aaronsil AT math.upenn.edu
dmytroy AT math.upenn.edu
The course will be varied, involving theory, computations, and examples. It is open to undergraduate students, both to math majors and others but is a serious, proof-intensive, math major-level course. Math 502 is a more advanced course in algebra than Math 370, and that course is open both to undergraduate and graduate students. That said, Math 370 is a math major course, and a major purpose of the course is to develop a sense of rigor and proof-writing skills. Problem sets will not merely recap what happens in class; they will require new ideas, creativity, patience, and time. A significant amount of effort is expected to be put into each week's problem sets. Do not fret though! They are designed to be challenging, but doable. Students are strongly encouraged to look at each problem set immediately when it is assigned, so that they may ruminate on it for as long as possible. There are ten problems assigned each week (with one or two exceptions), and problem sets are due Friday, at the earliest the week after they are given to give you ample time to complete each problem set. Homework is assigned, with due dates, on the syllabus.
Whenever I received a problem set, I looked at it immediately so my subconscious could begin to process it. I would usually start a few problems which looked easy, and left the rest of them for a bit later. I came back to them and realized that most were pretty easy, and some were actually difficult. I did the easiest ones when I could and pondered the difficult ones. Only at this point would it help me to talk about the problems with someone else if I still had problems. For once I had thought about the problem enough, I was able to see what was relatively easy, and where the true nature of the problem lay. Without that sort of distance, all steps seemed about the same level of difficulty, and it was difficult for me to see through to the "point" of the problem.
This is how I also do research mathematics. It may seem to you now that there is a lot of "material" to learn in this class. However, I would say that the "information" that we learn can be neatly summarized in a page or two — if you feel like you are memorizing a lot in this course, you are probably doing something wrong. The seeming bounty of material can often be illustrated in a compact and useful way in carefully chosen examples. Learning happens with the time you spend thinking about the objects we deal with to familiarize yourself with them and understand how they work. The theorems that are important (for instance, the Sylow Theorems and the Jordan Normal Form) stand out as syntheses of ideas and intuitions you should try to develop throughout the course.