Participate in teaching. Each student is required to teach as a teaching fellow two undergraduate courses by the end of their second year,
Pass a qualifying examination on their general applied mathematical knowledge (in algebra, analysis, probability and statistics) by the end of their second year, Find an adviser. By the time the student has passed the qualifying exam, he/she is expected to have found an adviser. The advisor has to be approved by the Director of Graduate Studies, and students are encouraged to find an advisor sooner rather than later. In collaboration with his/her adviser, the student prepares a dissertation prospectus, a short document that outlines plans for the research and demonstrates familiarity with relevant literature. The prospectus is expected to have been approved by the dissertation committee before the end of the third year.
A master's degree is not required to apply for a Ph.D. at Yale, although some programs give preference to applicants with post-baccalaureate training. Consult your program of interest directly for information on how it evaluates applications.
A statement of academic purpose.
Three letters of recommendation.
Resume/CV.
A list of all the prior colleges or universities you have attended, accompanied by unofficial transcripts from each school.