. These 28 experts play Cupid.

To do its work, the institute chooses some of the brightest mathematicians from disparate specialties and provides a focused but informal schedule packed with good food and libations, along with lots of small workspaces where no one is ever far from a whiteboard. Week-long workshops focus on problems from braid groups to keeping graduate students devoted to math.

The institute's executive director, Brian Conrey, jokes that the definition of an extroverted mathematician is one who looks at the other person's shoes in a conversation. But this is decidedly not the case with the 21 men and seven women who make up this week's group. Despite spending considerable mental time in places that would take half a book to explain, they laugh, chat, banter, and presumably mow their lawns much like everyone else, with nary a pocket protector in sight.

This particular group is puzzling a number of problems this week, the most famous of which is the Riemann Hypothesis. It was posited in 1859 by Bernhard Riemann, a German mathematician who was smart in the same way that Einstein was something of an idea man. Proving the hypothesis has become the math equivalent of hunting down the FBI's most wanted.

The good news is that the hypothesis involves an equation that can be written in the space of a postage stamp. The bad news is that it can take lifetimes to understand.

So let's say this: It involves a seemingly harmless set of numbers called prime – numbers divisible only by themselves and by 1. Where they occurred was unpredictable, until Riemann came up with his equation. Mathematicians believe that zeros will be the key to proving the equation, which is why they are so ardently pursuing the great goose eggs. Explaining it further would require that half a book. But here's something easy to grasp – whoever proves the hypothesis first will get a million-dollar reward.

Who says there's no real-world benefit to pure math?

The American Institute of Math was founded in 1994 by John Fry and Stephen Sorenson as a haven for pure math. These Silicon Valley businessmen made their money through retail computer stores, namely Fry's Electronics. But before that, they were math majors in college, and before that, they were all-American kids, playing team sports on gridirons and diamonds. In contrast to their childhood pastimes, math eschewed a team approach, favoring the lone-genius-in-the-closet phenomenon.

"Most of the other sciences have a high degree of collaboration," said Sorenson. "In math, working together was called cheating."

Fry and Sorenson wanted to change all that. So they bought a piece of land in nearby Morgan Hill and designed a $50 million crenelated castle with frescoes, fountains, and 12 marble lions, inspired by Spain's Alhambra. After 13 years of delays, approval processes, and environmental reports, the institute held a gala ground-breaking in May, replete with a Mediterranean menu and a decor full of dazzling Moorish geometric patterns.

In the meantime, they set up an interim shop in a plain, windowless space that was Fry's Electronics' first corporate headquarters and, in 1997, hired Mr. Conrey as executive director.

Conrey is a fit man with looks and charm that invite comparisons to Tom Brokaw. Unlike Mr. Brokaw, he is an analytic number theorist.

In nearly ten years, he has transformed the institute from an unknown, unfunded, vacant warehouse into a functional, comfortable setting that hums with math and has accrued support from donors including the National Science Foundation. "We've grown by factors of ten," said Conrey, being typically numeric. "We started with a $5,000 grant, and have built up to $5,000,000."

He has also built a program of five-year fellowships, awarded annually to "an absolutely first-rate PhD." And Conrey has put together a library of all the math papers he could lay his hands on – 100,000 and counting. Most of all, he works with David Farmer to bring in the best brains on the planet for week-long sessions. The idea is to foster group work and make connections – that math definition again – that will continue for months, years, or lifetimes.

Last March 19, one of these groups announced that they had hit the math jackpot. They'd solved a problem with a mystical-sounding name, "the exceptional Lie group E8." It took a platoon of 18 mathematicians four years – and two of those years were spent just realizing it could be solved. The team also needed 77 hours of a supercomputer named Sage, which did much of the grunt work – namely, the 204 billion entries that form the result. Writing it would require a piece of paper the size of Manhattan.

Even if the Riemann Hypothesis were the only focus of this week's group of 28, actual results would still take the mathematicians a great deal of time. "One year after the workshop, we'll e-mail them and that's when we find out what they've been up to" says Farmer.

Still, a half-dozen papers are likely to come out of this week of chasing zeros. And one day, Riemann will fall. Fear not, though: The institute will soldier on. "The Riemann Hypothesis is just the first of an enormous list of functions," said Farmer, "all of which need proving."