Daily Mathematics and Seven Unsolved Millennium Problems for the Twenty-First Century from the Clay Mathematics Institute

By Arthur W. Hafner, Ph.D.
January 10, 2001

I have always liked mathematics. It has a special kind of beauty, clarity, and simplicity. I find satisfaction in using it to work through a problem to arrive at an answer. As an undergraduate, I majored in math and later completed a Master's Degree in it with a minor in statistics. As a doctoral student in library and information science, I used both applied and theoretical mathematics, statistics, and computer sciences in my studies. As a practicing librarian working in library administration at Seton Hall University, almost daily I use mathematics and statistics to understand the underpinning applications of information technology as it expands and transforms the nature of libraries.

Mathematics is an indispensable tool that is part of daily life. We encounter it everywhere. We use its numbers and language without thinking when we view a calendar, read time on a watch or clock, or make change with money. We draw and analyze graphs to identify trends and reveal patterns. We use quantitative techniques to determine familiar relationships such as circumference, diameter, distance, length, miles per gallon, optimization, probability, randomness, rate of change, square feet, symmetry, temperature, volume, width, and so many others. We use mathematics' symbols and abstractions to create and describe models and to perform simulations. Mathematics is ubiquitous and at the heart of all technological development as a major force shaping today's and tomorrow's world.

As an administrator who also has an interest in economics, marketing, and finance, I regularly use mathematical and statistical tools. I wrote about descriptive statistics and several specific quantitative measures in my book, Descriptive Statistical Techniques for Librarians, second edition, published in 1998 by the American Library Association (ISBN 0-8389-0692-3). Analytic measures afford me techniques to identify data trends and to reveal patterns, and relationships that may be important or for evaluating, improving, and strengthening library programs to better serve the university community. In thinking about a new book that I would like to write, I recently prepared an inventory of the tools that I regularly use for data analysis. Among other areas, the techniques are drawn from algebra, calculus, computer programming for web technology and web design, descriptive and inferential statistics, database construction and analysis, linear programming, operations research and optimization, signal processing for collecting and analyzing data, simulation modeling, and spreadsheet analysis.

Today's jobs require a person to be a good problem solver. This ability requires depth in mathematics and breadth in related areas such as computer science, statistics, and information technology. By taking courses and/or working on data projects, one can learn concepts, acquire and understand vocabulary, and develops patterns of thought for addressing and solving problems. Undergraduates need to include these disciplines in their curriculum to be prepared to meet their future job challenges.

Mathematics as a discipline offers the interested reader many challenging problems. An exciting problem that has long challenged students is Pierre de Fermat's Last Theorem, which states, xⁿ + yⁿ  = zⁿ  has no non-zero integer solutions for x, y and z, when n > 2. In the margin of Fermat's book, he wrote, "I have discovered a truly remarkable proof which this margin is too small to contain." Fermat (1601-1665), the father of modern number and probability theory, likely posed this problem around 1630 but it was not until 365 years later in 1995 that the English mathematician Andrew Wiles demonstrated the proof.

I recently heard about the Clay Mathematics Institute (CMI), a private non-profit foundation in Cambridge, MA dedicated to increase and disseminate mathematical knowledge. They have identified seven mathematical problems for solution in the Twenty-first Century, called Millennium Prize Problems. These problems center on questions that are central to mathematics and that have resisted solution. You are invited to think about them and to try solving them. These problems are the following:

1. P versus NP 

2. The Hodge Conjecture

3. The Poincaré Conjecture

4. The Riemann Hypothesis

5. Yang-Mills Existence and Mass Gap

6. Navier-Stokes Existence and Smoothness

7. The Birch and Swinnerton-Dyer Conjecture

(Back to the top page)