PhD Requirements

The PhD Degree in Mathematics

The basic requirements for the PhD degree in mathematics include demonstrating a mastery of a broad area of mathematics and writing a dissertation making an original and substantial contribution to mathematics. Specific requirements for the PhD degree in mathematics consist of:

• Coursework
• Qualifying Exams
• Oral Degree Audit (a.k.a. General Exam)
• Dissertation and Defense

Coursework

• Completion of at least 54 semester hours of graduate credit.
• Completion of three of the Core Courses.
  
  

  Core Courses
  Math 7210 Algebra I
  Math 7311 Real Analysis
  Math 7510 Topology I
  Math 7382 Applied Mathematics

  
  
• One hour of Communicating Mathematics (Math 7001, 7002) is required each semester of the first year.
• Completion of the Breadth Requirement. In addition to three Core courses, four Breadth courses must be completed*.
  (.) One course must be completed from each of three distinct Breadth Areas.
  (.) At least two courses from the Basic Breadth Course list must be completed.
  (.) Each breadth course must be passed with a grade of at least B, and at least one must be passed with a grade of at least A.
  
  
  

  Basic Breadth Courses	Breadth Areas
  Math 7211 Algebra II	Math 72xx Algebra and Number Theory
  Math 7330 Functional Analysis	Math 73xx/77xx Analysis
  Math 7350 Complex Analysis	Math 74xx Discrete Math
  Math 7512 Topology II	Math 75xx Topology and Geometry
  Math 7550 Differential Geometry

  
  *It is implied that, if four Core courses are completed, one can count as a breadth course.
  
  Note: Students who started the PhD program before Fall 2023 can use either these requirements or those that were in place at the time they entered the program. In almost all cases, the present requirement allow more options.

PhD Qualifying Exams

The Written PhD Qualifying Examination consists of three qualifier exams, each of three hours duration, chosen from the four qualifier exams covering the material in the Core courses (the exam in Math 7382 is available starting in January 2024). A PhD student may take the PhD Qualifying Exam as early as feasible, but must pass it no later than the beginning of the fourth regular semester of study, except by permission of the Graduate Committee. Meeting this requirement is part of the definition of satisfactory progress---a requirement for holding a graduate assistantship or fellowship. The PhD Qualifying Exams are scheduled twice per year: normally the week before the beginning of the fall semester, and the week before the beginning of the spring semester. An incoming student who passes a qualifier exam at the PhD level before the first semester of studies is accredited with completion of the corresponding Core course.

Qualifying Exam policy is that at least 50% of the credit on each exam will come from the test problem banks below. There will normally be approximately 6 to 8 problems offered on each exam, and students will typically need to turn in approximately 5 of these.

Syllabi and test bank problems for each of the Qualifying Examinations:

• Algebra Syllabus and Test Bank
• Analysis Syllabus and Test Bank
• Topology Syllabus and Test Bank
• Applied Math Syllabus and Test Bank

Copies of past Qualifying Exams can be found in the Exam Archive.

Use this Registration Form to sign up to take a Qualifying Exam.

Oral Degree Audit (General Exam)

Description. The Degree Audit, a.k.a. General Exam, is an oral exam for Ph.D. students that tests if a student is prepared to begin dissertation research. This exam must be passed prior to the start of the fourth year of study, unless a postponement is allowed by the Graduate Committee. The exam is conducted by the student's Advisory Committee, which will consist of at least three Graduate Faculty Members plus a Dean's Representative appointed by the Dean of the Graduate School. The committee is to be chosen in consultation with the intended dissertation advisor, who will serve as Chair of the Advisory Committee, and must be approved by the Graduate Director. The Advisory committee shall determine whether the student passes the General Exam. If the student fails, the Advisory Committee will determine whether the exam can be retaken. The exam can be taken at most one additional time.

Content and Format. The specific content of the general examination is established by the individual student's Advisory Committee in consultation with the student, and it is formalized in the Exam Syllabus Form. The format of the exam will consist of an optional presentation by the student (typically no longer than 30 minutes), and an oral exam on the content of the exam syllabus. The exam should last at most two hours.

Note. The student must be enrolled in classes (at least one credit hour) in the term in which the oral exam is scheduled; this includes the Fall and Spring semesters and the Summer. The Fall and Spring semesters begin the first day of classes, not the week before.

Instructions on scheduling your Degree Audit can be found at the Instructions link.

Original Dissertation and Defense

Writing an original dissertation and passing the Final Examination which is the doctoral dissertation defense.

Instructions on the steps to graduation and the forms to be filled out the semester of your graduation can be found in the Instructions link.