jlin

Personally I won't be able to have extra time for maths when studying my PhD courses so I don't think your first way of proceeding would be viable. On the other hand, without refreshing Calculus and Linear Algebra, you won't be able to fully comprehend the YouTube course you suggested. My advice would be to refresh Calculus and Linear Algebra and try your very best to understand every proof (on top of knowing how to use the Theorems in concrete questions). After that, you could start with the YouTube videos. Most likely you won't be able to finish it, but it is still better than not doing it. I would not count on the Math Camp. Typically it goes very fast (depending on your institution of course). It is very hard to refresh unless you already had solid mathematics foundation and the only thing you forgot is the statements of some theorems.

Sorry, from your previous description I thought you were from the "top of the top schools" which were inarguably those three. And I was even thinking how could you get your Honours grade so early with February entry into the program. Of course, USYD and Monash are two best schools after those three. In my opinion, if you are heading towards Top 5 then you really should do your Honours in those top three schools (and get good grades for sure). But your target is Top 30, so an Honours from USYD should be good enough.

When you say you're from an Australian top uni, you must be from one of UNSW, ANU or UniMelb. Top 30 schools would already have the expectation that the graduate measure theory in one of these schools is very tough. A pass grade for that course is for sure not a positive signal, but may not be as bad as you think. The minimum mathematics is up to real analysis. Measure theory is more "advanced" than real analysis so it does not carry too much weight to begin with. Your research experience however, will carry more weights than a single measure theory course. Your advisor is from one of the three schools ,you have a great Honours thesis at one of these schools and you're RAed for your advisor. I think you're fine.

I get to a Top30 PhD program in the US directly from New Zealand after an Honours year, so I believe I may contribute some inputs. The Honours program at my university is the only one in NZ that is comparable to the Honours at the Group of Eight. But even so, I am placed at a significant disadvantage when applying for PhD in the US. Here are my thoughts and I hope they could be useful for future applicants from Oceania; it would not be useful for those, say, in Canada or the US, as things are so much different down here. 1. I do have the anecdotal observation that fewer and fewer Honours students are directly admitted to a PhD program in the US and UK. Most of the Honours cohort would work in the industry after graduation. 2. RA opportunities in Honours year are very rare and the amount is decreasing every year. You need to reach out to many professors, and only a handful of which offer RA jobs. Perhaps a better way is to work as a full time RA after Honours graduation, and certainly in both Australia and New Zealand there are relatively more RA job opportunities outside university, although these opportunities are considered to be limited by the US standard as well. 3. Choose your Honours school wisely, because in essence you are choosing who will write your recommendation letters. Ideally the writers of your recommendation letters should come from top US schools, but even within the Group of Eight this is not guaranteed. Try to locate the best professor from your Honours school and do the Honours thesis with him/her. 4. In my view, the best Honours school in the region is UNSW. In fact, I think one student from UNSW get to MIT this year (from the Profiles and Results 2019 page, if I'm not mistaken). I'm not entirely surprised by that, because UNSW has the best professors in the region, they have excellent RA opportunities, and their Honours program has always been rigorous. Other than UNSW, I would recommend ANU and UniMelb (maybe Monash and UniSyd marginally), but I think they are not at the same level as UNSW. Of course, this is only my subjective opinion, and is open to debate. 5. If unfortunately, you cannot secure a RA job because of limited availability and crappy quality, then perhaps the only way to get to a top PhD program is to try your best in Honours thesis, and become a mathematician. There are many threads discussing the ideal preparation you need to get to a top PhD program. In general, a successful applicant should have mathematical preparation up to real analysis, with extensive RA experience and excellent thesis in economics. However, if you cannot have RA experience, then you need to do more advanced maths courses after real analysis, and of course get good grades. Such remedy might be able to get you to Top 30, but for sure not Top 10. Without sufficient RA experience, if you only have an Honours thesis and study maths up to real analysis, you will be very lucky to be admitted to a Top 50 school. But of course, not everyone could do advanced maths, and they are not as attractive as RA experiences in the eyes of top PhD schools...