# Mathematics Major

The Department of Mathematics and Statistics offers the B.A. and B.S. degrees with a major in mathematics, as well as a minor in mathematics. All of our majors have the same basic requirements, but some students may lean towards our "applied" courses and others may lean towards our "pure" courses (or both!). See below for descriptions of our applied and pure degrees. Many of our mathematics majors also earn initial teaching licensure (see our Math Education Degrees page), and we also offer a concentration in computational sciences.

##### Applied Mathematics at JMU

Applied mathematics uses mathematics to solve problems that come from the sciences, engineering, computer science, business, and industry. While the motivation is often to solve particular problems, the theory developed can be of wide-ranging interest.

Much classical applied mathematics involves developing mathematical models of a problem and solving them, often in the context of solving differential equations or optimizing some process. This dates back to the development of calculus, and continues to be important today. Here at JMU, our entire applied mathematics group uses differential equations in some way to solve problems in a wide range of disciplines.

Another important strand of applied mathematics is computational mathematics, the study of algorithms and techniques to approximate the solutions to problems that can’t be solved exactly. Many JMU faculty use computational techniques to aid them in their research.

##### Courses

One of the main reasons for the calculus sequence is to represent situations where quantities vary continuously by differential equations, and solve them. As a result, the calculus sequence (MATH 235-7) followed by a course on differential equations (MATH 238 or MATH 336) is core to any applied mathematician. An introductory course on programming and computation for mathematics (MATH 248) also provides invaluable skills going forward.

At the intermediate level, there are a variety of courses with an applied bent. Methods of Applied Calculus (MATH 337) introduces a variety of techniques building on the calculus sequence for solving differential equations and modeling physical processes. Optimization (MATH 340) introduces a variety of techniques for finding the best possible solution, with applications in the sciences, economics and the social sciences. Nonlinear Dynamics and Chaos (MATH/PHYS 341) moves beyond classical (linear) modeling and looks at some of the strange (and chaotic) behavior of many problems. Mathematical Models in Biology (MATH 342) introduces and solves problems in biology using mathematics.

At the advanced level, there are two main sequences we offer. MATH 440-1 introduce partial differential equations, their applications and solution, and the qualitative theory of ordinary differential equations. MATH 448-9 introduce a variety of topics in numerical analysis, including numerical linear algebra and the numerical solution of systems of equations and differential equations.

Various faculty also teach special topic courses in areas including acoustics, solid mechanics, visualization and computer graphics, and computational applications in geometry and discrete mathematics.

##### Careers

At one level, an applied mathematician is a problem solver in any area that requires mathematics. The website WeUseMath.org includes a long list of potential jobs for mathematicians. The Society for Industrial and Applied Mathematics (SIAM) includes a site on Careers in Applied Mathematics.

##### Pure Mathematics

Mathematicians develop abstract structures to study complex realities. For example, it is often convenient to group like items together, as when the apples in the market are placed separately from the oranges and the pears, and mathematicians developed the notion of “set” to capture this. It sometimes happens that members of one set can be paired up with members of another set, and we use the concept of “number” to describe this tendency. In our daily lives we encounter numerous examples of related quantities, such as the relationship between temperature and the time of day, or between the distance we have travelled and the time during which we have been travelling. The notion of “function” is an abstract way of describing such relationships. Sometimes it is necessary to organize and manipulate large collections of data, and “matrices” are often useful for that purpose.

The pure mathematician takes the attitude that if abstractions like sets, numbers, functions and matrices are routinely useful for studying so many aspects of our daily lives, then they are also worth studying for their own sake. History records many instances of the usefulness of this approach. When Isaac Newton sought to understand the trajectories of projectiles, he found success not by studying projectiles, but by studying continuous functions. When Einstein was working out his theory of relativity, non-Euclidean geometry, a branch of mathematics developed for reasons having nothing to do with physics, proved to be indispensable. These are just two of many possible examples.

The usefulness of pure mathematics is only part of the story, however. There is also the tremendous beauty of the subject. It is hard to imagine an object more banal than the counting numbers, yet their structure is so complex that mathematicians routinely discover novel facts about them. Right triangles are all around us, but who would suspect that the square on the hypotenuse is equal to the sum of the squares on the other two sides?

It is this combination of beauty and usefulness that explains the importance and appeal of pure mathematics.

##### Courses

Students interested in studying pure mathematics should take MATH 245 as early as possible. This course provides an introduction to proofs and to topics in discrete mathematics. From there, our offerings can roughly be grouped into those covering discrete topics on the one hand, and those covering continuous topics on the other. For those interested in discrete topics, we offer elementary number theory (MATH 310), graph theory (MATH 353), abstract and linear algebra (MATH 430, 431, 434) and geometry (MATH 475). Continuous topics include real and complex analysis (MATH 410, 411, 360), and topology (MATH 435). We also offer a course in the history of mathematics (MATH 415).

##### Careers

Training in pure mathematics can lead to job opportunities in a variety of fields. In addition to teaching and research in mathematics, there are also fields like technical writing and investment banking to consider. Mathematics is also good preparation for those entering law and medicine. The Mathematical Association of America maintains a list of possibilities.