From the Director

Doing the Math and Making an Impact

Doug Arnold,

IMA Director

On May 18 I had the pleasure of delivering a commencement address for the math and stat graduation at the University of Illinois in Urbana-Champaign. I thought that the address might be of interest to some of Update readers, and so I present it here.

Wow! You did it! You really did it! With lots of hard work and concentrated effort, innumerable small advances and setbacks, invaluable support from Mom and Dad, undoubtedly a few memorable breaks to recharge batteries, and--oh yes--some pretty darn good classes and professors...you have earned that degree. And not only have you earned a degree of higher learning from a first rate research university, you have gotten a degree in math or statistics, the math sciences, among the most central--and toughest--fields there are. Congratulations!

What makes the math sciences so central? As Galileo put it, "The great book of nature can be read only by those who know the language in which it was written. And that language is mathematics." Math is the way to understand all sorts of things in the world around us. But only about 1 in 100 undergraduate degrees in this country is given in math and statistics and only a small portion of the population has any facility with that mathematical language Galileo spoke about. So you're a valuable commodity!

So what are you going to do now with all that math? What can you do? Well I was invited here to deliver some advice, so let me make some suggestions. How about competitive sailing?...Not sure what math has to do with winning a sailboat race? Well, a couple of months ago, Team New Zealand, the defenders of the America's Cup, were trounced by the Swiss team sailing a boat called the Alinghi. The Swiss team won the best-of-nine series five to zip, after having beaten their way through the previous rounds in similar style. This was no small upset--it marked the first time in the 150 year history of the race that it was won by a European team. Now you studied math, not geography, but you know that Switzerland is a small, mountainous, land-locked country. So how did the Swiss pull this upset off? Well Switzerland may not have a great sailing tradition (at least until now!) but it does have a very strong tradition in mathematics--Euler's picture appeared on a Swiss 10 franc note--and the Swiss team wisely brought this strength in math to bear on the America's Cup challenge. They enlisted a group of mathematicians specializing in mathematical modeling and numerical computation led by Professor Alfio Quarteroni at the national polytechnical university in Lausanne. The mathematicians used partial differential equations to model the flow of the sea around the hull, the dynamics of the air and the sails, and the turbulent interaction of the ocean, wind, and boat. They then applied advanced numerical algorithms to solve these equations on high performance computers. This allowed them to optimize such things as hull and keel design, sail geometry and placement, and so forth. Their work was essential to the design of the Alinghi, and so to the Swiss victory. They did the math and made a big impact.

Now I'm not claiming that the mathematicians alone were responsible for winning the America's Cup--certainly not. Other scientists and engineers made important contributions, for example, material scientists who developed the composites used in the hull. And of course there was the Alinghi sailing team which brought together sailors from a dozen countries under the brilliant Kiwi skipper Russel Coutts. Not to mention the generous financing of Swiss bankers. But the Alinghi example illustrates my main point here today: math and mathematicians make crucial contributions to all sorts of challenging problems. Math makes an impact. And you--being mathematically trained--oh the places you can go and the things you can do!

By the way, you may wonder what that Lausanne math modeling group does when it is not helping to win the America's Cup? Their primary research concerns the simulation of another sort of fluid flow--the flow of blood through the human circulatory system. Just as they achieved great results optimizing the Alinghi, they use mathematical modeling and numerical computation, in collaboration with doctors, to design surgical stents and other devices and to optimize a variety of medical interventions. If I had a difficult cardiac condition, I wouldn't want my mathematician friend Alfio Quarteroni operating on me. But I would sure feel better knowing that his kind of mathematical simulation had been fully utilized in the planning.

If you poke around a large research university like this one, you will see math popping up all over the place. There are long established linkages between math and the physical sciences, as in fluid dynamics example just presented. Increasingly math is making an impact in the life sciences as well, prompting biologist Rita Colwell, director of the National Science Foundation, to observe that "mathematics is biology's next microscope--only better." In their recent bio textbook Keener and Sneyd wrote that "teaching physiology without a mathematical description of the underlying dynamical processes is like teaching planetary motion to physicists without mentioning... Kepler's laws; you can observe that there is a full moon every 28 days, but...you cannot determine when the next...eclipse will be." And math increasingly reaches outside the sciences, to economics, sociology, and business for example. Yesterday I attended a workshop here for Illinois's new Applied Mathematics Program. It involves no less than 22 departments from bioengineering to linguistics. Clearly researchers all over this campus understand that math makes an impact.

