## Introduction

Welcome to the graduate program in mathematics at Temple University. This handbook, containing information specific to the mathematics graduate program, has been prepared to supplement the more general Temple University Graduate Bulletin.

Each graduate student in the Department of Mathematics has a faculty advisor. Initially the Director of Graduate Studies typically acts as the student's graduate advisor or assigns an advisor. At appropriate later stages in their studies, students choose advisors based on their research interests.

## Degree Requirements

### The M.S. Degree

The Master of Science degree requires thirty credits of courses at the 5000 level or above. The program of study must be designed in coordination with a mathematics faculty advisor and approved by the departmental Graduate Committee. With the approval of faculty advisor and Graduate Committee, relevant courses from departments other than mathematics may be included.

The M.S. degree is offered with an optional Concentration in Applied and Computational Mathematics. The concentration is designed for students interested in incorporating advanced study in mathematical and computational methods in the Master of Science program. Students pursuing this concentration will complete at least 15 credits of coursework in applied and computational mathematics within their 30-credit degree program.

Program completion. After satisfying the 30-credit course requirement, students may choose between the following three options for culminating events of their M.S. degree:

Students who have taken graduate courses at other institutions, or at Temple University prior to matriculation, may apply for advanced standing credit. The basic guidelines can be found here. The Department of Mathematics also has the following rules regarding advanced standing credit:

1. All applications for advanced standing credit are reviewed by the Graduate Committee of the Department of Mathematics, and may be denied if the committee decides that the courses involved are substantially inferior to similar courses offered by this department.

2. The credits must be equivalent to coursework offered at Temple with a grade of "B" or better.

3. No course that was completed more than five years before the date of the application will be awarded credit.

4. Credit for courses substantially similar to courses taken since matriculation will not be awarded.

5. Advanced standing credit is only available for graduate level courses in mathematics, or courses in related fields that have a substantial mathematical content.

6. Applications for advanced standing credit are not considered until the student has completed at least three graduate courses with a total of at least 9 credit hours.

7. The maximum number of credits a student in the M.S. program may transfer is 6.

### The Master's Thesis

Students who choose to submit a Master's thesis must select a faculty advisor and a thesis advisory committee. These arrangements are subject to the approval of the mathematics Graduate Committee. The date, time, and location of a thesis defense are set by the Graduate Chair in consultation with the student's advisory committee.

Formatting requirements for Master's theses can be found here.

### Time limit

Full time students must complete the requirements for the M.S. degree within three years.

### The Ph.D. Degree

Students are required to take at least sixteen graduate courses, comprising a total of 48 credit hours, and six additional credit hours of 9994, 9998 or 9999, with a minimum of two credits of 9999 (Dissertation Research). The courses, which are chosen with the advice and consent of the student's advisor, should include the foundational 8000-level courses for the topics in which the student plans to take the Ph.D. Comprehensive Examination. These courses should be taken during the first two years of graduate study. Students who have had graduate courses in these subjects prior to admission may omit some or all of the above courses with the consent of their advisor and the Graduate Committee.

Students are further required to obtain a Ph.D. pass on the Ph.D. Comprehensive Examination, pass the Ph.D. Preliminary Examination, and write and successfully defend a research dissertation.

### Eligibility for the Ph.D. Preliminary Examination

To be eligible for the Ph.D. Preliminary Examination, students must have obtained a Ph.D. pass on the Comprehensive Examination.

### Elevation to Candidacy

A student who has passed the Preliminary Examination is ready to begin working on his or her dissertation. A Dissertation Advisory Committee, or Doctoral Advisory Committee, consisting of the student's dissertation supervisor and two other faculty members, is formed. With the help of this committee, the student writes a dissertation proposal, specifying what the student intends to accomplish in his or her dissertation. When a dissertation proposal, initialed for approval by each member of his or her Dissertation Advisory Committee is filed, the student is a Candidate for the Ph.D. degree.

Upon completion of the dissertation, the Candidate gives a public lecture of one hour's duration, called the dissertation defense or oral defense. Prior to the dissertation defense, a Dissertation Examining Committee must be formed, including an Outside Examiner. At the conclusion of the defense, the Dissertation Examining Committee decides if the defense was successful.

### Time Limit

All requirements for the Ph.D. degree should be fulfilled within six years of graduate study at Temple University.

