Career advice - Getting a Math PhD admit in a good program with computer engineering background I have read other threads along similar lines but I am looking for slightly different advice. And I am posting this anonymously because I would rather my current employer did not find out my intentions!

So, I have a bachelors and a masters degree in computer engineering and currently work in one of the top microprocessor companies, which would otherwise be considered the "dream job" for someone with my degrees. However, every day I realize how much I miss Math and really want to go back and have an opportunity to work more with math, particularly in the field of numerical analysis, scientific computing, matrix algebra and the like. 

Supporting factors - 


I have always done well in math. I can get great recommendation letters from math professors I have taken courses with. 
I am a CRLA level 3 certified math tutor. I used to be one when in undergrad. 
I am preparing for Math GRE and am confident I can do well in it.
Good undergrad math and engineering GPA.
My masters thesis involved quite a bit of dealing with numbers, since I worked with various LINPACK benchmarks and linear algebra solvers. 


Negative factors - 


I am not from a very highly reputed school. 
I have not published any papers, even though I did write a thesis for my MS in Computer Engineering. 
My math courses are the basic math courses engineering students take, along with graduate level math courses in numerical analysis, scientific computing and the like. 


How do I go about getting a PhD admit in a reputed Math graduate school? How do I begin to convince professors / hiring committees that I am capable of doing a PhD in Math? I am only looking at quitting my company and switching to PhD studies around either of Fall 2013 or Fall 2014, so I have time. What are some extra-curricular 'outside my day job' activities I can pursue that would further solidify my application in the meantime?
A concern (raised by your background) that occurs to me is that do you really have a clear idea what graduate programs in math are all about? I don't think that any reputable grad school would allow you to study numerical analysis alone ... [Edit taking note of Willie's remark] within a graduate program in pure math - the scene is markedly different if you are applying into a program in applied and/or computational math. I apologize for not knowing that such possibilities exist. My smalltown background left me with the false impression that math and applied math always come together. Such programs may be better suited for you![/Edit] (continued rant) ... For the simple reason that doing research in math requires familiarity with a variety of tools and theories from adjacent, nearby and occasionally relatively remote areas of mathematics.

Is entering a computer science/computer engineering graduate program not an option? Probably you can specialize in scientific computing/numerical analysis in such a program. It may actually be easier to find an advisor in such a topic at a CS department? I don't know for sure, but it sounds like such a plan would entail less risk.

Our resident experts on numerical analysis/scientific computing can give you more useful advice. Below I will say my bit.



Before you burn your boats I would recommend, as an extra-curricular activity, that you take a look at what the obligatory 1st year courses of math programs have in store for you.
The core math at a typical US grad school (for 1st year grad students) contains at lest topology, abstract algebra, real analysis (sorry the link is only to measure theory, couldn't find a more fitting Wikipedia article at this time) and complex analysis. Some places would offer/recommend/require also mathematical logic. After the first year, you are expected to display a working understanding of the theories and results that those articles link to, and be able to reproduce their proofs on demand (ok, the committees will likely give you some slack on the more esoteric proofs, but don't count on it). The depth of those course probably depends on how much ivy the school has. I cannot give details on that for the simple reason that my experience is from a reputable but not top notch grad school, and I have only heard rumors about the others :-).

Only after having covered those basics can you start specializing on a topic that interests you the most.

I don't want to dampen your enthusiasm. I just had a few fellow grad students who were surprised by the graduate curriculum, and either dropped out or had a hard time making through the 1st year. My concern is that you may not fully appreciate how limited your exposure to math actually is (given your background). GRE is a joke, but it does test that you can speedily pick the correct calculus concept/theorem off the shelf in your brains and apply it.

So if going through those links just makes you hungry to learn more, then 'full steam ahead!', but otherwise you may want to reconsider.