Overview of Degree Requirements

Course Requirements for the Ph.D. Program

Students must complete 72 credits, of which at least 24 must be research credits.  The remaining credits should be of approved coursework in Mathematics and related disciplines.  The normal minimum grade for graduate courses is B.

The School of Graduate Studies expects all students to maintain satisfactory academic progress at all times.  Failure to maintain satisfactory academic progress may affect the student’s eligibility for financial support and awards, prolong the time to degree, and, if not remedied, may lead to academic warnings and possible dismissal. Programs must conduct periodic reviews of academic performance, including courses completed and grades, no less frequently than once per semester.  Academic review includes written warnings to any student who may not be maintaining satisfactory academic progress.    

Satisfactory academic progress requires all of the following:

• GPA of 2.5 or higher for students who have attempted 12 or fewer credits
• GPA of 3.0 or higher for students who have attempted 13 or more credits
• No more than one grade of “U” in courses that are graded S/U
• No more than two Incompletes that have been on record for two semesters, unless there are documented and acceptable reasons for the Incompletes along with a plan to complete the work.
• No more than two Permanent Incompletes on the transcript.
• No more than 9 credits of C or C+ may be used to meet degree requirements.

The program of courses should be chosen to provide the student with both breadth and depth in mathematics and/or its applications.

The courses 16:642:527-528 (Methods of Applied Mathematics), 16:642:550 (Linear Algebra and Applications), and 16:642:593 (Mathematical Foundations for Industrial and Systems Engineering) are intended as service courses for students in other graduate programs and are not approved for the Ph.D. program in mathematics.

Requests for transfer credit for courses taken at other universities are handled on a case-by-case basis, according to the rules of the department and the university.

Core Courses for the Ph.D. Program

The requirements for a Ph.D. in Mathematics include successful completion of an approved program. To be approved, a program should normally include the following requirements:

Students should complete the following 3 core courses during their first semester:

• 640:501 Theory of Functions of a Real Variable I (Offered every fall) (Outline of topics)
• 640:551 Abstract Algebra I (Offered every fall) (Outline of topics)
• 640:503 Theory of Functions of a Complex Variable I (Offered every fall) (Outline of topics)

Click here  https://mathanalysis.pages.dev/notes for comprehensive and self-contained lecture notes for Math 501 (Real Analysis I) and Math 503 (Complex Analysis).

Much of the syllabus of the written qualifying exam comes from 640:501,640:503 and 640:551 Besides teaching specific mathematical content, these courses are aimed at giving you considerable experience writing mathematical proofs at a level expected of graduate students. We attempt to give you considerable feedback on your proofs. Students who pass a Written Qualifying Exam in Algebra, Real Analysis, or Complex Analysis with a sufficiently high score may continue to request an exemption from taking the corresponding first semester required course (640:501, 640:503, or 640:551), which must be approved by the Graduate Program Director.

Students should complete 3 courses from a menu of additional courses within their first two academic years and not all chosen from the same area (Algebra or Analysis or Applied Mathematics or Geometry & Topology). The current Menu of Additional Courses is

Algebra

• 640:535 – Algebraic Geometry I
•  640:550 – Lie Algebras
• 640:552 – Abstract Algebra II
• 640:561 – Mathematical Logic
• 640:571 – Number Theory I

Analysis

• 640:502 – Theory of Functions of a Real Variable II
• 640:504 – Complex Analysis II
• 640:507 – Functional Analysis I
• 640:515 – Ordinary Differential Equations
• 640:517 – Partial Differential Equations I

Applied Mathematics

• 642:561 – Introduction to Mathematical Physics I
• 642:573 – Numerical Analysis I
• 640:577 – Introduction to Mathematical Probability Theory I
• 642:581 – Graph Theory
• 642:582 – Combinatorics I
• 642:583 – Combinatorics II (does not require 642:582 – Combinatorics I as a prerequisite).

Geometry and Topology

• 640:532 – Introduction to Differential Geometry I
• 640:540 – Introduction to Algebraic Topology I
• 640:548 – Differential Topology
• 640:549 – Lie Groups

Courses in the menu of additional courses have formal requirements that go beyond attendance (such as regularly assigned and graded homework or midterm exam(s) or a final exam or a final paper). Each menu course has explicit prerequisites to ensure that students have the required background for that course, and prerequisites for each course can be found under Course Descriptions www.math.rutgers.edu/academics/graduate-program/course-descriptions. Students with appropriate background may request approval from the Graduate Program Director  to instead take alternative courses that are not on the menu.

