artichoke

No, that's a murderously bad letter.

Teardrop's letter is strong. Analytical skills demonstrated stronger than some PhD students -- who are presumably doing OK themselves, and with benefit of more experience and coursework. Demonstrated success in writing, not just assisting, in research related to her application area. Massive determination, required to persist and make progress in a world where hardly anyone is your friend, some are your very devious enemies. These are clues for success. One can see these qualities resulting in a good job offer and eventual tenure at that institution. Admit.

I think 08Applicant's letter is weak because it adds little to what the committee can already see. It describes your work in one econometrics course as "stellar", but rather than discussing your stellar analytical mind, it goes on to say you attended every class and are an earnest young man. Damning with faint praise. They want people who could skip half the classes out of arrogance and still get an A. As you might have seen in studying information theory, the agent tries to show the principal that his performance is due to ability rather than effort. I think you should get some credit for the academic related work experience. Perhaps you didn't have a chance to contribute in a more meaningful way, but a committee might have been looking for a statement that you had incisive analytical insights. The summary "better preparation than most graduate school applicants" is far from "than most top school admittees" and yet farther from "than most top school PhD graduates". The overall impression is of someone who is maximizing his performance with hard diligent work, thus producing the numbers you have (whatever they are.) That's the problem I think. He never says you're brilliant, and if you're not coming from a top undergrad program, the prior is that you're not. They might be especially concerned if you're such a diligent student that you pass the qualification process for candidacy even without the ability to write a clever dissertation. (The qualification process is designed to weed out such people, if their interest is in theory.) But a lower tier program which flunks out a lot of students might be interested in admitting you to their obstacle course and using you for research assistance for two years, or longer if you make it.

In proving existence and uniqueness of economic equilibria, one typically uses some sort of fixed point argument. This can be in R^n, or a Hilbert space, or even more generally it can wind up being a function space, even infinite dimensional function spaces. The topological properties of these spaces become very prominent in the proofs. That's really economics but not econometrics, which is a branch of statistics dealing with economic problems (typically the independent variables in the estimation cannot be measured precisely.) Maybe there are some topological discussions in mathematical statistics that could be mentioned, but I don't know that much about math. stats. This sounds like it should be in the grad school forum (for economists, who are typically failed mathematicians, it's grad material) not the high school forum though.