Why Study Mathematics?
The study of mathematics can lead to a variety of exciting professional careers. Basic research, engineering, finance, business, and government service are among the opportunities open to those with mathematical training. Moreover, with the increasing importance of basic science and information technology, prospects for careers in the mathematical sciences are very good. Mathematical analysis and computational modeling are important for solving some of the most pressing problems of our time - new energy resources, climate change, risk management, epidemiology, to name a few. We must strive to maintain our technological edge; mathematical skills will be crucial to this effort.
Some more specific business positions include portfolio analysis, design studies, statistical analysis, computer simulation, software design and testing, and other areas of operations research. There are extensive opportunities for mathematics in finance, the actuarial fields, and economic forecasting.
Many laboratories, both government and private, maintain independent research staffs that include mathematicians. Their work often deals with the development of new technology, including research in basic physics and software development, as well as applied mathematics. Numerical simulation, such as weather and climate forecasting, depends heavily on the use of supercomputers.
Practical considerations aside, there is the pleasure of learning, applying, and creating mathematics. Real world issues pose problems that can be studied by formulating and analyzing mathematical models. In some cases applications may lead to new mathematics, and a new branch of the science is born. In other cases abstract theory finds unexpected practical purpose. Working on research problems is exciting; solving difficult problems successfully is, for many, satisfaction enough.
Graduate Study in Mathematics
While a career in mathematics can be very attractive, it takes time to acquire the necessary skills, particularly for basic research at the Ph.D. level. Graduate study is essential for most fields. The undergraduate course sequence provides a foundation upon which more advanced mathematics will be built. In graduate study, one or two further years of coursework completes this basic training. Thereafter, more specialized courses, often at the frontiers of research, are taken. Applied mathematics students will take courses in various application areas to acquire experience in modeling the real world, and to learn how mathematics can help with problems from the physical and biological sciences, and in finance.
The breadth and depth of work will depend on the degree level. With an M.S. degree, the student is prepared for many jobs in government, business, and industry; with the Ph.D. degree the choices are wider. Many Ph.D. mathematicians join the faculty of a university or four-year college, where they not only teach but also conduct research and publish their results in scholarly journals and books. Others take post-doctoral positions at various laboratories around the world, where work of interest to them is being done. Still others pursue careers in corporate research and management. With either an M.S. or a Ph.D., starting salaries are significantly higher than those of graduates with bachelor's degrees.
At both the M.S. and Ph.D. levels, graduate study in mathematics develops a number of important skills for solving problems suggested either by mathematics or by real world questions. Foremost is the ability to break complex issues into smaller, more manageable problems, until a model is reached which can be thoroughly studied and understood. Applied mathematics develops the art of extracting quantitative models from problems of physics, biology, engineering and economics. This ability comes from experience, such as that acquired gradually from examples studied in graduate courses.
An undergraduate student wishing to enter graduate study in mathematics should first satisfy the basic undergraduate requirements. The most essential courses are the calculus sequence (often three one-term courses and a course in advanced calculus) and a course in linear algebra. Courses in probability, statistics, and an introduction to computer science are also useful. Courses in algebra and topology can provide an introduction to more abstract mathematics. Students interested in applied mathematics will probably want to consider taking core courses from another department, such as physics, chemistry or biology. Introductory courses in ordinary and partial differential equations are useful. It is desirable to master at least one computer language.
Where possible, undergraduate students interested in applications should seek a broad scientific background. Understanding problems from the viewpoint of more than one specialty or application can help lead to a deeper mathematical understanding as well. The Courant Institute welcomes applicants with undergraduate degrees in other science fields, such as physics, biology, or engineering.