Hi everyone, I'm currently a 4th year applied math major who goes to a top 40 school and due to stupidity and poor class scheduling in my lower division years, has to stay an extra semester to fulfill graduation requirements. Here's the story: During my first two and a half years of college, I was lazy and unmotivated. I couldn't break away from my high school mentality of not needing to do work outside of class. I had passable but not stellar grades (I averaged about a 2.9 in my math and science classes) which I know is not near graduate school quality. During the transition between 3rd and 4th year, after a lot of thought and reflection, I found myself getting far more motivated to study mathematics and learning the material ceased to be a chore, but became fun and engaging. I changed my study habits considerably (I must've studied 5 hours daily all year long this year. I learned the hard way there is no substitute for sitting down for hours on end and doing the proofs on your own), taking year long introductory graduate sequences in both analysis and abstract algebra this year and getting A's in all of them. During my senior year, my grades went up to about a 3.9 taking all advanced upper division math classes, and I even was lucky enough to be accepted to a math research camp this summer at Williams College. I know there are several other factors to getting accepted to a say, top 20 PHD program, but would I have any decent chance assuming I had good recommendations from professors? Thanks for any opinions.