PlankNotPlanck

Hello there for those that are reading this, it my very first post ever! I am an undergraduate maths major with the hunger to apply to graduate school next fall. So here are the characteristics of my field of interest (pun all intended): 3.6 maths gpa, graduate courses in real analysis/abstract algebra/calculus of variations, maybe a reading course on topology/graduate course before applying, potentially a math-physics first author paper and a math (number theory) first author paper before applying to graduate school. I want to go to a top 50 school for a PhD in topology, number theory, or something alike. Assume I score a 650 or better on the Math GRE Subject test. What are my chances? Nearly two years ago I almost failed out of college two semester in a row, so on my personal statement I want to exhibit the sch-peel of "getting my act together" for a turn around that I have made since then. I use to want to do mathematical physics, but have switched due to my interests. My cumulative gpa by the time I graduate will be something like 3.2-3.0. I just do not know if I have a standing chance at a top twenty program in the US/UK PhD program let alone a top 50. Any thoughts? A fellow college student.