Written Qualifying Exam

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)