Careers in Mathematics

Association for Women in Mathematics

by Margaret Menzin and Robert Goldman

Mathematics is among the most fascinating of all intellectual disciplines, the purest of all art forms, and the most challenging of games. The study of mathematics is not only exciting, but important: mathematicians have an opportunity to make a lasting contribution to society by helping to solve problems in such diverse fields as medicine, management, economics. government, computer science, physics, psychology, engineering, and social science.

In this pamphlet we try to give you a flavor of some of the areas of mathematics in which there are many opportunities. At the end we provide a brief “road map” of mathematics.

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A bachelor’s degree in mathematics will prepare you for fascinating jobs in statistics, actuarial sciences, mathematical modeling, and cryptography; for teaching; as well as for graduate school leading to a research career in mathematics or statistics. A strong background in mathematics is also necessary for research in many areas of computer science, social science, and engineering.As you read this pamphlet, look for several themes:

• Mathematics is often done in conjunction with another field: biology, physics, economics, or a host of others.
• Mathematical modeling (explained below) is used to solve real problems in a variety of fields.
• Statistics is a growing field, particularly in those fields dealing with human behavior.
• Many topics in “pure math” have important applications in computer science.
• There is a national shortage of teachers in all the mathematical sciences (pure math, applied math, statistics, and computer science) at all levels, so any of these fields goes well with teaching and/or research.
• Mathematics is a field with a surprising variety of specialties which have different “feels” to them. You probably won’t like all of them equally, any more than most musicians feel the same about rock and classical music, or most English majors like all authors and periods equally. So if you come across a math course that isn’t your favorite, but there are others that you really like, it just means that you are getting to know math better and your taste is becoming more refined.

Statistics:

Let’s start by talking about statistics. Statistics is one of the few fields for which the U.S. Department of Labor forecasts a faster rate of growth in jobs than for computer science. The widespread use of statistics is very much a twentieth-century phenomenon. An increasingly important aspect of modern statistics is the design of experiments and surveys — as well as analyzing the resulting data — in areas of application ranging, literally, from archeology to zoology.Statisticians have also been responsible for the development of quasi-experiments in which, to use medical terminology, the researcher has no control over which patient receives which treatment. Examples include the numerous studies of the effects of smoking on health. Let’s look at an example.

Lung cancer rates are far higher for smokers than for non-smokers, but this does not “prove” that smoking causes lung cancer. People decide whether or not to smoke and it can be argued that some factor X, perhaps unknown to us, is responsible for producing both an urge to smoke and lung cancer.

An “ideal” but morally repugnant experiment might involve perhaps 1000 pairs of identical twins. One of each pair would be assigned at random to smoke from the age of sixteen and the other instructed not to smoke at all. Assigning the twins randomly in this way would ensure that the factor X — if it existed — would be distributed close to evenly between the two groups. Any differences in lung cancer rates between the smoking and non-smoking groups cannot be due to factor X but can be attributed to cigarettes.

Lacking such an “ideal,” statisticians have developed studies in which subjects are “matched” in as large a number of factors — age, sex, race, disposition, etc. — as possible so as to control for the effects of these factors.

Statisticians use surveys — for example, opinion polls and the Government’s Health Interview Survey — to predict the patterns of behavior of large groups based on relatively small samples. They ask questions such as: How can we be sure that what we predict from our small sample is true of the population being sampled? Questions like this are answered using probability theory. This is a branch of mathematics which is important in its own right, but which also provides the theoretical underpinning for statistics.

Related to both probability and statistics is queuing theory, which predicts what happens as people or things get in line. Typically you will know the average rate at which the things enter the system, but entries will be random otherwise. Queuing theory can be used to predict how traffic bunches up at toll booths, how lines grow and shrink at restaurants or in hospital emergency rooms, and how long your job takes to be printed out from a computer printer. If your local bank or post office is now using one long line that “feeds” to several tellers or windows, that is because queuing theory tells us that system will minimize the expected wait. So you can see that this part of statistics has many applications.

Another area in which there is a need to predict events is insurance. The people responsible for computing insurance rates are specialist statisticians called actuaries. To become an actuary you take a series of ten written exams that are given regularly by the Society of Actuaries. The first exam is on calculus and the second is on probability and statistics, which you can take as an undergraduate math major. The insurance companies offer courses to prepare you for the later exams and even offer you time off to study for them. Typically you can expect good raises and promotions as you pass these exams. (And because they are standardized tests, you know that you will not be discriminated against.) Actuaries have a lot of responsibility and, not surprisingly, are very well paid.