### Ph.D. Program Transfer Credit and Advanced Standing

Students who have taken graduate courses at other institutions, or at Temple University prior to matriculation, may apply for advanced standing credit. The rules for advanced standing credit toward the Ph.D. degree are identical to the corresponding rules for the M.S. degree above, except the maximum number of advanced standing credits that may be awarded is 30.

A current listing of graduate courses in mathematics can be found here.

• A student who receives more than two grades below "B-" or more than one grade of "F" will be dismissed for failure to maintain satisfactory grades.
• A minimum cumulative GPA of 3.0 is required in order to graduate.
• To remain in Academic Good Standing in the University, a non-matriculated or matriculated graduate student must achieve a semester GPA of at least 3.0 for each semester and maintain a cumulative GPA of at least 3.0 for all work completed at Temple University. The Academic Good Standing Policy operates in conjunction with preceding policies concerning substandard grades.

The department offer two examination options for the M.S. degree: A student can either pass the Master's Comprehensive Examination or obtain a Master's pass on the Ph.D. Comprehensive Examination.

## The Master's Comprehensive Examination

For students selecting this option, a comprehensive written Master's Exam will be composed by at least two departmental Graduate Faculty. The topics covered should correspond to the student's approved (by the Graduate Committee) program of study. The exam will be graded by at least two mathematics faculty members, with grades of either Pass or Fail.

Students interested in taking the Master's Comprehensive Examination are required to make a written request to the Graduate Chair at least four weeks in advance.

If the examination is failed, it may be taken again once, or the student may attempt a Master's pass on the Ph.D. comprehensive examinations.

### Algebra

Groups: definitions of subgroups, cosets, normal subgroups, quotient groups, homomorphisms, kernels, and automorphisms; automorphism groups, permutation groups, and more general actions of groups on sets; abelian groups, Lagrange's theorem, the fundamental theorem of group homomorphisms, the first and second isomomorphism theorems; familiarity with important examples, including cyclic groups, symmetric groups, alternating groups, the dihedral groups, the group of quaternions; commutator subgroups, and the center of a group; conjugacy classes and centralizers of group elements; the normalizer of a subgroup.

Rings: definitions of rings, integral domains, division rings, and fields; ring homomorphisms and their kernels; ideals and quotient rings; prime ideals, maximal ideals and their connection with fields; polynomial rings, Euclidean rings, principal ideal rings, and unique factorization domains; Chinese remainder theorem.

Linear Algebra: definitions, examples, and basic properties of vector spaces; subspaces, homomorphisms (linear transformations) and their connection with matrices; quotient spaces; linear span and linear independence; dimension; traces and determinants of linear transformations.

Fields: the characteristic of a field, field of fractions of an integral domain, extension fields, degree, algebraic extensions, roots of polynomials.

REFERENCES: 1. Dummit, David S.; Foote, Richard M., Abstract Algebra. 2. Hungerford, Thomas W., Algebra. 3. Jacobson, Nathan, Basic Algebra, I. 4. Jacobson, Nathan, Basic Algebra, II. 5. Herstein, I. N., Topics in Algebra.

### Complex Analysis

Complex numbers, analytic functions, Cauchy Integral Formula, Liouville 's Theorem, Uniqueness, Maximum Modulus, and Mean Value theorems for analytic functions. Morera's Theorem and the Schwartz Reflection Principle. Isolated singularities and Laurent expansions. The residue Theorem and evaluation of definite integrals by contour integrals techniques. Conformal maps.

REFERENCE: Conway, John B., Functions of One Complex Variable.

### Differential Equations

Ordinary Differential Equations Existence and uniqueness theorems for initial value problems. Linear equations with constant and periodic coefficients. Elementary Sturm-Liouville theory.

REFERENCE: Coddington, Earl A.; Levinson, Norman, Theory of Ordinary Differential Equations.

### Partial Differential Equations

Cauchy-Kovalevsky Theorem; its limitations. Classifications of second order linear equations. Well posed problems. Solution by separation of variables. Use of the maximum principles for the Laplace and heat equations. The method of characteristics. Green's function.

REFERENCE: Zachmanoglou, E. C.; Thoe, Dale W., Introduction to Partial Differential Equations with Applications.

### Numerical Analysis

Interpolation by polynomials. Numerical integration. Solution of linear systems of equations, and ordinary differential equations.

REFERENCE: Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis.

