Indiana University  Research & Creative Activity Spring 2002 • Volume XXIV Number 3


MATHEMATICS EVERYWHERE

by Daniel Maki

Mathematics has two faces. For thousands of years it has been an area of study and fascination as an abstract art that reveals the beauty of patterns in numbers and geometry. Mathematics is also an incredibly powerful tool, one that plays a key role in the sciences and in engineering. At the same time, it is often thought that mathematics is a dead subject, that all mathematicians do today is use what has been known for years, perhaps learning how to apply it in our modern world. But this picture of mathematics as a static body of knowledge couldn’t be more wrong.

Mathematics is alive and well, it is growing rapidly, and it shows up everywhere—yet most people know very little about it. Mathematics has always been known as the language of the sciences and as a tool for understanding the world and solving problems, but in recent years, both the power of mathematics and the scope of its applications have increased dramatically. Mathematics is the key to security on the Internet, it is the key to solving large scheduling and routing problems, it is a critical part of the search for genetic information in the human genome, and it is a crucial tool for understanding events such as the weather, lotteries, and political surveys and polls. Above all, mathematics is a creative activity that has been carried out for thousands of years and continues today.

Daniel Maki is professor of mathematics and chair
of the Department of Mathematics at Indiana University Bloomington.
photo © 2001 Tyagan Miller

In this issue of Research & Creative Activity, we showcase mathematics at Indiana University. We highlight a number of research areas, several applications of mathematics, and some important educational innovations that have originated on IU campuses

The mathematical study of many aspects of our world relies on the use of partial differential equations to model a particular area. In this issue we look at the work of Professors Jolly, Rubinstein, Sternberg, and Wang, who are all pursuing different aspects of the theory and applications of partial differential equations, including applications to ocean flows, superconductivity, optics, and turbulence. Their work shows that “applied” and “pure” mathematics are not two different subjects—in fact, the distinction is an artificial one. To use mathematics as a tool, we need new mathematical results to increase the abstract body of knowledge known as mathematics

Dynamical systems have been part of mathematics for at least 100 years; however, they did not become one of the “hottest” topics in mathematics until relatively recently. Professors Bedford, Misiurewicz, and Pilgrim blend mathematical analysis and computer graphics to study dynamical systems, including the notion of chaos, which has a precise mathematical meaning beyond its everyday non-mathematical one.

Theoretical mathematics is very important in the areas of probability and statistics, two areas that affect almost all aspects of our lives. Probability theory studies chance using mathematical tools and analysis. One active area for the applications of probability theory is the study of investments and the financial markets. Professor Goodman is working with others on the Bloomington campus to institute a new Mathematics of Finance option for the master’s degree in mathematics, and he is helping to develop new courses to encourage the study of the mathematics of finance.

Statistics uses mathematical analysis and probability theory to draw conclusions from data sets. In a surprising application of statistics, Professor Andersson has been working with Wallace Hooper on the problem of dating the various manuscripts of Galileo, the great 17th-century Italian mathematician and astronomer.

Among all the possible applications of abstract mathematics, quantum computing is perhaps the most intriguing. Many problems remain to be solved before the first quantum computer is built, but each day sees progress in that direction. There is a need for new mathematics and new physics, and researchers around the world are working to produce the needed results. Professors Larsen and Wang are at the forefront of research in the area of topological quantum computing.

Mathematics is, or should be, a part of the education of almost every student from elementary school into college. Because it is clear that quantitative skills are highly valued in the job market, there is great interest in the success of schools in teaching mathematics. The record is very mixed. Our country turns out a number of the best mathematicians in the world, while at the same time lagging badly in the mathematical education of a large number of students. Comparisons with other countries on certain standard tests show that the United States is behind in preparing students to understand and use mathematics and statistics, and the search is on for ways to improve mathematics education.

Indiana University has projects underway to improve instruction at all levels from middle school (led by Professor Frascella) up to the graduate and postdoctoral level (led by Professors Edmonds and Haile). Part of this effort involves collaborative work across the disciplines at Indiana University, and in this issue, we especially note the development of courses with the School of Journalism (led by Professor Voakes), and Speech and Hearing Sciences (led by Professor Kewley-Port).

These are exciting times for the study of mathematics. This issue gives a sample of this exciting and fascinating world at Indiana University.

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