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The graduate program in Mathematics is aimed at guiding students toward original research. At the end of the program (normally by the end of the fifth year of study) students are expected to submit and defend a PhD thesis that contributes new results to the body of mathematical knowledge. To successfully deal with problems arising in modern mathematics, graduate students need to acquire a considerable background. This is achieved through course work and independent study. The program provides for two sets of examinations to verify the progress: Preliminary Examination (normally passed by the end of the first year), Qualifying Examination (often passed by the end of the second year).