# 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. Many of these courses fall into categories of algebra or analysis. The distinction between algebra and analysis is instructive when trying to understand the relationships between various upper division courses.

###### Algebra 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. Algebra courses offered include

- Linear Algebra (MATH 211)
- Abstract Algebra (MATH 307)
- Graph Theory and Applications (MATH 332)
- Linear Algebra II (MATH 408)
- Number Theory (MATH 412)
- Abstract Algebra, II (MATH 430)
- Combinatorics (MATH 432)

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. Number theory is used to secure financial records across the internet. Combinatorics (MATH 432) and Graph Theory (MATH 332) are critical tools in operations research and computer science.

###### Analysis Courses

"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. Other analysis courses offered include

- Elementary Analysis (MATH 311)
- Intermediate Analysis (MATH 403)
- Complex Analysis (MATH 405)
- Functional Analysis (MATH 428)

###### Specialized Courses

Differential Equations (MATH 302) introduces a branch of analysis that is of enormous importance in applied mathematics.

Applied Statistics (MATH 351) introduces the basic concepts and techniques used in analyzing data collected from scientific experiments. It is intended to precede Probability (MATH 401) and Mathematical Statistics (MATH 452).

Operations Research (MATH 323 and MATH 424) introduces mathematical techniques for solving a variety of applied problems relating to the optimal allocation of scarce resources.

Numerical Analysis (MATH 413-414) is concerned with the mathematics of computation and the numerical algorithms that computers use to calculate various quantities of interest.

Topics in Geometry (MATH 416) is a modern geometry course whose content changes from one year to the next.

Vector Calculus (MATH 417) is a more thorough study of various topics introduced in Multivariable Calculus (MATH 212).

Topology (MATH 426) 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 Elementary Analysis (MATH 311). It may also include more geometric ideas such as the use of group theory in classifying surfaces in Euclidean spaces.