And it is not only in academics and research where math plays a central role. Problems which need mathematics for their solution also arise throughout industry. A quarter of US math doctorates and many more math bachelors go to work in industry. And a quarter of the thousand or so scientists who come by my research institute each year, the Institute for Mathematics and its Applications, come from industry. Why? I can't put it any better than was done in a strategic plan published last year by the British government, seeking to exploit research to improve the competitiveness of industry in the UK. The report stated: "Mathematics is the most versatile of all the sciences. It is uniquely well placed to respond to the demands of a rapidly changing economic landscape...Mathematics now has the opportunity more than ever before to underpin quantitative understanding of industrial strategy and processes across all sectors of business. Companies that take best advantage of this opportunity will gain a significant competitive advantage: mathematics truly gives industry the edge." And it is not only industry, but government as well. I can stay in Britain for a recent example of the impact of mathematics on government policy and its outcomes. In dealing with the hoof-and-mouth disease outbreak a few years ago, British decision makers relied heavily on studies based on mathematical epidemiology. This led them to the controversial decision to aggressively cull herds, which successfully contained the outbreak.

In the 1967 film The Graduate a family friend approaches the new graduate played by Dustin Hoffman and declares, "I just want to say one word to you--just one word--'plastics.'" Well I hope that my advice will be a bit more useful, but if I were forced to limit my advice to just one word--and "math" was not allowed--the word would be "data." Thanks to the Internet, to increasingly powerful and ubiquitous sensors, and to new storage technologies, many scientific fields and many companies and organizations have been able to build up huge stores of relevant data. We are awash in torrents of data, with terabytes more flooding in every day. The grand challenge we face at the beginning of this century is how to get the most out of all that data. For example, how can we exploit the world-wide network of seismic sensors to predict earthquakes? How can we mine the vast genomic databanks to advance biology and medicine? How can we sift through the massive amounts of text, video, web, and satellite data to detect terrorist events before they happen? Well, data means big collections of numbers--remember that text and images are digitized and stored as numbers--and data mining means discovering the patterns and structures hidden in those collections. That's practically a definition of mathematics: the study of structures and patterns in large numerical sets. So you can be sure that in the 21st century--the century of data--math will again have a huge impact.

As a small example of thinking mathematically about data, consider the outcome of a blood test. That's a data point described by 10 or 20 numbers, cholesterol 188, triglycerides 376, HDL 32, and so forth. So, mathematically speaking, a blood test describes a point in 20 dimensional space. Now my doctor tends to get worried if one of those numbers falls out of an expected range or starts to change. He thinks in terms of a collection of intervals. But blood is a complicated system of interacting chemical components and biological structures, and the collection of acceptable values almost surely form a complex region which reflects the equations relating the various consituents. A complicated geometrical region in 20 dimensional space, not just a collections of intervals. Now you've studied multivariate calculus and so you have some experience with surfaces described by equations in space. You know for example, that there are directions normal to a surface and directions tangential to it and that if a point in the surface moves in a normal direction then it will leave the surface much more quickly than if it moves tangentially. That's the kind of understanding that can make a big impact--in the future it may help to design a wearable sensor system, for example, which will warn us when our blood chemistry is getting out of whack or help us decide what to eat.

In concluding I want to emphasize that you don't have to become a professional mathematician or researcher to make an impact with the math you have learned. Wherever you go, whatever you do, you will surely be confronting problems. With math you learn to be a better problem solver. You learn to think logically, to reduce complicated situations to their essentials, to emphasize the relations between things over nonessential details of the things themselves, to abstract, to generalize. Cultivate this kind of thinking in yourself. Apply it whenever you can. Continue to develop it. Seek out important problems and think mathematically about them. Just because the people around you do not approach a problem logically, quantitatively, or mathematically--and most probably won't--should not stop you from doing so. To the contrary, as the mathematician in a team you bring something special. So my advice to you is simply: do the math and make an impact!