Mathematics and Mechanical Engineering and Materials Science
How do you start researching PhD programs?
US News and World Report's lists of top graduate programs can be a good starting point. Everyone needs to filter lists like these to suit their own interests and priorities, but it identifies some strong programs and shows what are active areas for current study. Digging further is done by looking at department and faculty websites, and even downloading recent papers that look interesting.
What should you look for in a PhD program?
Ideally programs should have sets of faculty that cover a spread of current research topics, but share some interests and produce students that are well-versed in broad areas and could potentially do interesting research in several directions. Look for the recent productivity of faculty and the career placement of students.
What questions should you ask in your campus visits?
Campus visits are great for forming impressions of quality of life issues, some things that might be intangible and don’t translate directly into email questions or web searches. Cost of living, housing and neighbors around the school can be an important consideration, just like department culture, office space, campus culture, and typical working week duties.
What are some common pitfalls to avoid in the application process?
Start the process early, and leave a lot of time for revising essays, getting letters of recommendation, submitting materials and preparing for GRE's.
Consider carefully who you get recommendation letters from—it's more impactful to get letters from professors that know you in the context of more advanced courses, or more individualized research projects. Make them aware of your goals for graduate school and your broader background (share your CV, essay and other application materials with them so they can write more holistically informed letters).
Write personal statement essays that give concrete and specific examples illustrating your experience moving toward research in the area. Knowledge of the field, contact with advanced topics in courses and showing commitment to struggling with challenging problems can carry more weight than narratives about early motivation and general formative experiences. Readers of your essay will be expert researchers—don't hold back on detail, seek to present your knowledge at a professional level.