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I am currently a Junior at an average state university with a not particularly well known math department. I'm pretty set on pursuing a Phd in math (my interests lie primarily in numerical analysis and PDE's), and would like advice on which programs would be realistic to apply to.

Background:

GPA: 3.95 (math GPA: 4.00 - but again, not from strong math department)

Upper Level Coursework: Linear Algebra 1 &2, Intro Analysis 1&2, Abstract Algebra, Applied Math 1 &2 (using David Logan's text), Probability Theory, Numerical Linear Algebra, Numerical DE's, Software Applications

Will take grad courses in Numerical Analysis, Real Analysis, and Topology next semester, as I've nearly exhausted the interesting undergraduate courses.

GRE: Q: 169 V: 160 W: 4.5 (Haven't taken the subject test yet)

Research: Very little. Did an REU in PDE's last summer. Won a small mathematical modeling competition. Hope to get involved with faculty during my senior year.

I'm considering staying at my undergrad institution for an extra year to complete a masters degree in mathematics and complete a master's thesis. Of course, I've consulted with professors regarding this decision, but have gotten mixed feedback. Some think I should apply for a couple top schools, others have recommended a more conservative approach. Where do you think would be reasonable to apply for a 'reach' school? Safety? Would a masters bolster my application (particularly if this led to more serious research)?

All advice welcome and appreciated!