Exemptions from taking core courses

A few entering students have already covered the material from one or more core course in sufficient detail that they may be exempted from taking the course. Students who wish to be granted such an exemption should contact the graduate program director (), explaining the reason for the requested exemption. Normally the reason for requesting an exemption is that you've taken a comparable course elsewhere. In this case, you would include with your request a syllabus for the course (including textbook, chapters covered and topics covered) as well as the grade received for the course. In evaluating such an exemption, we try to judge whether the mathematical content of the course taken is comparable to our core course, whether the course was taught at a level comparable to ours, and whether the course gave you sufficient mastery of writing proofs at the level expected of graduate students. You may be asked to provide some samples of written work (homework and/or exams) when you arrive at Rutgers, so please bring such material with you if you are requesting an exemption.

Receiving an exemption from a core course does not give you degree credit towards the 72 credits. There is a separate process for applying for transfer credit for graduate work completed elsewhere .

This comprises 3 two-hour exams on

1. Algebra
2. Complex Analysis and Advanced Calculus
3. Real Analysis and Elementary Point Set Topology

These exams cover the material contained in Math 16:640:551, 16:640:503, and Math 16:640:501, respectively.

Ph.D. Requirements - Written Qualifying Exam

The Written Qualifying Exams are offered twice a year, in late August and January.  The next sitting of these exams is as follows. 

• Algebra - Wednesday  1/14/26, 2:00-4:00 pm - Hill 705
• Complex Analysis and Advanced Calculus - Thursday 1/15/26, 2:00-4:00pm - Hill 705 
• Real Analysis and Elementary Point Set Topology - Friday, 1/16/25, 10:00-12:00pm - Hill 705

Each of these two hour exams consists of two parts. Part I has 3 problems, each of which is mandatory, and part II has 2 problems, of which the student is expected to do one. Each student is expected to submit solutions to all 3 problems in part I, and 1 out of the 2 problems in part II.

PhD Written Qualifying Exam Rules

The incoming students may attempt to take the  Written Qualifying Exams in the August sitting upon their arrival. They can choose to take all three exams, two, one or none.

Students who have yet to pass all three Written Qualifying Exams may choose how many exams (including none) they wish to attempt at the next exam sitting.

The deadline to pass all three written qualifying exams is the fourth semester (i.e January sitting in the second year before the start of Spring semester, is the last opportunity).

Students may attempt the Written Qualifying Exams as many times as they wish prior to the deadline for passing all three exams, but must pass at least one exam by the end of summer prior to the third semester.

Students who do not pass the Written Qualifying Exams on schedule will be allowed to complete requirements of the MS degree but will not be allowed to continue in the PhD program or receive financial support after the second year.

Syllabi of the Written Qualifying Exams

Note that keying written qualifying exam syllabi to specific sections of a standard textbook provides students with very precise information about what they are expected to know for the exam. However, the material covered in the exams can be found in many standard books on the topics. Students may use the book of their choice or consult additional books when preparing for the exam. The instructor of the core courses Math 16:640:551, 16:640:503, and Math 16:640:501, which cover the major part of  material of the written exams, may also teach the course based on different books/lecture notes.

Algebra

 Textbooks:

1. [1]  Paolo  Aluffi,  Algebra: Chapter 0, Grad. Stud. Math., 104, American Mathematical Society, Providence, RI, 2009. ISBN-13:978-0-8218-4781-7
   
2. [2]  Michael  Artin, Algebra, 2nd edition, Pearson, 2010. ISBN-13:978-0-1324-1377-0
   
3. [3]  Dummit and Foote “Abstract Algebra” book, 3rd edition ISBN: 978-0-471-43334-7
   
4. [4] Notes from Rutgers Algebra Bootcamp, https://sites.math.rutgers.edu/~asbuch/notes/algbootcamp.pdf
   