This completes our brief summary of statistics. We have tried to provide a flavor of the kinds of problems statisticians solve, working both singly and in collaboration with people in other fields. Part of the fun of being a statistical data analyst is the rich availability of opportunities to work closely with professionals in a wide variety of specialties: medicine, economics, business, science, and other fields.

Mathematical Modeling:

In mathematical modeling, you write down equations to describe how a real world system behaves. The “system” might be drawn from many different fields. For example, you might look at several populations living off the same food supply and try to predict how the populations grow and shrink. Or you might look at how an epidemic spreads and see how people move among the groups of “immune,” “infected,” and “at risk to get the disease”. In physics or engineering you might be interested in how heat is dissipated through the heat shield of a space vehicle, or how different planets exert gravitational pulls on each other. Or you might want to apply the laws of fluid dynamics to describe how blood flows in veins and what happens when blood pressure is increased. In economics you might want to predict how a strike in the steel industry will affect other parts of the economy, such as the unemployment rate and the gross national product.Building a mathematical model is usually a multi-stage process: you write down the equations, use them to predict what will happen, see if your predictions agree with experiments, modify your equations if necessary, make new predictions, and so on.

These models may use one of more of these kinds of mathematics: linear algebra (systems of simultaneous equations, as solved in high school algebra ), calculus (which deals with predicting a function’s values from knowing the rate at which the function is changing), and statistics.

The model may be solved exactly (you may be able to write down a function that tells you the values you want to know), or you may have to approximate the values because they can’t be found exactly, or you may have to simulate the model on a computer — i.e., let the computer imitate the real system to see what happens as you change some of the conditions, such as the amount of food available or how contagious the epidemic disease is.

One advantage of using a mathematical model of a “real world system” is that one can investigate the effects of changes in the system without altering the real world. For example, it is clearly more economical to discover from a computer simulation that a new two lane tunnel under Boston Harbor will be inadequate in three years than from the experience of actually building the tunnel.

Another advantage of making a mathematical model of a “real world system” is that you can use well-developed mathematical theory to build up theory in the system (rather than having to build it up from scratch), and to make predictions about the system. As usual, the power of mathematics comes from its ability to handle general abstract problems (such as solving simultaneous equations) and then to apply these general methods to an enormous variety of problems.

Let’s look at a mathematical model that may have affected you without your knowing it.

Have you been vaccinated against smallpox (the vaccination looks like a thumb-sized area on your upper arm)? Perhaps you have an older sibling who was vaccinated or a younger one who wasn’t? The decision to stop innoculating against smallpox was based on a mathematical model.

At the time the decision was made, about ten years ago, there was no naturally occurring smallpox in the United States, but there were children who died every year as a result of the smallpox vaccination program. (Typically the child who died was not the one who was vaccinated, but a sibling with eczema or some other skin problem who touched the vaccination then his/her own sores and got a massive case of smallpox.) It was very well known how many children died from the smallpox program.

Since there was no smallpox in the United States, it was fair to assume that any smallpox cases that occurred here would be introduced through a major port such as New York or Washington or Los Angeles. Estimates were made of how long it would take before such a case was detected. At that point, you would begin vaccinating in expanding circles around the city where the case appeared. Some people who came in contact with the infected person might go to another city, and cases would begin to appear elsewhere. You would then start to vaccinate around these other cities. Finally, it is possible to predict the expected number of deaths if all this happened, which turned out to be fewer than the known number of deaths under the vaccination program. So the decision was made to stop vaccinating. (Smallpox has since been declared a dead disease.)

There are other important models that deal with a person or thing that can move among a finite number of situations — for example, a person can be employed or unemployed, an electron can be in one of a finite number of orbits, a person can have cancer or not, a piece of tire on Route 128 can lie on the left shoulder, in one of the eight lanes, or on the right shoulder, a person can use Crest, or Colgate, or Brand X, or not brush his/her teeth at all. A simple but powerful model (based on high school algebra) can be used to predict what fraction of the population will end up in each state in the long run, and whether or not everything will eventually end up in one state (as tires do on the highway shoulders).

As you can see from some of the examples at the start of this section, mathematical modeling is a very important area of mathematics. It is the kind of work that can be done for its own sake, or in combination with another field, — biology, chemistry, economics, business, engineering, sociology, psychology, computer science, urban planning, or medicine. It is an area in which you can draw from several different branches of mathematics. It can be as abstract or as “hands-on” as you like.

Leontieff and Samuelson have separately won Nobel Prizes in economics for their mathematical models of the economy. (Leontieff’s model used linear algebra to predict how one segment of the economy affects other segments. Samuelson’s model was based on calculus.)