### Probability

Probability spaces, random variables, independence, distribution functions, conditional probabilities, Expectation, weak and strong convergence, Strong law of Large Numbers, Zero-One laws. Characteristic functions, Special distributions: Gaussian, Exponential, Poisson, and Binomial distributions.

REFERENCES: 1. Galambos, Janos., Advanced Probability Theory. 2. Rohatgi, V. K., An introduction to probability theory and mathematical statistics.

### Real Analysis

The real and complex number systems. Elementary properties and examples of metric spaces, including convergence, completeness, compactness, and separability. Continuous functions. Criteria for compactness in $\bf R$, in ${\bf R}^d$ and in $C[0,1]$. The Stone-Weierstrass Theorem. Diffeomorphisms in ${\bf R}^d$, and the inverse function theorem in ${\bf R}^d$. Lebesgue integration in $\bf R$ and ${\bf R}^d$.

REFERENCE: Rudin, Walter, Principles of Mathematical Analysis.

### Topology

Topological spaces. Continuous functions. Connectedness, compactness, and separation properties.Product and Quotient topologies. Urysohn Metrization theorem. Fundamental groups and covering spaces.

REFERENCE: Munkres, James, Topology: a first course.

## The Ph.D. Comprehensive Examination

The Ph.D. Comprehensive Examination is a written exam comprised of three separate sections selected from the following areas:

• Algebra
• Differential Geometry & Topology
• Complex Analysis
• Numerical Analysis
• Partial Differential Equations (discontinued)
• Real Analysis
• Applied Mathematics
Students can choose any three of these sections for their examination. Once exams in three different topics have been attempted, an exam in a fourth topic can only be taken under exceptional circumstances, and only after specific approval by the Graduate Committee. Each section is a three hour test based primarily on the corresponding two-semester 8000-level graduate course sequence. The separate section tests are given on different days, during one two-week period, twice a year -- once in August just before the beginning of the Fall semester and once in January during the two weeks prior to the beginning of the Spring semester. Students need not take all three sections of the test during the same two-week period. Each section test may be repeated once to obtain a higher grade.

Each of the three-hour section tests is further divided into two parts. Part I contains four questions, of which the student is asked to answer three; these questions are designed to test mastery of the facts of the subject. Part II contains three questions, of which two are to be answered; these questions test the ability to solve in-depth problems in the subject.

Students should begin taking the components of the Comprehensive Exam as soon as possible after finishing the corresponding coursework. Students are expected to complete and pass the Comprehensive Examination by August of their second year of study. While some delay in this schedule may be permitted, under exceptional circumstances, students not making good progress toward completing and passing their Comprehensive Examinations in a timely fashion will be asked to leave the Ph.D. program. Consult the timelines for specific information about time limits.

Incoming students may, with approval of the Graduate Chair, take up to three of the written PhD Comprehensive Exams once prior to their first semester of enrollment. Upon request be the student, any of these pre-enrollment attempts can be removed form the student's record.

Grading. Each section is graded independently by two faculty members, using a scale of 0 -- 25. The grades are compared, and reconciled in the event of a discrepancy. A total score of a least 60, with a score on each individual section test of at least 13, is required to pass. If a student falls slightly short of this standard, the Graduate Faculty may at their discretion recommend a grade of pass based on the student's entire academic record.

Master's Pass. A student who achieves a total score of at least 40 from the three sections of the examination, with no individual section below 8, has obtained a master's pass on the examination and has fulfilled the examination requirement for the M.S. degree. If one of the individual exam scores falls below 8 points, that exam may be repeated once, or the exam in a different topic may be attempted once, or the student may take the Master's Comprehensive Examination. As above, such arrangements are subject to approval by the Graduate Committee, and if a student falls slightly short of the required standards, then the Graduate Faculty may at their discretion recommend a grade of pass, basing their decision on the student's entire academic record.

### Syllabi for the Ph.D. Comprehensive Examination

All syllabi include the topics listed in the corresponding syllabi for the Master's Comprehensive Examination in Pure mathematics. In addition, the student should be familiar with the following:

### Algebra

Linear Algebra: eigenvalues and eigenvectors, minimal and characteristic polynomial of a matrix, canonical forms of matrices (Rational Canonical Form and Jordan Canonical Form), bilinear forms.

Groups: Sylow theorems, structure theorem for finitely generated abelian groups, solvable and nilpotent groups, simple groups.

Rings and modules: modules and submodules, structure of finitely generated modules over principal ideal domains, definition and basic properties of noetherian rings and modules.

