What are the differences between the math majors? And what are their benefits?

Dear Carlos,

Please allow me to assume you meant differences in math area specialization. Now, the actual answer depends on how things are set up at the university you are attending. I have worked in mathematical physics and computational mathematics as well know a number of people in applied mathematics, so I can say a few things from my experience. (Your mileage may vary.)

• Pure mathematics, is the study of fundamental problems in mathematics. Some examples are functional analysis theory, group theory, and number theory. As the name implies, the focus is not on applications but rather on finding new properties (theorems) of mathematics in the general sense. Mathematicians working in pure mathematics, in my experience, generally have a stronger ability to prove theorems. Having said that, there are plenty of applied mathematicians who are extremely good as proofs.


• Applied mathematics is, as the name implies, apply mathematics - usually something advanced to some other topics. For example, mathematical finance. Check out the website of the math department of my alma mater, https://science.rpi.edu/mathematical-sciences/research for examples. You will see all of the research topics there are of the applied math sort. Having an applied math background is useful in that from a pure industry point of view, an applied field may be more lucrative than that of pure math.


• One could think of computational mathematics as a subtopic of applied mathematics. (This was my Ph.D. topic, though the department I was in was Computer Science.) It is also known as numerical analysis. The idea is to develop or refine computational methods of solving equations or systems. Such development is based on mathematical principles and typically includes a mathematical proof that what one developed or refines works. For example, one may develop a new method to solve equations that determine how fluid moves about. Since this sits at the intersection of computer science and mathematics and whatever application one is applying the method to, one would have a breadth of knowledge that few will have.


• Mathematical physics can also be considered a subtopic of applied mathematics. For example, studying the mathematics of fluid dynamics was something I used to do some number of years ago. Another example is elementary particle physics theory, where there is a lot of advanced mathematics one studies to tackle the expansion of the current theory. One straddles the worlds of physics and math, so there is an additional perspective to be had. However, I will say from my own experiences, this is in some sense the least applied of the applied mathematics fields, unless one works closely with experimental physicists.


• Statistics and probability is something to consider. This is very applicable in a number of settings, such as actuary and data mining.

There are many other subtopics of mathematics. Please see https://en.wikipedia.org/wiki/Lists_of_mathematics_topics. They all have fun problems to work on. I think in the end, it boils down to what kind of mathematics you like to do. Best wishes!