I am applying to a number PhD programs and my options are limited due to my main interest in nonlinear dynamics/systems/chaos and related applications. At this point I was thinking of UC San Diego and Brown University as possible reach schools. I've done the research on grad school shopper and the school websites, but I'd like to hear some feedback (UCSD accepted applications average a PGRE score of 811 and Brown doesn't list any score data, but last year they accepted less than 10% of applicants with 14 of the 18 accepted being foreign students). Here are my specs: Domestic Student Final GPA: 3.50 with an upward trend. GRE: 600 verbal, 790 math, 4.5 Writing. PGRE: 650 or 44th percentile after bombing the test a year before with a 520. Research: 1 published collaborative paper in physics journal, 2 collaborative papers awaiting acceptance to mathematics journals. At this point I think these schools are out of my reach, but if anyone thinks otherwise then maybe I'll take the chance. Thanks.