Fields: existence of roots in an extension field, splitting fields, algebraic closure of a field, finite fields, types of field extensions (normal, separable, and Galois extensions), Galois theory (including the Fundamental Theorem of Galois Theory and the computation of Galois groups).

REFERENCES:

1. Dummit, David S.; Foote, Richard M.: Abstract Algebra.
2. Hungerford, Thomas W.: Algebra.
3. Jacobson, Nathan: Basic Algebra, I and II.
4. Lang, Serge: Algebra.

### Differential Geometry & Topology

Geometry content: smooth structures on manifolds; smooth functions and maps between manifolds; tangent and cotangent bundles; vector bundles; Frobenius integrability theorem; differential forms; tensors; integration and Stokes' theorem.

Topology content: fundamental group; van Kampen's theorem; covering space theory; singular homology; exact sequences: long exact sequence of a pair, Mayer-Vietoris sequence; de Rham cohomology and de Rham's theorem.

REFERENCES:

1. A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
2. J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
3. V. Guillemin, A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J., 1974.
4. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press; 2nd edition (August 19, 2002).
5. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, Chapter 11. Publish or Perish; 3rd edition (January 1, 1999).

### Complex Analysis

Power series, holomorphic functions, conformal maps, Cauchy's Theorem and Integral Formula, Morera's theorem; singularities, residues, Argument Principle, the Open Mapping Theorem, Rouche's Theorem, Maximum Modulus Theorem, Schwarz's Lemma, harmonic functions; spaces of holomorphic and meromorphic functions, Riemann Mapping Theorem; Weierstrass and Hadamard's Factorization Theorems; Picard's Theorems.

REFERENCE: John B. Conway: Functions of One Complex Variable.

### Partial Differential Equations

The Partial Differential Equations examination has been discontinued.

### Real Analysis

Functions of bounded variation, Riemann-Stieltjes integral. Lebesgue measure in ${\bf R}^d$, Caratheodory condition, existence of nonmeasurable sets. Measurable functions, convergence of sequences of measurable functions, Egorov's theorem, convergence in measure. Lebesgue integral in ${\bf R}^d$, relation between Riemann-Stieltjes and Lebesgue integrals. Fubini's theorem, applications. Lebesgue differentiation theorem, absolutely continuous and singular functions. $L^p$ spaces, Minkowski's inequality, metric properties of $L^p$ spaces. Maximal functions and approximations of the identity. Abstract measures and integration, absolutely continuous set functions, Lebesgue decomposition theorem, Radon-Nikodym's theorem. Abstract outer measures, Lebesgue Stieltjes measure, Hausdorff measure and dimension, examples. Riesz representation theorem of linear functionals in $L^p$ and over the continuous functions, duality, examples.

REFERENCES:

1. Wheeden R.; Zygmund A., Measure and Integral: An Introduction to Real analysis.
2. Stein, Elias M.; Shakarchi, Rami, Real Analysis: Measure Theory, Integration, and Hilbert spaces.
3. Makarov G., et. al., Selected Problems in Real Analysis.

### Applied Mathematics

Equations of continuum mechanics: derivation, non-dimensionalization, dimensional analysis, Fourier analysis, eigenfunction methods, separation, solvability, similarity, travelling waves, Sturm-Liouville theory, applications of Green's functions to physical problems.

Calculus of variations and optimal control: Euler-Lagrange equations, Hamilton's principle, Lagrange multipliers.

Dynamical systems: Floquet theory, phase plane analysis, stability, bifurcations, limit cycles and attractors, chaotic dynamics.

Asymptotic analysis: regular and singular perturbations, multiple scale analysis, boundary layers and matched asymptotics, asymptotic expansion of integrals.

REFERENCES:

1. V. Henner, T. Belozerova, and K. Forinash, Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions, A. K. Peters (2009).
2. I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall (1963).
3. M. H. Holmes, Introduction to Perturbation Methods, Springer (1995).
4. S. H. Strogatz, Nonlinear Dynamics and Chaos, Westview Press (2001).
5. H. F. Weinberger, Partial Differential Equations, Dover Publications (1965).

## Language Examinations

Beginning in Fall 2016, a language requirement is no longer in effect.

## Ph.D. Preliminary Examination

The Preliminary Examination is oral, and takes two hours. It should be taken by the end of the sixth semester, and must be passed by the end of the seventh semester. The student chooses the Chief Examiner with the advice and consent of the Graduate Committee, and with the consent of the proposed Chief Examiner. The Chief Examiner, in accepting his or her assignment, implicitly offers to be the student's dissertation supervisor if the examination is passed.

