# Detailed study guides for mathematical subjects from undergraduate courses?

Several years ago, I found this course schedule from Cambridge. I also found some synopses from Oxford here$[1]$. They have been very useful because they contain complete course guides, for example (Cambridge):

Notice that it is very detailed, it has the total number of lectures, the number inside brackets is the (approximate) number of lectures to finish the contents in each paragraph. There are also very good book recomendations with a cross to mark some book which is particularly fit for the course.

Aside from that, the only other material I found that looks similar is Garrity's All the Mathematics You Missed: But Need to Know for Graduate School, it doesn't have all the details but it has a nice conversation about the subjects. I also found the following website: How to Become a Pure Mathematician (or Statistician).

I found these guides to be extremely useful.

• Do you know more universities that provide such useful syllabi/synopses/schedules? I have been looking for some time, but it seems that Oxford/Cambridge is really unique with respect to this: Some of them just give the name of one book and doesn't detail it, other may have something like this but it's restricted for official students.

• Do you know other books/websites/etc such as the ones I mentioned?

$[1]:$ I am a bit confused by the bureaucratic usage of the words "syllabus", "synopses", "schedules", etc. When I first searched for it, the name of the document was "syllabus", now it is "schedule".

Do you know more universities that provide such useful syllabi/synopses/schedules?

As you said, these are usually restricted to enrolled students, or - more commonly - they don't seem to exist at all. If anything, it's not clear to me why Cambridge produces these, at least for Part IA courses, since I'm fairly sure most e.g. introductory analysis courses will cover roughly the same material; these might be useful for the course lecturers, or perhaps for students who choose to avoid the lectures / lecture notes entirely while learning, but not much else.

Are you aware that many Cambridge lecturers actually provide their own notes for free online? These are variable in quality, and not always easy to find, but they exist and are often very good. One that sticks in my mind is Keith Carne's Geometry and Groups notes (PDF): not only does his contents page give a breakdown of what's covered in each lecture, but it even gives a breakdown of what (almost) every individual theorem means!

Do you know other books/websites/etc such as the ones I mentioned?

You might find a few things on MIT OpenCourseWare - e.g. reading off the titles here.

Also see my note at the end of this answer.

I am a bit confused by the bureaucratic usage of the words "syllabus", "synopses", "schedules", etc.

You are right to call it bureaucratic. There is no standardised usage.

Here's a question from me to you: why do you want these? Specifically, why do you want guides produced by universities? After all, this is the purpose that textbooks are designed to serve - and textbooks are written by the same people who teach at universities! So, by avoiding textbooks, it seems to me that you're missing out on quite a lot.

"They're expensive" won't wash, either: Amazon will usually give free previews of the contents pages of at least some modern textbooks. For instance, here is Burkill's Analysis book - the one with a cross next to it in your picture. The contents pages are freely viewable, and about as detailed as the Cambridge syllabus you've posted above. (This is hardly surprising, given that Burkill was at Cambridge when he wrote it.)

Similarly, many universities don't produce syllabi precisely because their lecturers will be working from an easily available textbook. This textbook will sometimes even be written by one of the lecturers, but will almost always be written by a trusted lecturer at some university. It will usually have developed out of a course that the lecturer has taught in the past, too.

• I have found the lecture notes from Oxford, there is a big file somewhere (easily findable). I'm not necessarily looking for something produced by universities, I just pointed that it seems they produce it and if there is more, I'd like to see. I have also already noticed that they point out a lot of books by professors working there, but I don't think this is a huge problem: A lot of the books are good. – Billy Rubina Jun 27 '18 at 2:23
• "[...] it's not clear to me why Cambridge produces these, [...]" Two reasons: firstly, like any venerable institution, because that's how things have been done at Cambridge for a long time, so why change? Secondly, this allows the lecture much more freedom: because the schedules are minimal for lecturing and maximal for examining, the lecturer may cover the material however they want (within reason) and lecture whatever else they want: it is strictly their version of the course, not some textbook or other's. (1/2) – Chappers Jun 27 '18 at 13:00
• In my first year, I had lecturers who discussed the transcendentality of $e$ (in Numbers and Sets), and nonmeasurable sets (in Probability). In second year the Complex Analysis lecturer discussed some ideas he had about proving the Jordan Curve Theorem using groupoids (I don't think it worked, but that wasn't the point). In Part II Further Complex Methods, the lecturer explained some of his own research on integral transforms. (Of course, sometimes this doesn't stop lecturers setting inappropriate exam questions anyway.) (2/2) – Chappers Jun 27 '18 at 13:01
• @Chappers: I don't think I follow either of your comments. On the second: you're surely aware that lecturers at other institutions have exactly this freedom too? And in any case, this doesn't really address the question of why the university publishes these schedules; of course there always is a schedule, it's just normally kept inside the lecturer's mind or on a webpage available to students, not published online and en masse in the Reporter. I suspect the real answer is the one you first suggested - "we've always done x" - but I don't find that a convincing reason to do anything... – Billy Jun 27 '18 at 14:13
• Convincing or not, it's quite a common reason. And the Schedules aren't published in the Reporter; the Lecture List is, and the Class Lists, but not the Schedules. They're distributed to undergraduates, and made available on the Faculty website, as past Tripos papers are, and the Example Sheets are made available on the departments' websites. Some lecturers do guard their printed notes rather more jealously, often because people from other institutions have lifted from them without attribution (i.e. plagiarised them). – Chappers Jun 27 '18 at 14:54