• Groups and actions of groups on sets (Chapters 0, 1, 2, 3, 4, 5 in [3]): a) Subgroups, normal subgroups. b) Group actions, group homomorphisms. c) Symmetric groups. d) Direct and semi-direct products of groups. e) The Sylow theorems. f) The Jordan-H\"older Theorem. g) Free groups
• Rings (Chapters 7, 8, 9.1-9.5, 15.1, up to and including Corollary 5 in [3]): a) Ideals, maximal and prime Ideals. b) Polynomial rings. c) Hilbert Basis theorem. d) Principal ideal domains and unique factorization.
• Modules and linear algebra (Sections 10.1-10.3, beginning of 10.5, 11.1 -11.4 in [3], also [4] for linear algebra): a) Finitely generated modules over a PID, application to Jordan and rational canonical forms. b) Vector spaces and linear transformations, matrices, bases, change of bases. c) Bilinear and quadratic forms including inner product spaces, alternating and symmetric forms. d) Exact sequences.

Complex Analysis and Advanced Calculus

 Textbooks:

1. [1]  Paul Blanchard, Robert Devaney and Glen Hall, Differential Equations, 4th edition, published by Brooks/Col3
2. [2]  Stephen D. Fisher, Complex variables, 2nd Edition, Dover, ISBN 0-486-40679-2
3. [3]  Endre Pap, Complex Analysis through Examples and Exercises, Kluwer Texts Math. Sci., 21 Kluwer Academic Publishers Group, Dordrecht, 1999. x+337 pp. ISBN:0-7923-5787-6
4. [4] Walter Rudin, Principles of mathematical analysis, 3rd edition, McGraw-Hill Book Co., New York-Auckland-Du ̈sseldorf, 1976. x+342 pp

• Differential and integral calculus of one and several variables (Chapter 5, 6, 9 and 10 [4], Chapter 1, 2 and 3 [1]): Differentiability in one and several variables; Riemann integral and its fundamental properties in one variable; line, surface integrals including Green’s theorem, Gauss’ divergence theorem, Stoke’s theorem and Jacobians; implicit and inverse function theorems; elementary differential equations.
• Holomorphic functions (Chapter 1 and Chapter 2 [2], Chapter 3 [3]): The complex derivative; holomorphic functions on an open set; special functions such as the exponential, logarithm and trigonometric functions; Cauchy-Riemann equations; harmonic functions; the Cauchy inequalities and the maximum principle.
• Complex integration (Chapter 2 [2], Chapter 5 and 8 [3]): The Cauchy integral theorem and the Cauchy integral formula; the residue theorem and evaluation of definite integrals; Rouche’s theorem.
• Singularities and series (Chapter 2 and 3 [2], Chapter 6 and 7 [3]): Taylor and Laurent expansions of holomorphic functions; behavior near isolated singularities of holomorphic functions.
• Conformal maps (Chapter 3 [2], Chapter 4 [3]): The Riemann sphere and the extended complex plane; Riemann mapping theorem; mapping properties of holomorphic functions such as the exponential, the logarithm and the Mo ̈bius transformations

Click here  https://mathanalysis.pages.dev/notes for comprehensive and self-contained lecture notes for Math 503 (Complex Analysis) which provide the core material for the exam.

Real Analysis and Point Set Topology

Textbook: Gerald B. Folland “Real Analysis: Modern Techniques and Their Applications”, 2nd Edition, ISBN:978-0471317166.

• σ-algebras; Measures; Outer measures; Borel and Lebesgue measures on the real line;  Sections 1.1–1.5.
• Measurable functions; Integration of non-negative functions; Integration of complex functions; Modes of convergence (in particular convergence almost everywhere and convergence in measure); Convergence Theorems: Fatou’s Lemma, The Monotone Convergence Theorem and the Dominated convergence Theorem. Finite product measures: the Fubini-Tonneli Theorem; The d-dimensional Lebesgue measure and integral; Integration in polar coordinates; Sections 2.1–2.7.
• Signed measures; The Lebesgue–Radon–Nikodym theorem and the Lebesgue Differentiation Theorem; Uniform integrability. Complex measures; Differentiation on Euclidean space; Functions of bounded variation and absolutely continuous functions; — Sections 3.1–3.5.
• Point set topology; Topological and metric spaces; Continuous maps, Pointwise and uni- form Topology; Compact and connected spaces; Locally compact Hausdorff spaces; Metric spaces, sequential compactness, Dini’s Theorem, The Stone–Weierstrass theorem, Tychonoff’s Theorem; — Section 4.1–4.7.
• Basic Theory of Lp Spaces, including completeness and the density of nice functions; Hilbert spaces, the Riesz Representation Theorem, and weak convergence, the Projection Lemma, Bessel’s inequality, orthonormal bases; — Sections 5.1, 5.2, and 5.5. The Dual of Lp and weak convergence; Sections 6.1, and 6.2.