No student will be permitted to take the preliminary examination before passing the Ph.D. Written Comprehensive Examination.

Approximately half of the Preliminary Examination will be conducted by the chief examiner, who will ask questions in the area that the student has chosen as a specialty. The other half of the examination will be devoted to questions asked by other faculty members, on two or more topics related to the area of specialization. The exact description of the topics to be included in the examination is determined by the chief examiner, who will also be responsible for assigning examiners to cover these topics. The examination committee consists of the Chief Examiner, the examiners for the other topics, and any other faculty who choose to attend. All faculty in attendance may vote upon the outcome of the examination. The student will be considered to have passed if the Chief Examiner and at least half of the other faculty present vote in favor of passing.

The Preliminary Examination must be announced at least one week before it takes place. While any faculty member may attend, the Chief Examiner, the elementary examiners, and the Graduate Chair (or his or her surrogate) must be present during the entire examination. The examination will be moderated by the Graduate Chair, or a designated surrogate.

If failed, the preliminary examination may be repeated, in whole or in part, only once. A student failing the examination a second time is automatically dismissed from the program. In case of a repeat examination, the student is not required to have the same examining committee, or to have the same topics. Failure to retake and pass the preliminary examination within one semester will result in dismissal from the program.

A doctoral student who has completed all coursework for the degree but has not passed the preliminary examination, must register each Fall and Spring semester for 1 semester hour of course number 9994, "Preliminary Examination Preparation." The student must be registered for 9994 in the semester in which the examination is taken, including the Summer session. A student who is required to retake the preliminary examination in whole or in part must re-register for 1 semester hour of 9994 in the semester in which the examination will be retaken.

Further regulations concerning the preliminary examination may be found in the Temple University Graduate Academic Policies and Regulations Section of the Graduate Bulletin.

### Regulations Governing Departmental Examinations

Repetition of examinations: Students seeking to pass the Ph.D. Comprehensive Exam requirement may take each section of the exam at most twice. A student who has failed a section of the Ph.D. examination twice may still seek a master's pass on the Ph.D. examination or may take the Master's Comprehensive Examination.

## The Dissertation

The Ph.D. dissertation must be a substantial and original contribution to research in mathematics. It must consist of individual work, with only one author. Previously published work by the candidate may be included, if it represents research done while the student was enrolled in the Ph.D. Program in Mathematics at Temple University and was not used to obtain any other degree. Co-authored work can be included if the role of the candidate is clearly defined. Sections of joint work which cannot be attributed to the candidate alone must not be included in the body of the dissertation, but may be attached as an appendix. All work included in the dissertation must be logically connected and integrated into a unified whole, with a common introduction, conclusion, and bibliography. Existing copyright laws must not be violated. Furthermore, the dissertation must be written in a professional manner, consistent with standards found in current mathematical research publications. Consult the Temple University Graduate Academic Policies and Regulations Section of the Graduate Bulletin for further details.

The preparation and writing of the dissertation is supervised by the student's Dissertation Advisory Committee, also referred to as the student's Doctoral Advisory Committee. The committee must include at least three members of Temple University's Graduate Faculty. Two members of the committee, including a designated chairperson, must be members of the mathematics department. The chairperson is also referred to as the student's dissertation advisor or thesis advisor. Rules governing the makeup of the Doctoral Advisory Committee are listed in the Graduate Academic Policies and Regulations Section of the Graduate Bulletin.

The first step in preparing the dissertation is to write a Dissertation Proposal, which must be approved by the Candidate's Doctoral Advisory Committee. The proposal is kept on file by the department and the Graduate School, and if it becomes necessary to alter the proposal for any reason, the changes should be approved by the Doctoral Advisory Committee and Graduate Chair. These changes should then be filed with the original proposal.

The purpose of the proposal is to specify a set of results that can be reasonably expected to comprise a dissertation that the Doctoral Advisory Committee will approve. However, it may turn out, after the fact, that the research program outlined in the proposal is insufficient. In this case the student must develop a new research plan, in consultation with his or her thesis advisor and Doctoral Advisory Committee. A revised proposal, based on this new research plan, must then be approved by the student's Doctoral Advisory Committee and the Graduate Chair. The revised proposal is then filed with the original.

