Dissertation & Exams
Dissertation
When planning a thesis defense, keep in mind university submission deadlines, the threeweek minimum that the examining committee requires to read the thesis as well as time needed after the defense for revisions. Also review the GW Libraries website for guidance on dissertation format and submission.
Note: Schedules for the next exam period will be posted when available.
Dissertation Process
 Assemble the examination committee.

The thesis advisor, in consultation with the graduate committee, assembles the final examination committee. This committee is made up of the thesis advisor (director), the codirector (if applicable), two readers and two additional examiners; at least one examiner must be from outside the department. Also, another person is selected to serve as chair of the defense.
 Review final thesis draft.

The PhD candidate, with the assistance of the advisor, must make every effort to ensure that the thesis is free of errors before it is submitted to the final examination committee. The thesis must be submitted to the committee at least one month before the defense. Thesis defenses are open to the public; the advisor should ensure that an announcement is sent to the entire department at least one month in advance.
 Conduct the dissertation defense.

The defense consists of an hourlong talk by the candidate on his or her research followed by several rounds of questions by the final examination committee, excluding the director (and codirector, if applicable). Once the question session ends, the defense is over and the chair dismisses the candidate and the audience.
The committee then meets to determine the outcome, which can be:
 The dissertation is acceptable as is,
 the dissertation is acceptable subject to requested revisions (in this case, the requested revisions are conveyed to the thesis advisor, a subset of the committee is appointed to check and ultimately signoff on the revised thesis, and a date is chosen by which revisions should be completed) or
 the dissertation is unacceptable (in this case, the student may write and submit a new thesis).
The chair facilitates this discussion and, at its conclusion, sees that the Final Examination Committee SignOff Form is completed.
 Submit written thesis.

Following a thesis defense at which the thesis was accepted as is, or following the completion of the required revisions and the subsequent filing of the Final Dissertation Approval Form, the successful PhD candidate submits his or her dissertation electronically, following the instructions on the Electronic Theses and Dissertations (ETD) website. This website has links to the ProQuest/University Microfilm (UM) Form, which must also be submitted electronically. (See the graduate committee chair for the username and password needed for the UM form.)
Paper copies of two additional forms must be submitted to CCAS (Phillips 107): the ETD Access Approval Form (signed by the student and the thesis advisor) and the Survey of Earned Doctorates. It is highly recommended that all of these items be submitted well in advance of the CCAS deadline. After this material has been submitted, the graduate committee chair approves the candidate for graduation.
Qualifying Exams
The qualifying exams are offered twice a year: once in August, before the beginning of the fall semester, and once in January, before the beginning of the spring semester. A student may take each exam at most twice. Students have four total opportunities to pass the exams over the course of their first two years (first year August, first year January, second year August and second year January).
Questions about the qualifying exams should be directed to Joel Lewis, the exam coordinator.
In consultation with their advisors, students select exams from the four designated options:
 Algebra
 Analysis
 Applied Mathematics
 Topology
Exam Syllabi
 Algebra Qual Exam Syllabus

 Groups, subgroups, normal subgroups, quotient groups, homomorphisms
 Isomorphism theorems, Cayley’s theorem
 Normal and composition series, JordanHolder theorem, solvable groups
 Simple groups
 Direct products, structure theorem for finitely generated abelian groups
 Groups acting on sets, class equation, Sylow’s theorems
 Free groups, group presentations
 Elementary properties of rings, ideals, Zorn’s lemma, quotient rings
 Homomorphism and isomorphism theorems, field of fractions of an integral domain, polynomial rings, Noetherian rings, algebras
 Unique factorization domains, principal ideal domains, Euclidean domains, prime ideals
References
 Larry Grove, Algebra, Dover Publications, 2004.
 Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics Vol. 73, Springer Verlag, 2003.
 Nathan Jacobson, Basic Algebra I and Basic Algebra II, Dover Publications, 2009.
 Analysis Qual Exam Syllabus

The background for this exam consists of a good knowledge of advanced calculus, such as one might acquire in MATH 4239 & 4240 or MATH 6201 & 6202. This includes:
 Continuity, uniform continuity and differentiability of real valued functions of a real variable;
 Completeness of R and compactness of [0, 1];
 Complete, connected and compact metric spaces, including basic topological concepts like open sets, limit points, etc.;
 The continuity and derivatives of functions f : Rn → Rm;
 The implicit and inverse function theorems.
The remaining topics are mostly covered in MATH 6214. Any topics not covered in that class are the student’s responsibility.
 Lebesgue measure and integration in abstract spaces
 Fatou’s lemma; the dominated and monotone convergence theorems
 Constructing Lebesgue measure on [0, 1] and on Rn
 Functions of bounded variation on [0, 1], their differentiability (Lebesgue almost everywhere), and absolute continuity for real valued functions on [0, 1]
 The RadonNikodym and Lebesgue decomposition theorems
 Measures on product spaces, and the Fubini theorem
 Hölder’s inequality. The L^{p} spaces and their properties, including completeness
 Relations between different modes of convergence: uniform, pointwise almost everywhere, L^{p} and in measure
 Applied Mathematics Qual Exam Syllabus

