Handbook for Graduate Students
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.
Graduate Advisors
Each graduate student (in the Department of Mathematics) has a faculty advisor. At the start of their graduate studies, students have their faculty advisors chosen for them. At appropriate later stages in their studies, students choose advisors based on research interests.
Degree Requirements
The M.A. Degree
There are two tracks: pure and applied. Each track requires ten graduate courses, at least seven of which must be in mathematics. Students must also either pass a Master's Comprehensive Examination, obtain a master's pass on the Ph.D. Comprehensive Examination, or write an acceptable Master's Thesis. Courses are chosen with the advice and consent of the students' faculty advisors. Students in the pure track must take Math 8031 (Probability), Math 8041 (Real Analysis), Math 8051 (Complex Analysis), Math 8161 (Topology), and Math 8011 (Abstract Algebra). Remaining courses are chosen according to the students' individual interests. Many students begin the pure track by taking one or both courses in the sequence Math 5041 -- Math 5042 (Concepts of Analysis I -- II), to facilitate their transition to the 8000-level courses forming the core of the curriculum.
In the applied track, each student designs a program of study with his or her faculty advisor and is encouraged to take relevant graduate courses in other departments.
Students who have taken graduate courses at other institutions, or at Temple University prior to matriculation, may apply for transfer credit. The basic guidelines can be found here. The Department of Mathematics also has the following rules regarding M.A. transfer credit:
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Applications for transfer credit will not be considered until a student has completed at least one semester of full time graduate study, or the equivalent, if the student is studying part time.
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All applications for transfer 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.
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No course that was completed more than five years before the date of the application will be awarded credit.
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Credit for courses substantially similar to courses taken since matriculation will not be awarded.
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If a course was taken before the bachelor's degree was earned, it cannot be awarded transfer credit.
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Transfer credit is only available for graduate level courses in mathematics, or courses in related fields that have a substantial mathematical content.
Students in either mathematics track may satisfy the comprehensive examination requirement either by passing the Master's Comprehensive Examination for their track , or by obtaining a master's pass on the Ph.D. Comprehensive Examination.
The Master's Thesis
A student who chooses to submit a master's thesis must arrange for a department faculty member to supervise the writing of the thesis. All such arrangements are subject to the approval of the Graduate Committee.
Formatting requirements for master's theses can be found here.
Time limit
Full time students must complete the requirements for the M.A. degree within three years.
The Ph.D. Degree
Students are required to take a total of at least sixteen graduate courses comprising 48 credit hours, and six additional credit hours of 9999 (Dissertation Research). The courses, which are chosen with the advice and consent of the student's advisor, should typically include the following foundational courses: 8041 -- 8042 (Real Analysis I -- II), 8051 -- 8052 (Complex Analysis I -- II), and 8011 -- 8012 (Abstract Algebra I -- II). 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.
All students are required to present at least two seminar talks on topics related to their research. Approval of the Graduate Committee is needed to fulfill this requirement in other ways.
Students are further required to obtain a Ph.D. pass on the Ph.D. Comprehensive Examination, pass the foreign language 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, and passed the foreign language 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 seven years of graduate study at Temple University.
Ph.D. Program Transfer Credit and Advanced Standing
The rules for transfer credit toward the Ph.D. degree are identical to the corresponding rules for the M.A. degree above.
A student who has completed an M.A. degree at another institution may apply for advanced standing. Advanced standing credit is only given for course work involving graduate-level mathematics. Please consult the Graduate Academic Policies and Regulations Section of the Graduate Bulletin for further details.
Graduate Courses in Mathematics
A current listing of graduate courses in mathematics can be found here.
Grades
Please refer to the grading policies outlined in the Graduate Academic Policies and Regulations Section of the Graduate Bulletin for a full account. However, please note the following:
- 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.A. degree: A student can either pass the Master's Examination or obtain a master's pass on the Ph.D. Comprehensive Examination.
