This past year I was accepted into two mid range phd programs in mathematics without funding. Here's what my profile looked like:

Domestic White Male

Unknown state school in the midwest

Majors: Mathematics and Philosophy

GPA: 3.93

GRE:

Q: 168 (97)

V: 167 (97)

W: 5.0 (92)

M: not taken

Interests: Analysis, Medical Imaging/Modeling, Mathematical Physics

Major Coursework:

Complex Analysis, Discrete Structures, Applied Math (survey course mostly in Fourier methods and PDEs), Mathematical Statistics.

Recommendations:

One from a top applied mathematician who was very late in sending them out, one from a professor from the previous year who is well regarded but has not had much contact with me and one from a young professor who was fresh out of post doc.

Other:

I'm 30 years old. Some former work experience as a co-op student in engineering at a national lab, dean's list, normal stuff like that. I had only completed two semesters at my current school, though. About 6 years ago I was an engineering student at a different institution with poor grades and I did not finish my degree. Also mid to high level (lots of national, some international competition) as an athletic coach working with kids, teens and adults. Some international experience as an athlete as well. I also speak basic German.

Here's how my profile has changed:

Current GPA: 3.94 (.01 difference! However, this does mean 6 more As and 1 more A- to counterbalance my poor record from the early-mid 2000s.)

GRE Subject Score: 660 (52)

Additional Coursework: Formal Logic, Real Analysis, Advanced Linear Algebra, Intro to Abstract Algebra. This summer I am doing a course in number theory and an independent study in Galois theory.

Recommendations: I am going to give the late professor a much longer lead time this time around. He has said that he wants to help me but is always extremely busy. I will also be asking (and almost certainly receiving) a rec from the professor I am doing my independent study with.

Interests in pure math have shifted away from analysis, more into algebra. In applied, medical applications (but not bioinformatics) have taken the lead over mathematical physics.

Other changes: Medalist in school's math competition, math tutor in our honors college and privately. Taking a Spanish immersion course so I can list basic spanish on there as well.

I'll be taking the GRE subject again in October and am using saylor.org and MIT's opencourseware to review older subjects. I graduate in August and will be working for a year. I'm thinking about applying to teach at a high school or community college for that time. I also plan on taking one or two graduate courses per semester.

I plan on applying all over the place (>15 apps) compared to the 8 from last time. I would prefer to go to the west coast. Can anyone let me know if the mid level UCs (Irvine, Santa Barbara, Davis) are reasonable with what I've got now, especially since when I worked at that national lab I was technically a UC employee? Any other recommendations on schools or anything I can do over the next year to boost this?

EDIT: Oh, one other thing: I have no REUs, but is a good independent study with a strong recommendation from it a decent substitute?