# Discussion of Upper Division Courses

The upper division mathematics curriculum at William & Mary attempts to strike a balance between courses that provide a rigorous introduction to the fundamental concepts of modern mathematics and courses that study how mathematics is used to solve interesting problems in other disciplines. Both are important! Most mathematical theory was originally motivated by the desire to solve various applied problems, and solving new applied problems often requires developing new theory. Pure mathematicians are generally pleased when their work is found useful in other disciplines, and applied mathematicians need to know a great deal of mathematical theory.

Traditional mathematics majors at William & Mary are required to take Math 307 (Abstract Algebra) and Math 311 (Elementary Analysis), two courses that introduce concepts that are central to the study of modern mathematics. Majors in the applied track must complete at least one of these two courses. The distinction between algebra and analysis is instructive when trying to understand the interrelationships between various upper division courses.

"Algebra" is a term for mathematics that studies how mathematical objects can be combined, subject to precise operational rules. The objects of study might be real numbers, or complex numbers, or vectors, or matrices, or functions, and the operations might be adding and multiplying those numbers, vectors or matrices, or combining functions via composition. Math 211 (Linear Algebra) will be the student's first collegiate exposure to a branch of algebra. After completing Math 307 (Abstract Algebra), students will have a better idea of what modern algebra studies and will be able to see how high school algebra is a precursor of modern algebra. Other algebra courses offered by the department include Math 408 (Linear Algebra II), Math 412 (Number Theory), and Math 430 (Abstract Algebra, II). Although algebra is usually regarded as a branch of pure mathematics, it has many important applications. Linear algebra is the basic tool for solving systems of equations, algebraic or differential. Group theory (studied in Math 307 and Math 430) lies at the foundations of cosmology, coding theory, and crystallography. Combinatorics and graph theory (studied in Math 432) are critical tools in operations research and computer science. Number theory is used to secure financial records across the internet.

"Analysis" is the term used to describe those parts of mathematics that study continuously changing objects, often involving limits, functions, and derivatives of various kinds. Calculus is probably a student's first encounter with analysis. Math 311 (Elementary Analysis) studies the theory of calculus. Math 403 (Intermediate Analysis) looks at related concepts, in more general contexts. Other analysis courses develop aspects of calculus in the context of complex numbers (Math 405: Complex Analysis) and in more general situations such as Hilbert spaces (Math 428: Functional Analysis). Analysis is of fundamental importance in both pure and applied mathematics.

Math 307 and Math 311 were designed to prepare students to take 400-level courses. As a general rule, students should take Math 307 and Math 311 before progressing to 400-level courses. Notice, however, that students who opt for the applied mathematics track are only required to take one of these courses. Although we hope that all students of applied mathematics will elect to take both Math 307 and Math 311, we recognize that not all will. Which of these courses is more important for you will depend on what you choose to study. For example, a student who plans to take Math 401 (Probability) should definitely take Math 311, but might not take Math 307. Please consult with the mathemathics faculty to decide which courses will best suit your needs.

Other 300-level courses include Math 302, 351, and 323. Math 302 (Differential Equations) introduces a branch of analysis that is of enormous importance in applied mathematics. Math 351 (Applied Statistics) introduces the basic concepts and techniques used in analyzing data collected from scientific experiments. It is intended to precede Math 401 (Probability) and Math 452 (Mathematical Statistics). Math 323 - 424 (Operations Research) introduces mathematical techniques for solving a variety of applied problems, e.g. the optimal allocation of scarce resources.

Other 400-level courses not mentioned above include Math 413-414, 416, 417, and 426. Math 413-414 (Numerical Analysis) is concerned with the mathematics of computation and the numerical algorithms that computers use to calculate various quantities of interest. Math 416 (Topics in Geometry) is a modern geometry course whose content changes from one year to the next. Classical geometry studied shapes in the plane and in three dimensional space. Today the term ``geometry'' describes the study of more general shapes and spaces, ranging from geometries in finite sets, graphs, properties of convex sets in n-dimensional spaces, and the topology of abstract spaces. Math 417 (Vector Calculus) is a more through study of various topics introduced in Math 212 (Multivariable Calculus). Math 426 (Topology) studies the kinds of spaces in which most of analysis occurs and gives a far more general treatment of some of the topics found in Math 311 (Elementary Analysis). It may also include more geometric ideas such as the use of group theory in classifying surfaces in Euclidean spaces.