Where Mathematics Meets Computer Science:

As in the sections on statistics and modeling, we cannot cover the whole topic of mathematics and computer science. Rather, we’ll try to give some examples that convey some of the flavor of this area.One area that is particularly “hot” these days is cryptography. This is the art of making and breaking secret codes. (Strictly speaking, cryptography refers to making codes and cryptanalysis to breaking them, but we shall refer to them both an cryptography.) Of course, if you were asked to guess who is hiring cryptographers, you would immediately guess the CIA and other agencies concerned with spying and national defense. But there are other people who are very interested in cryptography.

For example, the cable TV companies do not like you to get their programs without paying for them, which can be done by putting a “dish” on your roof. So more and more they are encoding their signals before sending them out. You then need to rent their decoding device in order to turn the signals back into what you want to see on your television. Banks are also very interested in cryptography, especially since the advent of automated teller machines, which communicate with the banks by radio. Would you want someone to stand near one of these machines with a radio receiver and listen to your name, password, and bank balances? Clearly not. So these transactions are encoded, and the code changes for each transaction.

One kind of code, which many organizations are interested in, can be broken if you can factor its key number into primes. For example, if we ask you to factor 15, you quickly reply that it is 3 times 5. But what if we gave you another number, stating that it is the product of two primes, each of which is approximately 1000 digits long? If you want to use a computer you may. How long do you think it would take? Using the largest computer currently available, it would take approximately a million years! So, as you can well imagine, this area of pure math (number theory) is attracting a lot of interest these days.

Everyone knows that computers are good at doing arithmetic, but arithmetic on a computer can be tricky. For example, with a pencil and paper

1 + (.04 + .04) = (1 + .04) + .04

But on a computer something different can happen.Let us suppose that we have a very small computer, one that can carry only two significant digits. When the left side of the equation is evaluated, the .04 + .04 is added to get .08. The “1″ is turned into 1.0, the 0.08 is rounded to 0.1, and the 1.0 and 0.1 are added together to get 1.1, which is the correct way to round 1.08 to two significant places. On the other hand, when the right side of the equation is evaluated, the first thing is to find 1.0 + 0.04. The 0.04 is rounded to 0 and the sum is 1.0. This is added to the other 0.04, which is also rounded down to 0, and the sum of the whole thing is 1.0 — which is a different answer from the first case, and not quite what you would expect. Since computers do many thousands or millions of arithmetical operations per second, these rounding errors can become very important very quickly.

The area we have just been talking about is called numerical algebra. It is clearly related to the very old area of numerical analysis, which is concerned with how to approximate the value of a function (and know the range of your error) if you can’t find the value precisely.

As a final example of the math/computer science interface, let’s look at a non-numerical problem.

One common operation on a computer is sorting. For example, you may have to sort all the names of people who applied for a Social Security Number last month, or all the checks that came to a bank by account and check numbers. There are many well-developed techniques for sorting and searching (seeing if someone’s name is already on the list of people with Social Security Numbers). In fact, one entire volume of Knuth’s The Art of Computer Programming is devoted to sorting and searching, and new methods continue to be published.

If you have to decide among several methods for sorting a file of names, you would like to know which one to choose: which uses the fewest number of steps and the least computer memory. The number of steps is described as a function of n, the number of items to be sorted. This area of mathematics is called analysis of algorithms. If you are analyzing an algorithm, you might find yourself using such facts as

1 + 2 +…+ n = n(n + 1)/2

If you enjoy problems like this you might also be interested in complexity theory, which asks: For what problems is the function that gives the number of steps it takes a polynomial?

This is only the very briefest of excursions into areas where math and computer science interact. There are many others involving computer graphics, geometry, graph theory, abstract algebra, probability and statistics, calculus, differential equations, translation of computer languages, and many more.

Research and Teaching

Research in mathematics is as varied as mathematics itself. It may be done alone or with other mathematicians. It may be done with nothing more than pencil and paper, or it may require a computer. It may be done with applications in mind, or, very frequently, it is done just for the fun of it. Many mathematicians like the beauty of mathematics, and are not particularly interested in applications. Sometimes the applications appear later, and sometimes they don’t.One famous example of a purely abstract problem was the question of whether or not Euclid’s Fifth Postulate (through a point outside a given line, one and only one line may be drawn parallel to the given line) could be proven from the first four postulates. This question was of purely intellectual interest as everyone believed that Euclid’s postulates described space as it really is. Nonetheless, many famous mathematicians tried to prove that the Fifth Postulate could not be proved from the other four.

