Hands down, the course I am most thankful to have taken before coming to BSM is my Discrete Mathematics course.
I’ve been really surprised at how much I am relying on the things I learned in discrete math. The only official pre-rec to BSM is having taken either Abstract Algebra or Real Analysis, and that’s really just to ensure you have learned enough math and are at a high enough mathematical level for the courses offered.
Yes, I’m very glad I took Real Analysis– I use the thought processes developed in that class all the time. Real Analysis was the first math course I took in which I genuinely struggled; it taught me how to work through confusion and persevere. It began to teach me what it actually means to do math as opposed to just learn math.
But I use the topics covered in Discrete Math every single day at BSM. After this week, I have officially done some degree of graph theory in ALL FOUR of my math classes! My friend’s response upon hearing that from me was simply: “Welcome to Hungary.” (:
For example, in Abstract Algebra, we’re studying and doing problem sets on the symmetires of graphs.
Every single bioinformatics model I’m studying is a type of graph.
The matrices I am researching are bipartite directed multigraphs.
Our game theoretic models are…yes…graphs.
I knew that Hungarians were famous for their teaching of combinatorics, but I couldn’t have anticipated the extent to which graph theory is utalized as a mathematical tool here in Hungary.
I’m finding myself doing proofs in which I need to interpret the problem as a graph, then use graph theoretic theorems to solve it! I learned how to do this in discrete math; I would really be struggling if I hadn’t taken that class. And discrete is actually not even a direct requirement for my math major back home.
Some other discrete math topics I’m so grateful to know:
- Counting, counting, counting: I use a lot of combinatorics counting arguments in abstract algebra. For example, last week we needed them to answer the question that there were, of course, 7 choose 3 divided by three times 4 choose three divided by 3 all divided by 2 unique cycle permutations in some group. I found the counting argument much more difficult than the algebra part of that problem!
- Modular Arithmetic: Many of the groups we use as examples in Abstract Algebra utilize modular arithmetic in some way. We did a short lesson on it in the beginning of the abstract course, but it was incredibly helpful to already have an understanding of the properties of the operation and to have practice adding, multiplying, finding “fractions” and using inverses in modular sets. Even though we did a small unit on modular arithmetic during this course, our homework sets require a more through understanding of modular arithmetic than was covered in class and one that I only have thanks to my discrete math course back home.
- Set Theory: In Game Theory, we are constantly using power sets, and everything is turned into a set of actions, a set of player payoffs, etc.
- Probability: My bio research professor told me that many American students he meets seem to fear probability. “It’s just some value between 0 and 1. That’s it.” In my research group we’re working with uniform probabilities– we are using lots and lots of Markov Chain Monte Carlo processes, and I wish my probability background was stronger than it is.
- LaTeX: It’s actually just coincidence that I learned to use Tex in discrete math, but I’m going to include it on this list anyway. (: I TeX all of my problem sets for Game Theory. It’s by far not required, yet by far the preferred method of receiving problem sets by my professor.
Additionally, there is this social hierarchy that exists in the math community surrounding the use of LaTeX (just see #1 on “Ten Signs a Claimed Mathematical Breakthrough is Wrong” for proof). For us, it’s an unspoken understanding that the students who regularly turn their problem sets in using TeX are the ones you’re trying to measure up to. The math world is a learn to typeset in LaTeX or be an outsider kind of place.
I am using all of my previous math knowledge in some way this semester: sequences and series sometimes come up, properties of the real numbers are important, I might (rarely…) take a derivative, matrices are great, but, I am currently thanking the math gods that I’ve had a semester of discrete mathematics.
I am still updating my BSM Tips page! Don’t forget to check it out if you’re considering spending a semester or two at BSM.