Looking back at the three years I was an undergraduate student, this was one of the happiest periods of my life. For the first time I could study only the subjects I chose for as long as I wanted. Yes, the dorms were basic, the food wasn't very good, not all the courses and professors were great. We had to study hard, prepare many assignments, final exams were sometimes hard and stressful. However, for the most part I enjoyed being a student and I was good at it.
My major subject was physics, with a minor in mathematics. Very quickly though I found myself drawn to mathematics much more than physics. The main first year courses were calculus, linear algebra, mechanics and electricity. The math courses were all new and exciting material and the teachers were really good, at least I liked them. In physics we mainly repeated the high school material, much faster and going into more detail. There were no surprising laws of physics I didn't know about, the main difference was to apply the same laws in more difficult settings, and this usually meant using advanced mathematical tools.
The physics department was of course aware that the students needed tools that they were going to learn about much later if at all in regular mathematics courses. As a result there was a physics course called "mathematical methods for physicists". And so while we were getting to know the definitions of a vector space and a limit of a function in math courses, in physics we were already computing line integrals, solving ordinary and partial differential equations, and using 3-dimentional differential operators. For physics you needed to solve the technical problem, it rarely mattered if you understood what you were using, whether you knew why the solution is valid and under what conditions. In mathematics, exactly the opposite: definitions, conditions and proofs, solving a concrete problem was rarely important.
I imagine the physics teachers were frustrated sometimes, because they had to get into mathematical details for the students to really understand what was going on. There is only so much you can do with hand waving about physical principles. It was sometimes awkward to hear about math concepts from physicists, especially if it happened after we learnt about them from mathematicians. For example a lot of the material in the 3rd year course we had on quantum theory depended on graduate level algebra, it was very difficult to really understand what was going on, and I doubt the teacher could teach the subject properly.
Physics had a clear advantage over mathematics regarding analysis and functions. When dealing with physical functions we could always assume they behaved properly: continuous, differentiable to any degree, defined wherever we needed. After all in nature reality dictates that everything is defined and smooth. In mathematics nothing was given, and you could construct functions that defied logic: a continuous function that is nowhere differentiable, a curve that fills a square, a monotonous continuous function that has a derivative 0 almost everywhere but manages to increase from 0 to 1. Analysis courses were filled with mathematical monsters that could make you fail exams if you weren't careful.
In the 1980's personal computers, cell phones and the internet were in their infancy, and the main source of information was books. All students had to use the library from time to time to find additional information beyond the course textbook or for special assignments. The physics books were in the general science library, a large place that was sometimes hard to navigate in, I tried to avoid it if possible. The mathematics department had a separate library, a main hall with big tables in the middle surrounded by book shelves on the four walls. This was one of my favorite locations in the university, I liked to sit there and read while absorbing the atmosphere. All the mathematical subjects were covered, from the elementary basics to the current research, from the old to the new, clearly divided into sections. I even found there was a side room for the applied mathematics section, which turned out to be physics with less hand waving.
I spent a lot of time reading about elementary number theory, the basic theory of integer numbers. This was a subject that required no prior mathematical knowledge, I could read textbooks and understand most of the material. I already encountered prime numbers as a child, now I learned that mathematicians have been asking themselves the same questions as me: what makes a number prime, how many prime numbers are there, how can you find the prime divisors of an integer. I learned about Fermat's little theorem, the Chinese remainder theorem, and quadratic reciprocity. I also realized that in number theory some statements are easy to understand but very difficult or impossible to prove. Famous examples are the Goldbach conjecture and Fermat's last theorem (with the infamous 'margin' remark, actually proved in 1994 by Andrew Wiles). Mathematicians have been working on some of these problems for hundreds of years, and a lot of new theory was created in these efforts.
(While writing this entry I realized I don't remember how to prove quadratic reciprocity. For a few days I read up on the subject and followed this proof. It was good to be in the presence of the greatness of Euler and Gauss that contributed to the original proof. This is the kind of mathematical proof that I like, a clever way to compute something directly without using any modern and abstract structure. On the other hand such a proof does not give an answer to 'why is this true'? other than 'because that is the result of the computation'. You expect a more profound reason for such a fundamental result. Perhaps this is why mathematicians kept coming back to quadratic reciprocity, Wikipedia claims there are hundreds of published proofs.)
At first I found all math courses equally interesting and exciting, but over time I developed preferences. There were many courses related to analysis: calculus, functional analysis, differential equations, probability-measure theory. These subjects had a practical aspect when used in physics, but I found that I was less interested in the theoretical questions that were discussed. I was more drawn towards algebra, definitions of the various algebraic objects and the rich structures that are the result of these definitions.
We had three semesters of linear algebra where we studied vector spaces. In physics vectors in two or three dimensions were used all the time as quantities with size and direction, such as forces or velocities. In mathematics vector spaces were abstract, consisting of objects that could be added and also had a scalar multiplication, with scalars from any abstract field and not just the Real numbers. Using the concept of linear dependence we proved the existence of a basis, and that any basis has the same size - the dimension of the vector space. Thus any abstract vector space is identical to n-tuples of field elements. We learned about homomorphisms (=structure preserving mappings), sub-structures and quotient structures. We learned to represent homomorphisms by matrices, I was fascinated by matrix multiplication which was the first time I encountered a non-commutative operation (AB not equal to BA). There was a lot more to know, linear algebra has many practical aspects that were important for physics and other math subjects. It is still useful to me in my professional life outside the university.
There were only two more algebra courses. In field theory we learned about field extensions, Galois theory and how it is related to solutions of polynomial equations. My favorite though was group theory. To define a group you need a multiplication operation, not necessarily commutative, and that's it. The wealth of possible structures is incredible: permutation groups, matrix groups, free groups, symmetry groups, to name a few. When I learned about the Sylow theorems I got excited: if a prime power divides the size of a finite group, there exists a subgroup of this prime power size. Prime numbers were important for the structure of finite groups, wow! Groups also have a composition series, with factors which cannot be broken up further called simple groups. Commutative simple groups are just cyclic of prime size (i.e. addition modulo p), but there are also non-commutative simple groups. The classification of finite simple groups was more or less completed at the time, a joint effort of many that took decades and spans thousands of pages of published articles. One of the famous steps was the fact that a group of odd size could not be simple and non-commutative, easy to state but very difficult to prove. All this was mind-boggling for me then, still is now.
Not everything was about studying in those years. For the first time in my life I was surrounded by people with similar interests, and some became my friends. There was no television or computer or tablet or phone in my room, I read a lot, saw movies in the cinema and listened to music on a small boombox playing cassettes. One book became important to me, "Godel, Escher, Bach" by Douglas Hofstadter. My brother had this book and I tried reading it when I was in high school, but only after I learned mathematics I could really understand and enjoy it. Escher's drawings were familiar to me from home, they are not a main subject in the book but it is amusing when Hofstadter puts characters inside such an impossible reality. I really liked the use of Bach's music in the book, in particular the dialogues that illustrated musical concepts verbally (e.g. a recurring theme or a canon). I became a huge Bach fan ever since. The Godel incompleteness theorem is explained in half of the book, after learning basic set and model theory I understood most of it. The book raises many questions: is the brain a finite state machine, what is consciousness, how can machines learn, are humans more than machines since they understand Godel incompleteness. A lot has happened in the field of machine learning since then, some of the book seems archaic but still very interesting stuff. Above all, the humorous word play, all the intertwining concepts coming together in unexpected ways, I consider this book a truly great achievement.