A student who has passed the preliminary examination but has not filed an approved dissertation proposal with the Graduate School by the last day to Drop/Add in the semester must register each Fall and Spring for course number 9998, "Pre-Dissertation Research."

The dissertation proposal must be filed with the department and the Graduate School within 30 days of sigining. The proposal must include an official Proposal Transmittal Form.

Further regulations concerning the Dissertation Proposal can be found in the Graduate Academic Policies and Regulations Section of the Graduate Bulletin.

After the dissertation proposal is filed, following the above guidelines, the student becomes a candidate for the Ph.D. Candidates should be registered for one or more credit hours of 9999 (Dissertation Research) during every semester until the dissertation is complete, unless they are on an approved leave of absence.

It is the candidate's responsibility to ensure that the dissertation manuscript conforms to Temple University guidelines, and the candidate should obtain the Dissertation Handbook from the Graduate School. A collection of LaTeX files with macros for producing a document conforming to these guidelines can be found here.

### The Dissertation Defense

When the dissertation is deemed complete by the candidate and the Doctoral Advisory Committee, a Dissertation Examination Committeeis formed.

The Dissertation Examining Committee is responsible for evaluating the quality of the dissertation and conducting the oral Dissertation Defense. The committee must include the members of the Doctoral Advisory Committee and at least one Outside Examiner not previously involved with the dissertation or the Doctoral Advisory Committee. The Dissertation Examining Committee must have a chairperson, and this chairperson must not be the chair of the Doctoral Advisory Committee.

The Outside Examiner may not be a faculty member in the candidate's degree program. This examiner must have a doctorate, and if the examiner is not a member of the Temple University faculty he or she must be approved by the dean of the Graduate School at least 2 weeks prior to the dissertation defense.

If any of the Examining Committee members are not members of the Graduate Faculty of Temple University, the chair of the Doctoral Advisory Committee must request approval by submitting the request form and a curriculum vitae to the dean of the Graduate School at least 4 weeks in advance of the defense. Approval must be received prior to posting announcements of the defense.

Further regulations governing this committee can be found in the Graduate Academic Policies and Regulations Section of the Graduate Bulletin.

The Dissertation Defense must be announced in writing at least ten days in advance, following the guidelines in the Graduate Academic Policies and Regulations Section of the Graduate Bulletin.

All members of the Dissertation Examining Committee must be physically present for the defense, except in the case of an emergency. The dean of the Graduate School may, in serious circumstances, give prior written approval for no more than one member to be absent. The candidate and chair of the Doctoral Advisory Committee must, however, both be present for a valid defense.

The Dissertation Examining Committee will meet at the conclusion of the Dissertation Defense and decide, by majority vote, if the candidate was successful.

Doctoral candidates who pass the oral defense may be required by the Examining Committee to make revisions to the dissertation as a condition of completing the degree. The Chair of the Doctoral Advisory Committee is typically responsible to review and approve revisions, although any member of the Dissertation Examining Committee may require the candidate to submit a final draft for approval. The final revised dissertation must be submitted to the Graduate School within 30 calendar days of the dissertation defense -- if not the defense is nullified, and another defense must be scheduled.

Further regulations governing the Dissertation Defense, the revision process, submitting the dissertation, and filing the dissertation can be found in the Graduate Academic Policies and Regulations Section of the Graduate Bulletin.

## Timelines

### Ph.D. degree

Comprehensive Examination: The comprehensive examination requirement should be completed by the end of the second year of studies. This notwithstanding, to be considered in good standing, students must take two of the three sections before the beginning of the fourth semester with a total score of at least 30, and pass all three components before the start of the sixth semester.

Preliminary Examination: The Ph.D. Preliminary Examination should be passed by the end of the sixth semester. It must be passed by the end of their seventh semester.

Dissertation Proposal: Students should present their Dissertation Proposal no later than the end of their forth year, but must do so before the end of their ninth semseter of studies.

### Financial Support

The majority of the graduate students in the mathematics department are supported by Teaching Assistantships. A smaller number of students are supported by Research Assistantships and Fellowships. Very rarely, students attend without financial support from Temple University.

Research Assistantships are primarily funded by external grants held by faculty supervisors. Assistantships and Fellowships cover tuition and provide stipends. Further details can be found in the Financial Aid Section of the Graduate Bulletin.

It is a Temple University regulation that a Teaching Assistant or Research Assistant must maintain a GPA of 3.25.