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The Field of Mathematics

Mathematics begins with simple questions in arithmetic. This has led to harder and harder questions involving a huge array of techniques. Perhaps the best way of understanding the scope of mathematics is to look at some examples of questions that mathematicians have worked on and are working on.

A prime number is an integer that cannot be factored into the product of two smaller integers. Every number can be written as the product of primes. Thus, primes are the building blocks of the integers. It is easy to tell if you have a prime number, but can you give a method for deciding whether a number, say one with 200 digits, is prime that works quickly? Can you give a method that works quickly for factoring a number into prime factors? These are simple questions but efforts to answer them have led to much elegant and deep mathematics. And the answers to these questions are useful. Many of the encryption devices we use every day are based on the fact that we can quickly tell if a number is prime and we cannot quickly factor numbers.

Is the planetary system stable? In other words, taking only gravity into account, will the planets keep revolving around the sun, or will they fall into the sun, or will they move farther and farther away from the sun? We assume that the sun does not change and that there are no visitors to the planetary system. Efforts to answer this question have led to the study of chaos and fractals.

Diseases sometimes appear in geographical clusters. When do these clusters indicate that the disease is caused by something in the environment near the cluster? More generally, one can ask how does exposure to a certain substance affect the probability of an individual developing a certain kind of cancer? Often one can find two explanations that describe a data set equally well. Which is the better explanation? These are examples of a broad array of questions having to do with using imperfect and incomplete data to understand the behavior of complicated systems.

The rapid advance in genetics has led to a plethora of problems that have a large mathematical component. We describe one technique and a pair of problems arising from this technique. A micro-array tells us which genes in an organism are being expressed at a single instant. This tells us roughly what proteins are being manufactured at that time. For example, from a single yeast cell, we obtain information about the productions of 5,000 kinds of protein. If we repeat this experiment 12 times, we seem to have the information needed to get a picture of the biochemical pathways of the organism. This knowledge will help us understand, for example, how undifferentiated stem cells become blood cells or muscle cells, or how diseases harm an organism.

But there are several challenges to overcome. First micro-array readings are prone to noise, which could come from tiny differences in the measurement methods or initial conditions (for example, ambient temperature). One needs to mathematically model this noise in order to compensate for it. The second challenge is to develop methods of handling the huge amount of data produced by micro-arrays. How does one find patterns in this set of data. This is called "data mining." Clearly techniques for analyzing a huge data base can be used in many endeavors—for example, ecology.

Mathematics Majors

The goal of the mathematics program at the University of Massachusetts Amherst has three aspects. First, students learn basic material, such as linear algebra, differential equations, and statistics, that are needed to successfully attack a wide range of problems. Second, they learn to think with rigor. Lastly, they learn to approach apparently unsolvable problems by studying simpler problems, doing experiments, and bringing together different concepts.

All majors must complete a calculus sequence and courses in linear algebra, modern algebra, and analysis. Each major has wide freedom of choice in upper-division courses and can, with the assistance of a faculty advisor, tailor a program to their interests and career goals. For example, one can prepare for a career in actuarial work, statistical analysis, computer programming, data processing, industry, government, or secondary school teaching. One can also prepare for graduate study in mathematics, statistics, computer science, and other fields or professional programs in business, law, medicine, and education.