One way to prove that the Fifth Postulate could not be proved from the other four would be to exhibit a system which satisfied the first four postulates, but not the fifth — either because there were at least two parallel lines through the given point or because there were none. In the first half of the nineteenth century Lobachevsky, Riemann, and several other mathematicians finally settled the issue by showing that geometries could be constructed for both cases.

The geometry for the “no parallel lines” is easy to visualize. Take a sphere, and wherever Euclid’s postulates refer to a “line” use a great circle on that sphere; wherever they refer to a point use a pair of antipodal points (points directly opposite each other on diameter of the sphere, like the North and South Poles). Then the first four postulates will be satisfied, but the Fifth Postulate will not be.

This settled the question forever — though probably a great many people thought that mathematicians were crazy to worry about such issues when everyone knew that in the real world space was Euclidean, that is, satisfied all of Euclid’s postulates. Or at least everyone knew that until Einstein published his relativity theory, in which he used a non-Euclidean geometry and showed that “space is curved.” So you see that sometimes it is hard to see what the applications of some piece of pure mathematics will be.

The Road Map

Here is a very brief description of some of the major areas of mathematics, and a picture to show how they are related to each other. The boundaries between some of the areas are very fuzzy (for example, there are algebraic geometers and analytic number theorists). You should not take the sizes on this map too seriously — they have more to do with fitting in names than anything else.

Real and Complex Analysis:

This is the general area that includes calculus. Analysts think about what kinds of functions can be integrated and differentiated (and they have different varieties of integrals too); the functions can have real numbers or complex numbers as their variables. Analysts also think about something c alled measure theory, which suggests how to generalize the ideas of length and area. This is perhaps the single largest area of pure mathematics.

Abstract or Modern Algebra:

Algebra is concerned with objects and operations on them that are like arithmetic operations. Two of the most important kinds of objects in algebra are groups and rings. Rings are objects with two operations on them, usually called addition and multiplication: e.g., integers, real numbers, or clock arithmetic. Groups have only one operation, such as all the ways to flip around an equilateral triangle. The concepts of “commutative,” “associative,” etc. all come from algebra. Recently, algebraists have been studying boolean algebras and automata, both of which are very important in computer science.

Geometry:

You have all met Euclidean geometry in high school, and we’ve talked about non-Euclidean geometries above. There are lots of other geometries, too, many of them on curved surfaces and quite complex. And, of course, you can always do geometry in four or more dimensions.

Topology:

Topology is the study of what properties of an object stay the same as you stretch and shrink it. A standing joke is that a topologist is someone who dunks his coffee cup in his donut, because a coffee cup and a donut are “topologically equivalent” (see the illustration below) Topologists ask how many holes of what dimension an object has.

Number Theory:

Number theory is the study of positive integers and modular arithmetic (“clock a rithmetic” is modular arithmetic with the base 12).You have already learned some important theorems of number theory — for example, that there are an infinite number of primes, and that every positive integer may be factored into a product of primes, and this factorization is unique aside from the order. One of the most famous unsolved problems in mathematics comes from number theory. In geometry you saw “pythagorean triples” — numbers like 3, 4, 5 where

The question is, can similar triples be found for an exponent higher than 2? “Fermat’s Last Theorem” says that for n >2 it is not possible to find x, y, z all non-zero with

This is called “Fermat’s Last Theorem” because he wrote in the margin of a book that he had a very interesting proof of this — and shortly thereafter he died. Mathematicians have been trying to find his proof for the last 300 years!

Linear Algebra

This is another area that you already know something about — it is solving systems of m linear equations with n unknowns (“linear” means no square terms). All the equations are things like

3x + 5y – 7z = 9 .

Linear algebra is used in many other areas of mathematics, and especially in mathematical modeling. Sometimes there are hundreds of equations, and then the problem becomes how to solve the system of equations without letting round-off errors ruin you.

Logic and Set Theory:

Set theory is important for many areas of mathematics, including analysis, probability, and topology. You have done some set theory when you drew Venn diagrams and proved such theorems as

(A U B) U C = A U (B U C)Logic is very important in the foundations of computer science — in complexity theory, which we mentioned earlier, and in recursive function theory. A recursive function is one which is defined in terms of itself. For example, the factorial function n! = n 4 (n -1) 4 … 3 4 2 4 1may be defined as

n! = n 4 (n – 1)!Recursive functions are very important in some computer languages such as Pascal.