Applied mathematics focuses on mathematical techniques that yield practical information on the problems of the natural world. The fundamental processes in the natural world are to a large extent described by partial differential equations.
This syllabus focuses on a number of basic partial differential equations and some standard techniques used to analyze these equations. We see how some first order equations can be solved along characteristic curves, how explicit solutions may be found for the wave equation in the entire space, and how the Fourier transform is used to solve linear evolution equations. We learn to analyze the Laplacian operator on a bounded domain.
Afterwards we will solve the linear wave equation, the heat equations and Schrödinger equation via various methods. Calculus of variations is a rich source of problems from physical sciences and other branches of mathematics and we learn how this approach leads to differential equations.
Basic References
 Applied Mathematics by David Logan
 Partial Differential Equations (Graduate Studies in Mathematics) by Lawrence C. Evans.
1. Onedimensional boundary value problems
 Eigenvalues and eigenfunctions
 SturmLiouville theory
 Green’s functions, integral equations, Arzel`aAscoli theorem
2. First order equations, characteristics and shock waves
 First order equations, the method of characteristics
 Failure of the characteristic method, weak solutions
 Shock waves
3. Linear elliptic and evolution equation
 Fourier Transforms
 The Heat equation, Schrödinger equation and kernels of linear operators
 Separation of variables
 The Laplace operator on a bounded region
 Green’s functions
4. Calculus of variations
 Variational problems, EulerLagrange equations
 The classical harmonic oscillator, the pendulum, minimal surfaces
 The Dirichlet boundary condition, the Neumann boundary condition
 Variational problems with constraints
 Topology Qual Exam Syllabus

 Topological spaces, basis, subbasis, open sets and closed sets, quotient topology, product topology, continuous maps and homeomorphisms, finite topological spaces.
 Connectedness, path connectedness, compactness, compactness in metric spaces, local compactness, paracompactness.
 Metric topology, equivalent metrics, complete metric spaces.
 Product topology, Tychonoff Theorem, Alexander subbase Theorem.
 Separation axioms and countability axioms, Urysohn lemma, Tietze extension theorem.
 Brouwer fixedpoint theorem via Sperner Lemma.
 Fundamental groups, homotopic maps, strong deformation retracts, Fundamental groups for Rn, Sn, graphs, torus and link exteriors.
 Covering spaces, universal covering, classifications of covering spaces.
 The first homology group as the abelianization of a fundamental group.
References
 A. Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002. xii+544
 J. Munkres, Topology: A First Course, PrenticeHall.
 O. Ya. Viro, O. Ya. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary Topology Problem Textbook, AMS, 2008
Details and Scheduling
 Scoring and Results

Each exam is graded out of 30 points. The student’s subject score in an area is the highest qualifying exam score received in that area. The student passes the qualifying exams if the total of any three subject scores equals 50 or higher.
For a fulltime PhD student, failure to pass the qualifying exams by the start of the fourth semester in the PhD program results in the termination of his or her PhD program. Students who are receiving funding as graduate teaching assistants must pass the qualifying exams by January of their second year to be eligible for funding in their third year.
 Exam Format and Scheduling

The qualifying exams must be completed successfully before the specialty exam may be attempted. Students should let the graduate committee know which exams they intend to take by the middle of the semester preceding the desired exam session.
The qualifying exam in each area is
 Two hours long
 Scored out of 30 points
 Based on the corresponding graduate core course
The exams are given on two days during the week preceding the first week of classes each semester. On each of these two days two exams are given: one in the morning and one in the afternoon. It is not required that all the exams be taken for the first time during the same exam session. Students should consult their advisors to develop a sound plan for when to take the exams.
 Past Exams

While the graded exams remain in the department files, students can ask the department for copies of their own graded exams. Obtaining copies of one's exams and discussing them with the graders is especially important if an exam has to be retaken.
Copies of past exams are also available for download. Note that some older exams may contain problems on subjects no longer required by the current qualifying exam syllabi.
 Algebra
 Analysis
 Applied Math
 Calculus/Linear Algebra
 Combinatorics
 Differential Equations
 Numerical Analysis
 Topology
If you require these documents in an alternate format, please contact the Department of Mathematics at [email protected].
 Fall 2024 Schedule

 Algebra: Tuesday, August 13, 10 a.m. – 12 p.m.
 Analysis: Tuesday, August 13, 1 p.m. – 3 p.m.
 Topology: Thursday, August 15, 10 a.m. – 12 p.m.
 Applied math: Thursday, August 15, 1 p.m.  3 p.m.
Specialty Exams
The specialty exam is an oral exam in the student's intended research area. It is conducted by a specialty exam committee made up of three faculty members (usually all in the department), including a prospective dissertation advisor. Starting soon after the completion of the qualifying exams, students typically spend the better part of a year preparing for and then taking the specialty exam.
 Preparation

The prospective advisor prepares a syllabus; in addition to listing the exam topics and recommended sources, this document sets a target completion date and names the exam committee. Early in the exam preparation process, the graduate committee is informed of the arrangements and the syllabus is filed in the student's departmental file. The exam committee (especially the advisor) assists the student in preparing for the exam.
 Committee

The advisor chairs the exam, which is conducted by the exam committee and is open to attendance by all mathematics faculty members and graduate students. The advisor should ensure that an announcement is sent to the entire department at least a week in advance.
 Scoring and Results

The exam committee meets privately immediately after the exam to determine the outcome, which can be Pass with Distinction, Pass, Low Pass or Fail. A student may not take the specialty exam more than twice. The second attempt, if needed, must be scheduled within one semester of the first.
Soon after passing the specialty exam, if not earlier, the student starts working with a thesis advisor. A student who wishes to change thesis advisors may do so at any time; all that is required is the consent of the new advisor. The new advisor may require the student to retake the specialty exam.