I'm looking for advice on schools, since my profile is a bit unconventional. TL;DR: I have high test scores, a low GPA, and one publication in a science journal that has been receiving a good amount of attention from researchers in the field, but it's NOT pure math.... not sure if that will matter.

**Undergrad Institution:**Highly ranked liberal arts, well-respected math dept.

**Major:**Mathematics

**GPA:**3.02 (I was on the varsity crew team, and had to wake up at 5 am 6 days a week... not sure if that's a good enough excuse, but it definitely didn't help with grades..)

**Type of Student:**White male, domestic

**Graduate Institution:**Middle of the road large state school

**Major:**Pure Math

**GPA:**3.35 (No excuse for this. I had initially planned on staying for PhD here, and I did really well my first year. I intended to study algebraic geometry, but once I took a course on the subject and read more and more, I realized that the classical algebraic geometry I was familiar with was far different from modern research in the subject. Despite my best efforts, I couldn't make myself feel the passion for the subject I had once had. This was very disappointing to me, since I quite liked the professor I was planning on taking as an adviser, and he is pretty well-respected in the field. After this, I realized I needed to take a step back and reevaluate my plans. I also had some very upsetting things happen in my personal life at this point which were a major distraction, but to be honest I'd rather not disclose any of this information in my applications since it's very personal).

**GRE Scores:**

Q: 170

V: 164

W: 5.0

M: 900

**Program Applying:**Pure Math (Representation Theory/Number Theory or Algebraic Topology)

**Research Experience:**Worked at a meteorology/oceanography research company during the summer of my masters degree (as an alternative to TAing, mainly since they were offering double what the TAship would have paid). My research led to one publication (I'm 2nd author of 5) that has been receiving very good attention from researchers in the field. The paper is NOT pure math, which I think is important. Rather, it's an application of basic metric space theory to a problem in modelling.

**Awards/Honors/Recognitions:**Nothing noteworthy

**Pertinent Activities or Jobs:**Aforementioned research job, TA for two years in masters, and currently working as an engineer at a defense contracting research company for the past 2.5 years.

**Any Miscellaneous Points that Might Help:**I've been studying independently quite a bit since I started my most recent job, and I feel like I am

*significantly*better-prepared for a PhD program than I was coming out of undergrad.

**Any Other Info That Shows Up On Your App and Might Matter:**Nothing noteworthy.

Long story short, I'm going to have a lot of difficulty conveying to admissions committees that I actually know my stuff since my application will do little to inspire confidence..... I've spent the last 2 years while at my current job studying in my free time, and I know that I could at the very least pass the qualifiers from the school where I did my masters rather easily. In terms of background, for what it's worth, I've covered the following books essentially cover to cover (unless otherwise noted) and have done as many of the exercises as possible, and I'm extremely comfortable with the material (far more than I was after undergrad):

**Linear Algebra:**Strang, Lang's

*Intro to Linear Algebra*, and Axler's

*Linear Algebra Done Right*

**Algebra:**Artin's

*Algebra*, Vinberg's

*A Course in Algebra*, Dummit and Foote (used in my graduate program, haven't read it much since. I strongly dislike the "chatty" writing style. Truly excellent choice of topics, but I just can't stand the style. Just feels uninspired and dull to me).

**Analysis:**Pugh, Spivak's

*Calculus on Manifolds*, Needham's

*Visual Complex Analysis*, Ahlfors, Royden

**Topology:**Munkres, Hatcher (I need to review cohomology more, but I'm comfortable with homotopy theory and homology)

I've used other books of course, but these are the ones I've read recently and tried to really know inside and out.

My questions to you are as follows:

1.) What caliber of school would I have any chance of getting into?

2.) I keep TeX files of proofs to many of the problems from these books (I have terrible handwriting... even I have trouble reading it ) Would it help my case to include these in my applications so I can demonstrate my writing style and make up for my terrible GPA's?

3.) As I mentioned above, I was enrolled in a graduate program a few years ago, so I have a lot of graduate course experience and I've been studying as much as possible for fun on weekends or after work when I have time. As such, I can pass the qualifiers at many schools. Would it help to include solutions to schools' qualifiers in my applications?

4.) Will my publication actually help me much? As I said, it's been well-received in the oceanography community, but the actual math I used wasn't anything above some basic topology.

Thanks for any help anyone can give me!