by Bradley Knockel
I recommend taking the college classes you enjoy. More specifically, take the classes that you would enjoy working hard for. If what you enjoy is not very practical, minor in it, or consider minoring in something that is practical—or at least take a few practical classes! Luckily, technical majors like math and physics serve an important role in modern economies and can be very practical.
If you study math, you will first flesh out calculus: vector (3D) calculus, complex analysis, and differential equations. Calculus is the language of nature. For example, one of the first things learned in physics class is velocity, which can only be described with calculus—vector calculus to be exact. Complex analysis then explores what calculus looks like when complex numbers (when √-1 exists) are allowed to be independent variables. Complex numbers are governed by beautiful theorems and have interesting applications. Differential equations mix algebra and calculus (for example, y′′ + y′ + y = x2 is a differential equation that can be solved giving y as a function of x). Many of the fundamental laws of nature are differential equations whose solutions describe reality!
If you are more interested in applied math, you will need to get good with computers. You may learn advanced numerical techniques (that is, how to use computers to do math) to solve important problems such as understanding complex features of solutions to differential equations such as shock waves. Or, you may learn the very useful field of statistics, which can allow you to study the quickly growing fields of data science, big data, and data analytics, which heavily use statistics and computers.
If applied math does not sound interesting and you are more interested in what is called pure math, get ready to work on proofs of theorems (and to sell your calculator because you won't be calculating anything). You will ponder the deepest logic to infinity and beyond. In a field known as real analysis, you will construct what numbers and algebra are using axioms and definitions, from which theorems can be proven using the strict rules of deductive reasoning. You will do this by writing strange symbols in what resembles paragraphs more than numbers and calculations. From there, all sorts of interesting definitions can be made to extend math. One of the most useful definitions have been matrices, from which the very important field of linear algebra arises. People also define fascinating logical structures such as sets, categories, rings, groups, manifolds, etc. One cannot describe the curved spacetime of general relativity without this type of math!
If you study physics, you will gain a progressively deeper and more complete understanding of the four core subjects: classical mechanics, quantum mechanics, statistical mechanics (thermodynamics on steroids), and electrodynamics. Relativity is often included in any of these four subjects, and it can be studied as a separate topic. All of this will require much math to fully grasp, but, once grasped, you will know the fundamental laws that describe nature, laws that often go against common sense!
You can then build on this foundation to learn optics (lasers!), particle physics, solid-state physics, quantum information (quantum computers!), biophysics, geophysics, astrophysics, cosmology, etc.
In each of these fields, you can be either an experimentalist who obtains and analyzes data (often using cool "toys"/technology) or a theorist who tries to explain the data using mathematical models and calculations (often using sophisticated computers). Regardless, being knowledgable in applied math should be very helpful (however, if you are extremely smart and want to be a string theorist who must invent new math, then learning pure math will be very helpful).
At the end of the day, physics is about understanding how things work. You will learn ways of analytically thinking about the world. These ways of thinking will seem unintuitive at first, but they will eventually be powerful tools.
If any of this sounds fascinating, then study it! I personally knew that I couldn't die happy without understand various things I mentioned above. Just hearing about them would have made me curious about them! I had no choice but to follow what I wanted to do after taking my first physics classes in college. It will certainly be challenging, but my years in college were the best years of my life. CNM can provide a great foundation to go to UNM (or any other university) to get a degree in physics or math!
After your degree, simply follow the opportunities. I have learned a lot about myself by seeing which opportunities I let pass by and which opportunities I pursue. Trying to sit down and think about what I want to do is not productive compared to exploring opportunities. You will probably end up in research, teaching, and/or industry, especially if you very wisely get experience in these areas while you are a student. A bachelor's in physics shows that you know general technical things such as math and problem solving, so you can easily branch out from there!
If no opportunities arise, alter your course (for the near future at least). You will always know you tried, and you will always have the experience to remember. We often end up somewhere we never expected, and we never would have begun to explore this new thing if the old thing hadn't been already explored.