Mathematics, New Trends in

What is new and different in the world of mathematics? Where is mathematics going in the future? What questions are mathematicians asking and exploring?

Tools Used in Calculation

Let's compare two of the instruments of calculation widely used during the last 50 years of the twentieth century. The slide rule is a portable calculator once carried by engineers so that they could quickly perform complicated calculations. Compare the slide rule, now a collector's item, to the graphing calculator used by high school students taking basic algebra. The graphing calculator can be used to do much more—more accurately and more easily.

The use of powerful calculators and computers will be an integral part of the mathematical problems and questions investigated in the future. Some classic problems, such as those involved with prime numbers , the geometry of soap bubbles, and the four-color theorem about how many colors are needed to distinguish neighboring colors on a map, are being extended to more complex questions, some involving three or more dimensions. Other problems of the future involve newer themes, such as chaos theory and how it can be applied to model various systems and computers and how they can be used to generate proofs.

The Direction of Theories and Study in Mathematics

The advances in chaos theory made in the twentieth century will probably be extended into many areas of investigation as mathematical understanding of dynamical systems in biology, physiology, and clinical practice is increased. For example, according to Barry Cipra in What's Happening in the Mathematical Sciences, a yearly publication that reports on the latest mathematical research, mathematicians are working with scientists in a variety of fields to address many of our modern challenges, including how the human immune system works and how to deal with hazardous wastes. Chaos theory is also being used in questions about waves in all their forms, along with turbulence, complex fluid flows, and computational fluid dynamics . Ecology is another active field that is benefiting from the mathematics of chaos.

Mathematicians continue to explore the nature of proof itself, and now this involves exploring whether a computer can develop a proof. The abilities of computers, the possible ability of computers to recreate themselves, and the bioengineering of computers continue to fascinate those at the cutting edge of developing technology and mathematics.

Throughout the history of mathematics, games and game theory have fascinated mathematicians, and this trend continues today. A computer has defeated a human chess champion, but people continue to search for the "perfect" play in various games. In addition to studying games, mathematicians and many other people continue to work with codes and cryptography as they seek complete security for all types of messages—particularly those transmitted over computer networks and the Internet.

The Influence of Mathematics Beyond the Field

In the 1990s, Andrew Wiles solved what has been called "The World's Most Famous Math Problem" when he proved Fermat's Last Theorem. This accomplishment inspired people beyond the world of theoretical mathematics. Marilyn vos Savant wrote a book about Wiles's work in which she quotes a poem about Fermat's Last Theorem; there was a play presented in New York called Fermat's Last Tango ; and there was a Fermat's Last Theorem Poetry Challenge.

Mathematics provided the subject matter for other achievements in the arts: The 2001 Pulitzer Prize for drama went to Proof, a mystery about a famous mathematician, and the book and movie titled A Beautiful Mind, which tells the story of John Nash, the mathematician who won a Nobel Prize in 1994 for his work on game theory.

So, solemn or frivolous, mathematics will continue to be used to model situations in every field of human endeavor where patterns and predictability pose challenges. And, at the same time, the search for useful models will continue to expand and benefit from the world of theoretical mathematics.

see also Chaos; Computers, Future of; Fermat's Last Theorem; Games; Gardner, Martin; Minimum Surface Area; Puzzles, Number; Slide Rule.

Lucia McKay

Internet Resources

American Mathematical Society. <http://www.ams.org/>.

Eric's Slide Rule Site. <http://www.sliderule.ca/>.

"John F. Nash." Cepa.Newschool.Edu <http://cepa.newschool.edu/het/profiles/nash.htm>.

QUIRKY MATH TOPICS

Mathematicians often enjoy exploring the mathematics of some rather peculiar topics. Some recent interests include finding the quickest way to untie a knot, figuring out how fish swim, and developing strategies to solve various puzzles.