Doctoral Degree Requirements

The PhD in Mathematics requires completing at least 36 credits of graduate coursework beyond the bachelor’s degree, preliminary written and oral examinations, 24 thesis credits, and a doctoral thesis with final defense. 

PhD students in Mathematics are expected to earn an MS degree upon completion of their preliminary oral examination.


Entering students

  • Required to have completed an undergraduate degree prior to matriculation into the program. 
  • Some students enter with previous graduate coursework which can include having completed a Master’s degree.

Course requirements for a doctoral degree

  • 36 credits of coursework — Students normally complete the coursework for the PhD at the University of Minnesota. Students may transfer up to 18 credits of previously completed graduate coursework toward their PhD in consultation with their advisor and the Director of Graduate Studies upon submitting their degree plan. 
  • MATH 8001: Preparation for College Teaching — All PhD students must complete MATH 8001 in the Fall semester of their first year.


  • 18 credits of core coursework — Courses in Mathematics that are the foundation for advanced doctoral work in Mathematics. 
  • 18 credits supporting coursework — Mathematics coursework in your research area or that supports this research. Students may take a maximum of 12 credits outside of Mathematics. Coursework used for a formal minor can be counted towards this requirement.
  • 24 thesis credits — Taken upon the completion of the preliminary oral examination. Students are expected to complete their thesis credits no more than two semesters after successfully completing their preliminary oral examination.

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Core courses

Core courses

  • General Algebra (MATH 8201/02)
  • Manifolds and Topology (MATH 8301/02)
  • Mathematical Modeling and Methods of Applied Mathematics (MATH 8401/02)
  • Numerical Analysis and Scientific Computing (MATH 8441/42)
  • Real Analysis (MATH 8601/02)
  • Theory of Probability Including Measure Theory (MATH 8651/52)
  • Complex Analysis (MATH 8701/02)

Preliminary written exam

Students must demonstrate proficiency in basic areas of mathematics by passing the preliminary written exams.


Students are expected to complete their preliminary written exams by the end of the second year of study. Extension to this timing may be approved through a meeting with the academic advisor and the Director of Graduate Studies.

What’s required

Students must pass two written examinations by either:

1. Taking the written examinations.

Given Fall and Spring semester in Real Analysis, Complex Analysis, Algebra and Manifolds/Topology.

2. Passing the final examination in the second semester of any of the core courses.

What to expect

  • Students with previous graduate coursework may transfer up to 18 credits to fulfill core course credit requirements upon passing preliminary written exams.
  • Students without previous graduate coursework who complete core coursework requirements by passing exams will take up to 36 credits of other major coursework.  

Preliminary oral exam

Students complete a preliminary oral examination to demonstrate proficiency in their primary area of study and supporting program or minor program. The exam also serves as a final exam for a Plan B Master’s in Mathematics. 


  • Typically completed after the third year of study.
  • Must be completed prior to the last day of Spring semester in the fourth year. 
  • Changes to this timing may be approved by meeting with the advisor and the Director of Graduate Studies.

What to expect

The preliminary oral examination concentrates on the primary area of study and the supporting program or minor. 

  • Examination committee — The examination committee of four faculty members consists of the major area advisor and three additional faculty members. Preliminary Oral Exam Committees require a faculty member from outside of Mathematics OR Outside of the University of Minnesota. Students who have declared a minor in another department are required to have a committee member from the department they are minoring in.
  • Advisor — The advisor recommends coursework in the basic material, as well as expository and research papers for individual study.
  • Expository paper — The study, including the recommended research papers, will be summarized in an expository paper (approximately 10 pages), with a substantial bibliography. This will demonstrate a knowledge of the definitions and results in the area, and indicate open problems which may form the basis for the PhD thesis.
  • Registration and credits — Students taking their preliminary oral examinations during Fall or Spring semester are expected to register for six pre-thesis credits (MATH 8666). 


  • The student should choose their primary area and decide on an advisor who will support them through these exams by the end of fall semester in the second year.
  • In the semester prior to the oral exam, students should work with their advisor to determine a committee and expectations for their exam. 
  • Students are expected to have begun work on their expository paper by the beginning of the semester in which they are planning to complete this exam. 


See the PhD degree (steps 1-7) and the Plan B Master’s degree (the preliminary oral examination serves as a Final Exam for a Plan B Master’s in Mathematics) completion steps:

Degree completion steps


Final exam/dissertation defense

Mathematics PhD students typically complete their degree in the sixth year of their program. The examination is a defense of the thesis that is the culmination of the student’s work in the program. 


  • The doctoral final exam committee consists of four members, including the advisor(s).
  • The chair of this committee must be a full mathematics faculty member who is not the student's advisor.  
  • If there is a formal minor, this committee member must represent that program. In the case of multiple minors, there must be a separate thesis reviewer for each minor.


The doctoral final exam must include a public presentation of the candidate’s dissertation to the doctoral final oral examination committee and the invited scholarly community, followed by a closed session for questions by the examiners. 



See the PhD degree completion steps (steps 8-15):

Degree completion steps