This question is interesting to me because it reflects several aspects of our cultural understanding of mathematics. The average person in our society has very little knowledge of what math looks like after high school. We all live through the progression from Algebra I to Geometry to Algebra II. Some take a class called Pre-calculus or Trigonometry or Advanced Functions. Even fewer make it through Calculus in high school. In any event, the high school math student lives in a world where Calculus is the pinnacle of all mathematics. Only the smartest seniors take Calculus. How could there be math that is harder than Calculus? What would that math even look like?

The reality is that Calculus is just one more step in a long progression of subjects that together make up the body of modern mathematical knowledge. These subjects are usually taken in the designated order because understanding the concepts taught in one course requires mastery of the concepts and techniques taught in all of the preceding courses. You have to have mastered the tools from Geometry and Algebra II in order to have any hope of really understanding Pre-calculus. Similarly, you have to have completed a couple semesters of calculus before jumping into Differential Equations, which often follows Calculus.

Below is a diagram laying out the progression of math courses that a typical math major will take before finishing college. Some schools will offer additional classes or call these ones by different names, but the basics are shown below. Most graduate schools offer deeper versions of each of the top rung classes listed here, as well as very specific research courses that delve into narrow corners of Algebra, Analysis, and Topology.

Linear Algebra is the study of vector spaces, which is a fancy name for the usual 1-, 2-, and 3- dimensional spaces that we learn about in high school. We discuss the properties inherent to such spaces, and how they can be generalized to higher dimensions and new types of "vectors". The tools of the trade are matrices. We use matrices to efficiently solve very large systems of equations and to create linear approximations for more complicated functions. Computer programmers use matrices for numerous applications, including most three dimensional animations. Any animation in which you see the scenery move as if you are turning, looking up, etc. is using matrices to simulate those changes in perspective.

Differential Equations is the intersection of Calculus and the solving of equations. The equations that we learn to solve in differential equations have variables that themselves represent functions or the rates of change of functions.

Consider a 50 gallon tank full of pure water. Suppose that we open a spigot that pours into the top of the tank a mixture that is 10 % salt by volume, at a rate of 5 gallons per minute. From a second spigot at the bottom of the tank we extract the salt mixture at a rate of 5 gallons per minute. The volume in the tank remains constant, but the salt content changes with time. How much salt is in the tank after 10 minutes? After 50?

Questions such as these are very easy to ask, but very difficult to answer without a solid understanding of calculus. Differential equations are the bread and butter of modern engineering. Without the tools and strategies learned in this course we could have no fancy bridges, jet airplanes, or small hand-held electronic devices.

Proof-writing is the course in which students learn the art of writing truly rigorous math proofs. While the math content covered in a proof-writing course will change from school to school, the emphasis is always on helping students make the transition from computation-oriented courses such as Calculus and Differential Equations to the proof-based nature of the higher level courses. This course is generally where students first encounter math research in any meaningful way.

Abstract Algebra is the study of number systems and how they work. We discuss the structures that different types of number systems must have in order to to be consistent. We also look at exotic examples of number systems that behave very strangely. For instance, we might examine a number system in which the multiplication operation fails to be commutative.

Real Analysis is the technical study of the details behind Calculus. So yes, this is like really hard calculus. In this course we develop rigorous proofs of the concepts of limits, derivatives, and integrals. We then develop versions of these concepts for more abstract number systems.

Complex Analysis brings the same tools of limits, derivatives, and integrals to bear in the study of the complex numbers. This is also like really hard calculus. Complex calculus.

Topology is the study of shapes when we remove the notion of distance. Put another way, topology studies the properties of shapes that remain when we make them of rubber that we can stretch and smash to our hearts content. We also define in very rigorous terms what we mean by geometric notions such as open and closed sets, connectedness, compactness, etc. We then have fun applying all of these notions to exotic spaces where things do not turn out to work like we might expect them to.

Non-Euclidean Geometry is the study of geometric structures in which our basic postulates from Euclidean geometry are tweaked slightly. Exotic spaces result, and many of the theorems that we take for granted in Euclidean geometry begin to fail. For instance, we no longer get the sum of angle in a triangle adding up to 180 degrees.