[jefferickson]

Yep, all this. Let me add two more points:

- Own your past mistakes. They happened. Don't pretend they didn't. Figure out the underlying cause of those mistakes, and gather EVIDENCE that you've resolved that cause.

(In my case, I was a LAZY undergrad. I'd never had to work in high school, and so I didn't know how to work in college. And then I got a real job, and it was either do the damn work or it'll be there tomorrow only the boss will be there in my office wondering why the hell I'm costing the company hundreds of thousands of dollars a day and why can't you just get this shit DONE already. And so when I applied for grad school the second time I had "smart but lazy" letters from my old professors, stellar GRE scores, and "smart and works hard" letters from my managers.)

- APPLY WIDELY. You are at a significant disadvantage compared to other students with stronger backgrounds. Do not imagine that your passion and good intentions and maturity are enough to get you into the top programs, or even into any particular program. You're playing a lottery that's stacked against you; buy more tickets.

[Topolomancer]

That's very good advice! One point in particular resonated with me as it seems to be applicable to science in general:

> You're playing a lottery that's stacked against you; buy more tickets.

I take this to mean that is makes more sense to apply for many things, re-submit publications often (not without considering the feedback of reviewers, obviously), and so on.