Group math
What mathematicians really do and why they have to meet to do it

BY JADE BOYD
Rice News staff

The halls on the fourth floor of the Herman Brown building were abuzz this week as faculty in Rice’s Department of Mathematics prepare for the largest academic meeting the department has played host to in at least twenty years.

The Ahlfors-Bers Colloquium, which runs March 24-27, is a triennial meeting devoted to the mathematical legacy of two notable mathematicians, Lars Ahlfors and Lipman Bers. The meeting features nine plenary talks and 45 seminars and is expected to draw more than 200 participants from more than a dozen countries on four continents.

	MIKE WOLF

Rice News spoke recently with colloquium organizer Mike Wolf, professor of mathematics, about the important role that meetings play in mathematics research.

Q: I read a
story in the Wall Street Journal
a couple of years ago that said being a mathematician was the most desirable job in the country. I imagine most people think that professional mathematicians spend their days teaching and working out arithmetic problems, which sounds pretty boring. Where’s the disconnect? What do mathematicians really do?

A: It’s true that mathematics research is far less visible than research in, say, biomedicine or physics, but the process itself is largely the same. There are problems we don’t understand, and we use the tools of the discipline — or we invent new tools — to understand them. Of course, the tools of mathematics aren’t experimental. So there are no animal models or lasers, but there are some well-used mathematical methods that take their place. And, just like in the other disciplines, usually the solution of one problem immediately presents five more.

Q: What sort of mathematics problems are not yet understood?

A: You know, this seems to be the aspect of mathematics research that the public understands least of all: Many people are surprised to hear that all of mathematics is not already known! Maybe that’s because the results in math are so definite and eternal. They are not revised and refined by increasingly sophisticated experimentation.

But since you brought up arithmetic, maybe that’s a good place for an example. Most people learn in school about prime numbers — those numbers that can’t be factored as two smaller numbers multiplied together. Those numbers in math are analogous to the elements on the periodic table in chemistry: They are the basic building blocks of more complicated numbers. Now these primes are distributed seemingly sporadically in the whole numbers, and we know something about how often to expect one to appear. But even with this short an introduction, I can tell you a problem we have no idea about. Primes sometimes arise in pairs, like 11 and 13 or a bit further along, 59 and 61. Is there an infinite number of such pairs of primes, just two places apart? No one knows. Our knowledge and tools just aren’t sophisticated enough yet to decide. But humans are working on this problem and will eventually figure it out, I think.

Q: What’s an average day like for a mathematician?

A: Much like other scholars, I expect. We wake up stuck on yesterday’s problem, but we get a new idea in the shower. Then, encouraged, we work feverishly on it, trying our new tool or the connection we’ve made with something we read once in the literature or what we heard once at a conference. (Conferences are especially important, as ideas get honed in conversation with other experts; mathematics is a surprisingly social profession.) It stays on our minds as we talk to our students or teach our classes: Like the interactions at conferences, sometimes those are triggers as well. At lunch, we imagine a much simpler problem whose solution we know very well, and maybe our new approach works there. All mathematicians know dozens of examples, and we try our new idea on those, hoping it will work on the examples it should work on and fail when it should fail. At the end of the day, we’ve made some progress, and we fall asleep thinking about the problem in a slightly different way.

Q: Is the day-to-day practice of mathematical research analogous to another task that people might be more familiar with, like cooking or painting or surgery or architecture? What’s the best analogy for what you do and why?

These computer-generated random pathways were created by mathematicians conducting basic research in probability theory.

A: All professions are about solving problems, and I doubt we’re much different. Artists solve problems through creative use of what came before channeled though their personal perspectives, and there are surgical problems that require new insights as to how the systems of the body interact and what techniques could be best exploited or invented to address those issues. That’s what we do, too.