Differential Equations and Numerical Analysis:

Many people properly think of these as part of analysis. Differential equations relate a function’s value to the rate at which the function is changing, or they relate the rates at which two or more functions are changing. If you’ve studied calculus, then you’ve come across the differential equation

where c is a constant and p(t) is a function.Depending on your interests, you can use this equation to describe either population growth — p(t) is the population at time t — or radioactive decay — p(t) is the amount of radioactive material left at time t. Numerical analysis is concerned with approximating functions whose values cannot be found exactly. Frequently these functions are solutions to differential equations, but numerical analysis can also, for example, help us find the square root of 2 accurately and efficiently.

Discrete Mathematics:

This area is really several areas that have a similar feel to them. It includes combinatorics (How many ways can you form three letter words from all the letters? How many ways can you get a 7-2 decision on the Supreme Court?) and graph theory. (Can you draw Figure A without picking up your pencil?).How many colors do you need to color all the possible maps of the world if any two countries that have more than a point of border in common are given different colors? The map in Figure B shows four countries, all of which border each other, so this map requires at least four colors. It has been known for a long time that five colors would always be sufficient.

This is known as the “four color problem.” A few years ago two mathematicians, with the help of a computer, showed that four colors are always enough.

Classical Applied Mathematics:

This field is concerned with applying mathematics to solving problems in physics and engineering. Applied mathematics draws heavily on analysis, differential equations, and linear algebra to solve problems in statics (making sure the beams are strong enough to hold up the bridge), mechanics (describing how a pendulum moves), electricity and magnetism (calculating the magnetic field induced by an electric current), and many other areas. Indeed, most physicists know almost as much mathematics as they do physics, so intimately are the two fields entwined.

Operations Research:

This area of mathematics came into existence during World War II. It uses mathematics (primarily linear algebra and probability/statistics) to solve problems that occur in business and managerial situations.Game theory, a part of operations research, was used to select a strategy for the Battle of Midway, a turning point in the Pacific arena during World War II. The U.S. Navy was on one side of Midway Island, and the Japanese Navy on the other. We calculated our probability of winning in the four cases of our going north of the island or south of it, and the same for the Japanese. Game theory was then used to select the winning strategy.

Linear programming (the other major part of operations research) was also used at the start of the war to determine how many of our pilots (we had very few) should be used to fight in the war and how many to train other pilots.

Computer Science:

An adequate description of all the parts of computer science would take almost as many words as you’ve read already — and we’ll spare you that. Suffice it to say that computer science also overlaps with almost all of mathematics. Algebraic objects are used to build programs that translate computer languages, graph theory is used to design computer networks, statistics is used to improve the efficiency of large computer systems, and differential equations are used in the design of computer hardware.

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Sources of Information on Careers in
Mathematics and Computer Science:

The Actuarial Profession, A description of the responsibilities, training, and opportunities of the actuary in the United States and Canada. Society of Actuaries, 500 Park Blvd. Suite #44D, Itasca, IL 60143, (312) 773-3010. FREE.Careers for Women in Mathematics, Association for Women in Mathematics, Box 178, Wellesley College, Wellesley, MA 02181, (617) 235-0320 ext. 2643. Up to 10 copies FREE.

Careers in Applied Mathematics and Profiles in Applied Mathematics, Society for Industrial and Applied Mathematics, 117 South 17th St, Suite 01400, Philadelphia, PA 19103. (215) 564-2929. 1 copy FREE.

Careers in Mathematics, American Mathematical Society, PO Box 6Z48, Providence, RI 02940, (401) 272-9500. 1 copy FREE.

Careers in Operations Research and Educational Program in Operations Research, Operations Research Society of America, Mount Royal & Guilford Ave, Baltimore, MD 21202, (301) 528-4146. Up to 3 copies FREE.

Careers in Statistics and Women and Statistics, American Statistical Association, 1429 Duke St, Alexandria. VA 22314, (703) 684-1221. Up to 25 FREE.

I’m Madly in Love with Electricity — and Other Comments About Their Work by Women in Science and Engineering. Career booklet with quotes by over 70 women mathematicians, scientists, and engineers. Very lively. $2.00 plus 50″ postage from EQUALS, Lawrence Hall of Science, University of California, Berkeley, CA 94720, (415) 642-1823. EQUALS also puts out career activities (math-oriented activities) kits. Write or call for more information.

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Many other aspects of modem mathematics could be included in this pamphlet. We hope it has given you some sense of the scope and power of mathematics, and of all the interesting, intriguing, and varied problems that are out there, waiting for YOU to solve them.
Margaret Menzin
Robert Goldman
Simmons College, Boston MA
November, 1987
updated Feb. 1993 Available from:
Association for Women in Mathematics
4114 Computer and Space Sciences Bldg.
University of Maryland
College Park, MD 20742-2461
301/405-7892
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