 Click here  https://mathanalysis.pages.dev/notes for comprehensive and self-contained lecture notes for Math 501 (Real Analysis) which provide the core material for the exam.

 Sample exam in current format

A complete solution to the August 2016 exam is posted here to serve as a model for students to learn how much justification and detail one should strive to provide --- these solutions are not worked out in a timed setting as the solutions to the real exams are, so can afford to contain more complete arguments; students can get full or close to full credits when their solutions contain all the key ingredients, with perhaps less detail than those contained here. A complete solution to the January 2011 exam, which is in the older format, is posted here.

Please note that this exam, like all exams before Summer 2014, followed a different format. A sample exam in the current format, culled from problems of earlier exams, can be found here in pdf format.

Prior versions of the exam (New format)

The Mathematics department offers a TA Training program each spring semester. All students in the Ph.D. program are required to participate in this program during their first year in the program. Exceptions can be made at the discretion of the graduate director.  Satisfactory completion of the teacher training program is required for the Ph.D. program and for receiving a TA appointment (with corresponding funding) after the first year of studies.

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An Oral Qualifying Exam (80 to 120 minutes in length) administered by a committee of four faculty on a specialized syllabus selected by the student in consultation with the Committee. Each student must select an Oral Qualifying Exam Committee Chair within one (1) calendar year of passing their Written Qualifying exams. In consultation with their Oral Exam Committee Chair, each student is strongly encouraged to schedule their Oral Qualifying Exam as early in their program as possible and no later than March 1 of their 6^th semester in the program. The Oral Exam Committee Chair will usually become the student’s Research Advisor after passing the Oral Qualifying Exam and being admitted to Candidacy. Please email the Graduate Administrative Assistant () and ask her to send you an Oral Qualifying Exam application packet. Return the completed application and exam syllabus to the Graduate Administrative Assistant via email no later than one month before the exam.

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A student is designated a "Candidate for a Ph.D. degree" upon application to the graduate school, after completing the written, oral and foreign language exams (if required by the program; our program no longer has a foreign language requirement). This marks the official beginning of the student's dissertation work. A total of 72 credits with a minimum being 24 Research credits are required to receive a PhD degree.

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A student must satisfactorily complete 24 specially designated "research credits".

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The doctoral dissertation (or thesis) is completed under the direction of a thesis advisor, who must be a member of the Mathematics graduate faculty.

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A written dissertation based on original research in Mathematics, completed under the direction of a member of the graduate faculty

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Traditional Option

For the traditional option, there are two options explained below with the credit requirements.  Your courses should be chosen with the approval of the Graduate Director. At least 18 of these credits must be in courses offered by the Graduate Program in Mathematics. Specifically required are (i) one of the courses 16:640:501 Theory of Functions of a Real Variable I, 16:640:503 Theory of Functions of a Complex Variable I, 16:640:515 Ordinary Differential Equations, and 16:642:516 Applied Partial Differential Equations, (ii) 16:640:551 Abstract Algebra I and (iii) a course in computer science, statistics, or some area of applied mathematics within the department.

There are two Masters Degree options:

Option 1 - Section A. with Thesis  - write a thesis, 24 course credits, 6 Research credits, oral defense with 3 Graduate faculty members as your committee (this includes your advisor). 

Option 2 - Section B - Writing Requirement (Non-Thesis) - Write an essay, 30 course credits.  An oral defense is not required, but can be requested by the Graduate Director.  

Requests for transfer credit toward their degree for courses taken at other universities are handled on a case-by-case basis, according to the rules of the department and the university.

General requirements for graduate degrees at Rutgers are governed by the rules of the Graduate School - New Brunswick and are listed in the current catalog. These include how and when credit can be transferred and how many credits can be taken each semester.