So, I took the general GRE and got 161V, 165Q, and 3.5W. I'm somewhat concerned about the writing score, so would it be worth it to retake or is this good enough to go forward?
Also, I'd like to ask for some advice on how to do well on the subject test. In particular, I'll need to have a crash course on stuff like ODEs and Calc 3, that I never really learned about (also I'm not quite the fastest person at this point in Calc 1-2 tricky questions). What score should one be aiming for, and what's a good way to decide whether to take the October one before knowing September?
Finally, here is the current state of affairs, in part if it informs the advice to the above, and also if anyone has other comments on where to look at/general advice.
I currently go to one of the "top 6" math schools. I've taken 16 math classes (equivalent of 10 semesters), one of which was a reading course. Had a B in one, A in the rest.
The people I hope to ask recommendation letters from all names in their field.
I did an REU ("apprentice") first summer, a kind of analysis camp second (basically a class where we gave the lectures ourselves), and an REU third.
I've written some expository papers (one on Sylow theorems for the apprentice REU, one on elliptic curves for the REU third year, and one on Dold-Thom for the reading course).
The hope is to work in the general area of algebra/number theory, possibly topology, though most of my background so far is in analysis (in particular, I've taken 3 grad classes in it and none in other subjects, though I am hoping to take grad AT this fall).
Current list is Harvard, Stanford, Michigan, UCLA, UIUC, and Northwestern.