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Doctoral mathematics courses offer the opportunity to develop in-depth knowledge of a specific subject area. PhD course topics cover all areas of pure and applied mathematics but choice varies by institution.  You will attend classes and seminars as well as undertaking extensive independent research. Your thesis must demonstrate innovative thought and make an original contribution to the field. You need a good undergraduate degree or research masters in mathematics or other related science subject.  There are many career options or you can continue with academic research.