Ph.D. students at Washington University are required to demonstrate proficiency in algebra, real analysis, complex analysis, and geometry. Most students satisfy these requirements by taking a yearlong course in each subject, capped by a final exam that serves as a "qualifier." Students with strong backgrounds may be excused from some of these courses.
The result of this broad and uniform requirement is two-fold. First, graduate students have an opportunity when they first arrive to share common concerns and to become acquainted. One of the most attractive features of our program is the friendly and supportive atmosphere among the graduate students. Second, advanced courses in the Washington University Math department can build on the common background shared by all students. As a result these courses are richer and nearer to the level of Ph.D. work than advanced courses regularly taught at other good mathematics departments.
Typically, it takes two years for a student to complete the written qualifying exam phase of the program. By the end of the second year, the student usually has some idea of which area of specialization to choose. By that time the student is also acquainted with several faculty and feels comfortable asking one to direct his/her research.
Once the qualifying exams are passed and a thesis advisor engaged, the next step in the program is for the student to prepare a "minor oral presentation" and a "major oral presentation" (see the section, Degree Programs and Requirements, for further details on these oral presentations). Topics for these orals are chosen in consultation with their thesis advisor, and culminate in two public lectures. These should be completed by the end of the year following completion of the written qualifying exams.
After these preliminaries, the essential part of a student's graduate work---the thesis---begins. Unlike class and exam work, a Ph.D. thesis in mathematics involves producing a substantial piece of new research. The work is difficult, protracted, and frustrating. But in the end the rewards are great.
While working on their orals and their research, students have the opportunity to take a variety of advanced courses. These vary from semester to semester. Offerings in Fall 2003 include: Topics in Algebra (Kumar); Geometry and Manifold Theory I (Roberts); and Advanced Probability (Sawyer).
When the research is completed the student prepares a thesis, which is the detailed writeup of the research results. This writeup may range from fifty to a few hundred pages and is the formal record of the student's achievement in the graduate program. The final official step is for the student to give "defense" of the thesis both in a public lecture followed by questions from a panel of six appointed faculty members of the University in a closed session.
Students typically complete the Ph.D. program in five years. A student who comes here with advanced preparation may finish in less time. On the other hand, some students find that it is advisable for them to take preparatory work before attempting the qualifying courses. In special cases, the time schedule may be lengthened accordingly. Students should plan to develop a close relationship with their thesis advisors so that they may have a realistic idea of how they are progressing.
Graduate study in mathematics is not for everyone. Entering students usually find that the time and effort required to succeed goes well beyond anything they encountered as undergraduates. Success requires both ample mathematical ability and the determination to grapple with a subject for many days or weeks until the light of understanding shines through. The experience can be daunting. Those who continue in their studies are largely those for whom the pleasure in attaining that understanding more than compensates for the required effort. For such persons, the life of the mathematician can be richly rewarding.
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