Acceptance into graduate math programs

I have been hearing different views about my likelihood of getting accepted into a math graduate program.

Which of the following factors contribute most to the acceptance of a non-math undergraduate into a math graduate program (please order them)?

1. GPA


2. Recommendation letters


3. Publishing math papers

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Since I am an engineering student I only registered basic math courses (such as calculus, linear algebra, etc.). However, I studied a lot of other more advanced courses by myself. My GPA is about 3.5. A lot of people have told me that I need to raise my GPA in order to get accepted into a decent math graduate program and this is worrying me.

Courses that I self-studied: Undergraduate abstract algebra, real analysis (with an introduction to measure theory), first 4 chapters of Munkres' Topology, elementary number theory, graph theory

Courses that I am self studying: Hungerford's abstract algebra, algebraic topology

Courses that I plan to study: algebraic geometry, algebraic number theory, complex analysis.

I also managed to finish 2 math papers. So the question is:

Where do I stand now?

Answer

Since you are an engineering major who took only the standard math courses, I infer that your overall GPA is driven by your engineering coursework. I don't think that graduate admissions will learn much from your grades in engineering, good or bad. The GPA of 3.5 is not that bad in engineering, especially if (as it seems) your heart was not really in it.

Self-studying is great, but the knowledge acquired from it needs to be evaluated by someone. Acing GRE Math would send a signal that you indeed learned something. If during your studies you kept in touch with math professors in your school, and they know enough about you to write an informed letter full of specifics, that would be even better.

The opinions on undergraduate papers in math are divided: e.g., not everyone considers them a good way to spend time as an undergraduate. I would not expect the admission committees to seriously read papers sent with an application, though they will glance at them.

So, my order is: letters, GRE math, papers, GPA.