Mathematics - Doctor of Philosophy

Candidates for the Ph.D. degree in the Department of Mathematical Sciences must pass

  • a qualifying examination,
  • three comprehensive written examinations,
  • a comprehensive oral examination,
  • a series of courses, and
  • a final oral doctoral thesis examination.

These are briefly described below. For more information, see the Graduate School requirements in this catalog, and the Mathematics Graduate Student handbook at

Qualifying Examination

Every student admitted to the Ph.D. program must complete the Ph.D. oral qualifying examination. Its purpose is to determine the areas in which the student shows strength or weakness, as well as the ability to assimilate subject matter presented at the graduate level. Students who complete their mathematics master's degree at NMSU may request, at the time of applying for their master's oral final examination, that the Master's examination also fulfill the Ph.D. qualifying examination requirement. In all other cases, towards the end of the student's first semester in the Ph.D. program, the student and his or her advisor will convene an oral examination with three examiners, the examiners being the advisor and some of the student's current or past instructors. As a result of the Qualifying examination, the department will take one of the following actions:

  1. admit the student to further work toward the Ph.D.;
  2. recommend that the student's program be limited to a Master's degree;
  3. recommend a reevaluation of the student's progress after the lapse of one semester; or
  4. recommend a discontinuation of the student's graduate program in mathematics.

Written Comprehensive Examinations

Candidates for the Ph.D. degree must pass written comprehensive examinations in three of the seven areas of algebra, complex analysis, differential equations, logic and foundations, real analysis, statistics and topology. To ensure adequate breadth, a combination of three comprehensive examinations must include real analysis, and at least one of algebra and topology.

The seven examinations are based on the following comprehensive examination sequence courses:

MATH 525Advanced Linear Algebra3
MATH 581Algebra 13
MATH 582Algebra II3
Complex Analysis
MATH 517Complex Variables3
MATH 591Complex Analysis I3
MATH 592Complex Analysis II3
Differential Equations
MATH 518Fourier Series and Boundary Value Problems3
MATH 531Ordinary Differential Equations3
MATH 532Partial Differential Equations3
Logic and Foundations
MATH 504Mathematical Logic3
MATH 557Axiomatic Set Theory3
MATH 585Universal Algebra3
Real Analysis
MATH 527Introduction to Real Analysis I3
MATH 528Introduction to Real Analysis II3
MATH 593Measure and Integration3
MATH 594Real Analysis3
Probability and Statistics
STAT 562Foundations of Probability3
STAT 571Continuous Multivariate Analysis3
MATH 541Topology I3
MATH 542Topology II3

Full time students should complete the comprehensive written exams in the first two years. Those who have not made substantial progress towards completion of their written exams at the start of the fifth semester may be removed from the program. Students who have not completed the written exams by the start of the sixth semester will normally have any departmental funding revoked.

Exams are offered every August and January. A student must register to take exams in the semester prior to taking the exams. A student has three consecutive examination periods to complete the written comprehensive exam requirements (Example: if s/he starts in August, s/he has the August, January and August examination periods to complete the exams). This does not extend the time limit mentioned above. Students will normally not be given more than two attempts at any one exam. 

Course Requirements

Before graduation, a student must pass a total of four comprehensive exam sequences, but needs to take the comprehensive examinations in only three of them. In addition, a student must pass four more (one-semester) MATH/STAT courses from the seven comprehensive exam sequences listed above.

A student may pass any of the four comprehensive examination sequences before enrolling as a Ph.D. student, but the four additional courses have to be passed after enrolling as a Ph.D. student.

The following courses will not count towards the course requirements:

Any course below MATH 501
MATH 511Fundamentals of Elementary Mathematics I3
MATH 512Fundamentals of Elementary Mathematics II3
MATH 513Fundamentals of Algebra and Geometry I3
MATH 516Calculus with Hands-on Application3
MATH 563Algebra with Connections3
MATH 564From Number to Algebra3
MATH 566Data Analysis with Applications3
MATH 567From Measurement to Geometry3
MATH 568Using Number Throughout the Curriculum3
MATH 569Geometry with Connections3
MATH/STAT 540Directed Reading1-6
MATH 599Master's Thesis15
MATH 600Doctoral Research1-15
MATH 700Doctoral Dissertation1-15

Students and advisors are encouraged to consider further courses beyond this minimum.

Oral Comprehensive Exam

The student must take this exam at the end of the semester after completing the written comprehensive exams. The student should present a proposed direction for thesis work.

Final Oral Exam

This should be an exam over the student’s thesis and administered by the same committee of the oral comprehensive exam.