ucbalumnus,
Thanks for the information. I have read both UCB and UMich’s pure math major requirements and course catalogs and found them startlingly different in style.
At UMich, one may take the Honors Math Sequence 295-296-395-396 from freshman through sophomore year. As long as one is familiar to and passionate for pure math’s abstract theories and proofs, in two years one can obtain a firm knowledge of the entire undergraduate real analysis, linear algebra (very theoretical and advanced) and get a taste of various other topics like differentiable manifolds. With the sufficient knowledge from 295 and 296, one can study things like differential geometry and abstract algebra concurrently with 395 and 396. After that, the 400 and 500-level courses seems to provide a very smooth transition into graduate study with their sufficient difficulty and breadth of materials.
In contrast, at UCB, with my AP credit, I still need to spend my entire freshman year on Math 53, 54 and 55, which seem to focus a lot on computational aspects and lacks the rigor and abstractness of modern math that I love. Only after getting through them could one enter the upper-division courses, but it seems to me that most of these 100-level courses are not as satisfactory as courses in UMich on the same topics. For example, in UCB one has Math 113 (Intro to Abstract Algebra) for undergraduate abstract algebra, but from the course description it seems that it only covers the most rudimentary aspects, without topics like Sylow’s Theorems, module theory and Galois theory at all (which are all must-know subjects for graduate math study). And after that, the next course on abstract algebra would be the graduate course Math 250A, which is a very abstract and hard treatment on the topic; the leap of difficulty between these two courses just seems a little large.
I think one particular reason why the UMich system seems more appealing to me is my previous experience in mathematics. Although I haven’t formally taken any math courses in university, I self-studied from real analysis and linear algebra to advanced probability theory and random processes in high school (I finished a paper on stochastic processes). I think I’m already very used to theoretical and rigorous math, as well as reading long and abstract proofs and constructing proofs. In contrast, I really don’t quite like computational things (e.g. indefinite integrals). However, my knowledge and problem-solving skills are still not strong enough to directly delve into things like differentiable geometry and complex analysis, so I still need to consolidate them through some courses, but as fast as possible. For these reasons, the entire UMich course catalog seems very suitable for me while the UCB system seems a little dull and unsatisfactory.
Anyway I don’t mean to be biased towards UMich at all, only giving my thoughts honestly. Am I right about Berkeley’s math? UCLA’s math major requirements and course catalog seems quite similar to Berkeley’s. Thank you so much.