Maybe the best analogy is just to math homework. It’s actually something of a historical accident that the problems you did as a kid had already been done years before. Just as you faced them for the first time with the tools you had learned, so do professional mathematicians face problems for the first time with the knowledge they have. The only difference is that our problems haven’t ever been solved, but the feeling of “How will I ever do this?” is the same.

Q: When the average person thinks about how science is done, they probably remember grade-school lessons about the scientific method: Based on what’s already known, a scientist makes predictions and then tests those predictions with observed facts. Does research in mathematics work the same way?

A: Perhaps, in some rough sense. We do understand many very special mathematical situations, and we can use those “facts” to test ideas about what single general phenomena might be behind all of those examples. But maybe the more responsive answer is that here the experience in math diverges a bit from the experimental sciences. Mathematical theories, once established, are eternally and completely true and not subject to revision or clarification; unlike many scientific results, mathematical facts are not approximative in any sense. For example, the lengths of the legs and the hypotenuse of a right triangle exactly satisfy a²+b²=c² today as precisely as they did three millennia ago.

Q: I understand the meeting you’re organizing this month is relatively large. What can you tell me about it?

A: The Ahlfors-Bers Colloquium is named after two mathematicians that did seminal work in an important modern area. This conference has been held every three years since their deaths in the 1990s and is now perhaps the most important meeting in this subfield. We’re expecting more than 200 participants from around the U.S. and also from many countries in Europe, Asia and South America. Ahlfors and Bers were great supporters of young mathematicians, so we’ll have 50 or so talks by early career mathematicians as well as nine plenary talks by the folks who have done the most important work in this area in the past three years. The National Science Foundation has been very generous with its support.

Q: I’ve heard it said that the majority of the breakthroughs in mathematics result from meetings like this. If that is true, why?

A: Sometimes it’s hard to trace where your thoughts come from. Certainly it’s great to get the chance to hear new ideas and to try your own ideas out on other experts. I know I have some questions for some of the participants that would be very difficult to track down just in the literature, as sometimes the insights aren’t recorded in a paper in exactly the way that’s most useful for what you’re thinking about at the moment.

Q: What’s the process of discovery at these meetings? Do people walk out of lectures and say, “Aha, I’ve got it!” or is it subtler than that? Do the answers usually emerge right away, or do they have to incubate?

A: My experience is that it depends. I’ve written a paper with a colleague over lunch at a conference, but that’s really very unusual. Often you just decide that maybe there’s a problem that together you and another participant have a new approach to, or on your own, from a chance remark, there flashes the possibility you never saw before for a new attack on a problem you’ve struggled with for years.

Q: What are the applications of all of the mathematics at this conference?

A: If you mean direct applications, like building a better bridge or curing a disease in the next few years, then I doubt there are any. But in fact, science really doesn’t usually work in the way the question imagines — rarely does a scientist look all the way to some end application and then try to learn or create all of the math, physics, chemistry, biology and engineering one would need to accomplish that task from scratch. Experts in math create math, physicists understand physical phenomena and over time, across continents and through fields, knowledge gets transmitted and combined, creating advances useful for all of us. Crucial to the development of a useful CAT scan in the 1960s in the U.S. was a purely theoretical math study in France after World War I of how one could recreate an object from the shadows it cast; that mathematician a hundred years ago was just curious about a problem he thought up. At the core of the Google search algorithm is a century-old piece of math about tables of positive numbers.

Q: Do most mathematicians find arithmetic as boring as the rest of us?

A: I love arithmetic! Of course, if you mean adding up lots of long lists of numbers, then yes, that gets dull after a while — though most kids will tell you it’s just plain fun to get the right answer. But like most subjects, the further you get into arithmetic, the more engaging it becomes. It’s beautiful how the primes arrange themselves in the whole numbers. And when I tell you that distribution was first understood only a century ago and only by leaving multiplication facts behind and applying the sort of calculus used in electrical engineering, then arithmetic has a richness and beauty that’s as awe-inspiring as your first glimpse of the Grand Canyon.