Should I be worried that I am doing well in analysis and not well in algebra? [closed]

I attend a mostly liberal arts focused university, in which I was able to test out of an “Introduction to Proofs” class and directly into “Advanced Calculus 1” (Introductory Analysis I) and I loved it. I did great in the class. I was not very mathematically mature at the time, but I studied hard and started to outpace many of the senior level students who had a least a good year or more of experience than me. Furthermore, the professor teaching the course was apparently known to be particularly difficult, but I loved his course. I enjoyed the challenge and wound up with a B+, the 2nd highest grade given in the class. I took Advanced Calculus 2 and loved it even more. The professor even suggested that I take a graduate complex analysis course in the Fall. (Just a side note here, the undergraduate complex analysis course at my school does not use any theorem’s or proofs. The grad version is similar to say, an honors undergraduate course at more traditional math program.) I took this as a high complement, and a verification that I was in fact doing well. I know I am not very deep into analysis, but I feel comfortable with the subject, even with the more abstract parts.

However, I am really struggling with abstract algebra. I can’t understand why. I study the material really hard. I am doing better than most in the class, and I am maintaining a solid B average, but I really have trouble thinking about algebra like I do analysis. I feel like I am mostly just regurgitating theorems and techniques just to pass the exams. I know I can pass the course, but I also know that this mindless memorization will eventually come back to haunt me later on in my mathematical career. Algebra is truly one of the pillars of math which is why I really feel terrible that I don’t understand it.

Is this a sign that I simply don’t have what it takes to succeed in math? I would love to go on to graduate school and hopefully get a PhD. In fact, a professor actually said to me, “I think it would be a shame if you didn’t go to grad school for math.” He told me that before I took algebra, but now I feel like my world is “crashing down” in a sense. Before I was a “good” student; now, I feel like a zombie in the back of the room. Any input is greatly appreciated, but what I really want to know is, has this happened to anyone who has gone on to succeed in a Ph.D math program?

Answer

I believe that I may be of some consolation.

I had a very similar experience to you. I started doing “serious” math when I was a senior in high school. I thought I was very smart because I was studying what I thought was advanced analysis–baby Rudin. My ego took a hit when I reached college and realized that while I had a knack for analysis and point-set topology, I could not get this algebra thing down! I just didn’t understand what all these sets and maps had to do with anything. I didn’t understand why they were useful, and even when I finally did grasp a concept I was entirely impotent when it came to those numbered terrors at the end of chapters.

I held the same fear that you do. I convinced myself that I was destined to be an analyst–I even went as far to say that I “hated” algebra (obnoxious, I know). After about a year of so, with the osmotic effect of being in algebra related classes, and studying tangentially related subjects, I started to understand, and really pick up on algebra. Two years after that (now) I would firmly place myself on the algebraic side of the bridge (if there is such a thing), even though I still enjoy me some analysis!

I think the key for me was picking up the goals and methods of algebra. It is much easier for a gifted math student to “get” analysis straight out of high-school, you have been secretly doing it for years. For the first half of Rudin while I “got it”, this was largely thanks to the ability to rely on my calculus background to get why and how we roughly approached things. There was no such helpful intuition for algebra. It was the first type of math I seriously attempted to learn that was “structural”, which was qualitative vs. quantitative. My analytic (read calculus) mind was not able to understand why it would ever be obvious to pass from ring X to its quotient, nor why we care that every finitely generated abelian group is a finite product of cyclic groups. I just didn’t understand.

But, as I said, as I progressed through more and more courses, learned more and more algebra and related subjects, things just started to click. I not only was able to understand the technical reasons why an exact sequence split, but I understood what this really means intuitively. I started forcing myself to start phrasing other parts of mathematics algebraically, to help my understanding.

The last thing I will say to you, is that you should be scared and worried. I can’t tell you how many times in my mathematical schooling I was terrified of a subject. I always thought that I would never understand Subject X or that Concept Y was just beyond me. I can tell you, with the utmost sincerity, that those subjects I was once mortified by, are the subjects I know best. The key is to take your fear that you can’t do it, that algebra is just “not your thing”, and own it. Be intrigued by this subject you can’t understand, read everything you can about it, talk to those who are now good at the subject (even though many of them may have had similar issues), and sooner than you know, by sheer force of will you will find yourself studying topics whose name would make you-right-now die of fright. Stay strong friend, you can do it.

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