Improving my PhD application for math programs 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 (organ/system modeling and imaging more than 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?

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

Thanks for taking the time to read my post.
In reading through your comments, I couldn't help but get the feeling that you are "all over the map" when it comes to what you're looking to do in graduate school. If this is similar to the way that your personal statement read to the committee, it could help to explain some of the challenges you faced in applying to graduate schools.

For instance, after reading what you have written, I can't tell if you're looking to do algebra or bioinformatics—or if you're in an "I'm happy with either" situation. If it's the latter, this can make it harder for a committee to accept you: algebra readers might think they'll admit you, only to have you wander off into applied math, and vice versa. You may find it better to concentrate on one area—or, better yet, show how the different topics you're interested in are actually related to one another.

Another potential issue is the quality of your letters of recommendation. Is there anyone who can help you to evaluate your letters for you? Some colleges have career offices or academic counselors who can comment on the suitability of your letters for different kinds of programs (academic, industrial, etc.), although they probably won't "rank" them for you. If you had a very late letter, it's a good question as to how helpful it would have been, even if  it was from a big name (or perhaps that should be "in spite of" it being from a big name).  By this, I mean that the letter would only have value if it actually discusses your talents and skills in a meaningful way. A pro forma letter isn't all that helpful, particularly if you are from an "unknown" program. (This latter can also make your job tougher, since your school might not have  a track record at many graduate schools).

Finally, it also sounds like you have a rather untraditional academic history. This needs to be explained clearly somewhere on your application, in a way that shows thattaking you into a graduate program now would not be a risk. (Not doing this could again feed into the notion that you're flittering around from one thing to the next—which is not a good impression for an admissions committee to have of you.)