Ph.D. Programs

Continuous mathematics	Discrete mathematics	Computational mathematics
Complex Variables (MA 513)	Linear Programming (MA 505)	Computer Algebra (MA 522)
Analysis (MA 515)	Linear Algebra (MA 520)	Uncertainty Quantification (MA 540)
Control (MA 531)	Abstract Algebra (MA 521)	Graph Theory (MA/CSC 565)
PDEs (MA 534)	Combinatorics (MA 524)	Modeling (MA 573)
Probability (MA 546)	Matrix Theory (MA 523)	Numerical Analysis (MA 580)
ODEs (MA 532)	Algebraic Geometry (MA 526)
Topology (MA 551)
Manifolds (MA 555)

Students who have not had the equivalent of MA 405 (Introduction to Linear Algebra and Matrices), MA 407 (Introduction to Modern Algebra), or MA 425–426 (Mathematical Analysis I and II) must make up these courses. Students who have not had the equivalent of MA 426 may take it for graduate credit by registering for MA 591M.

Select three one-year sequences from among the following.

MA 515--715	Analysis
MA 520--720	Linear Algebra and Lie Algebras
MA 521--721	Abstract Algebra
MA 522--722	Computer Algebra
MA 523--723	Applied Matrix Theory
MA 524--724	Combinatorics
MA 531--731	Systems and Control
MA 534--734	Partial Differential Equations
MA 546--747	Applied Probability and Stochastic Process
MA 555--753	Geometry and Topology
MA 573--574	Modeling
MA 580--780	Numerical Analysis

• For students without a prior masters degree in Math or Applied Math: Such students must attempt as many exams as necessary to complete the requirement by August before the start of the third year. If they do not pass all exams at that time (and have allowed retakes left, see below), they have until January of the third year to pass all remaining exams.
• For Students with a Prior Masters Degree in Math or Applied Math: Such students must attempt as many exams as necessary to complete the requirement by August before the start of the second year. If they do not pass all exams at that time, they have until January of the second year to pass all remaining exams. A student with a prior masters degree who does not pass any exams when entering the program should enroll in 3 qualifying exam sequences in the first year so that they can attempt 3 exams in the August before the start of the second year.
• Each exam is written and graded by two faculty members.
• At the end of the Spring Semester, the two exam writers for a qualifying exam will come up with a study guide for the qualifying exams that year. That study guide will be given to the students in the course, and will be given to the DGP so that any students who wants to take the exam and who were not in the course will be able to prepare for the exam.
• Each exam is a three hour long written exam.
• One retake is allowed for each of the three examinations (with exceptions described below). Students are advised to schedule retakes as soon as possible; retakes must be done within 12 months of the date the examination is first taken. If a student fails an examination twice, he/she is considered to have failed the written qualifying examinations.
• The retake does not have to be the same exam as the one initially failed. However, if an examination for a specific sequence is failed and retaken later, the second examination must be considered a retake of the first.
• Incoming students are allowed to take qualifying exams in the August that they arrive. A fail at that time does not count against the student.
• The number of examinations taken at any given exam period cannot exceed the remaining number of passes need to reach a total of three.
• January exams are reserved for retakes, or in rare instances, special circumstances (please consult DGP). If a student takes a January retake in the first year or second year (first year only for students with a previous master’s degree), fails that happen at that time do not count against the student.
• It is recommended that students attempt at least one to two exams by the end of their first year. By the end of their second year (first year for students entering with a MS in Mathematics), students have to attempt a number of exams equal to the number of exams they have yet to pass.
• It is possible to “drop” an exam one has registered for any time up to two weeks before the exam (no questions asked). After that, we recommend students talk to the director or the administrator of the programs. Dropping an exam only applies to students who do have the option to “wait”, i.e., are not required to take the exam at that exam session.
• For part-time students, years will be counted using credit hours, with one year equal to 18 credit hours. For full-time students, calendar years are used. Students who start the graduate program in the spring semester should have a statement from the DGP put into their file specifying the date by which their exams must be taken. Depending on prior coursework, whether they are a transfer student, etc., it will be after three, four, or five semesters.

Faculty have provided these “suggested paths” of courses that students can take, depending on their area of interest.  Note, however, that the particular courses most relevant for an individual student will vary depending on which faculty the student works with, and what project the student works on.  Hence, it’s important to talk to faculty early to figure out the best courses that are right for each student.

Interest	Qualifying exams	Important courses (if not taken as quals)
Algebra	MA 520-720 (Linear/Lie Alg) MA 521-721(Abstract Algebra) and one of MA 524-724 (Combinatorics) MA 555-753 (Geometry/Topology)	MA 725 (Representation Theory)
Analysis and differential equations	MA 515--715 (analysis) MA 534--734 (PDEs)	MA 532 (ODE) MA 716 (Functional Analysis) MA 719 (Vector Space Methods) MA 732 (ODE II) MA 748 (Stochastic DE)
Biomath	MA 573-574 (Modeling) MA 580-780 (Numerical Analysis) MA 531-731 (Control Theory) MA 523-723 (Matrix Theory)	MA 771-773 (Biomath) MA 540 (Uncertainty Quantification) ST 511-512 (Statistics for Biology)
Combinatorics	MA 521-721 (abstract algebra) MA 524-724 (combinatorics) and one of MA 515-715 (analysis) MA 520-720 (Linear/Lie Theory) MA 555-753 (geometry/topology)	MA 726 (Algebraic geometry) MA 725 (Representation theory)
Financial Math	MA 546--747 (probability) MA 534--734 (PDEs) and one of MA 580--780 (numerical analysis) MA 531--731 (control)	MA 515 (Analysis) MA 547 (Stochastic Calculus in Finance) MA 548 (Monte Carlo Methods for Financial Mathematics) MA 549 (Financial Risk Analysis) MA 580 (Numerical Analysis) MA 748 (stochastic differential equations)
Geometry and topology	MA 555-753 (geometry/topology) MA 521-721 (abstract algebra) and one of MA 515-715 (analysis) MA 520-720 (Linear/Lie Theory) MA 524-724 (combinatorics) MA 534-734 (PDEs)	MA 515 (analysis) MA 726 (algebraic geometry) MA 754 (algebraic topology) MA 755 (Riemannian geometry)
Modeling and control	MA 531--731 (control) MA 573--574 (modeling) and one of MA 515--715 (analysis) MA 523--723 (matrix theory) MA 534--734 (PDEs) MA 580--780 (numerical analysis)	MA 515 (analysis)
Numerical analysis and scientific computing	MA 580--780 (numerical analysis) and two of MA 515--715 (analysis) MA 573--574 (modeling) MA 546--747 (probability) MA 534--734 (PDEs)	MA 515 (analysis)
Probability	Main : MA 546 - 747 (Probability) MA 515 - 715 (Analysis) Others : MA 523 - 723 (Theory of Matrices) MA 531 - 731 (Dynamical Systems) MA 534 - 734 (PDEs)	MA 544 (Computer Experiments in Mathematical Probability) MA 547 (Financial Mathematics) MA 716 (Advanced Functional Analysis) MA 748 (Stochastic Differential Equations)
Symbolic computation	MA 522-722 (Computer algebra) and two of the followings: MA 520-720 (Linear/Lie Theory) MA 521-721 (Abstract Algebra) MA 523-723 (Applied Matrix Theory) MA 580-780 (Numerical Analysis)	CSC 505 (Design and Analysis of Algorithms) MA 505 (Linear Programming) MA 706 (Nonlinear Programming) MA 726 (Algebraic Geometry)