The Master's Examination
The Master's Comprehensive Examination is taken by M.A. students (who choose this option) at the conclusion of their graduate study. Each exam is prepared and graded by at least two members of the Graduate Faculty. Students are given a grade of either Pass or Fail.
There are pure and applied versions of the exam.
Subject Areas:
- Applied Mathematics track
- Differential equations
- Numerical Analysis
- Linear Algebra
- Probability
- Statistics
- Pure Mathematics track
- Algebra
- Complex Analysis
- Real Analysis
- Topology
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 masters pass on the Ph.D. comprehensive examinations.
Syllabi for the Master's Examination
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 R, in Rn and in C[0,1]. The Stone-Weierstrass Theorem. Diffeomorphisms in Rn, and the inverse function theorem in Rn. Lebesgue integration in R and Rn.
REFERENCE:
Rudin, Walter, Principles of Mathematical Analysis.
Statistics
Random variables, Transformation of random variables. Order statistics. Sufficiency and completeness. Point estimation. Hypothesis testing. Confidence regions.
REFERENCES:
1. Roussas, George, G., First Course in Mathematical
Statistics
2. Wilks, Samuel S., Mathematical Statistics.
3. Rohatgi, V. K., An Introduction to Probability Theory and
Mathematical Statistics.
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: Algebra, Complex Analysis, and Real Analysis. 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 week, twice a year -- once in August during the week prior to the beginning of the Fall Semester and once in January during the week prior to the beginning of the Spring Semester. Students do not have to take all three of the section tests during the same one 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 Algebra, Real Analysis, and Complex Analysis 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.
Students who begin their study at Temple already having successfully completed graduate coursework in one or more of the section subjects are encouraged to begin taking the Comprehensive Exam within their first year here.
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 fails the examination with a score of at least 40, with no individual section below 8, has obtained a master's pass on the examination and has fulfilled the examination requirement for the M.A. degree. As above, 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.
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, canonical forms of matrices (particularly 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, Galois theory (including the Fundamental Theorem of Galois Theory), 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.
4. Jacobson, Nathan, Basic Algebra, II.
5. Herstein, I. N., Topics in Algebra.
6. Lang, Serge, Algebra.
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.
Real Analysis
Functions of bounded variation, Riemann-Stieltjes integral. Lebesgue measure in Rn, Caratheodory condition, existence of nonmeasurable sets. Measurable functions, convergence of sequences of measurable functions, Egorov's theorem, convergence in measure. Lebesgue integral in Rn, relation between Riemann-Stieltjes and Lebesgue integrals. Fubini's theorem, applications. Lebesgue differentiation theorem, absolutely continuous and singular functions. Lp spaces, Minkowski's inequality, metric properties of Lp 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 Lp 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.
Language Examinations
The foreign language requirement for the Ph.D. degree is a demonstrated reading knowledge in one of the following three languages: French, German, or Russian. A student whose native language is one of these three will be given credit without examination. For other students, language examinations are arranged by request. The exam consists of translating approximately two pages of mathematical text in the examinee's field of interest. Use of a dictionary is permitted. A time limit will be imposed by the examiner, who may take into account the length of the passage to be translated.
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.
Ph.D. Preliminary Examination
The Preliminary Examination is oral, and takes two hours. 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 and satisfying the foreign language requirement.
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 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 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.
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.
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.
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.
Offices for Graduate Students
The department will attempt to provide office space for all full-time graduate students. All Fellows, Teaching Assistants, and Research Assistants will be provided with a desk and access to a computer, but other graduate students may be asked to share desk space. The graduate student offices are primarily for studying, either individually or in groups. Students in their offices are asked to respect the needs of the other students in the same or adjacent offices.
Cooking is not permitted in the offices, although students may heat water for coffee, tea, etc. No one is allowed to remain overnight in his or her office for any reason.
All Temple University facilities are designated as non-smoking.
An office key will be issued to each graduate student occupant. The offices have high security locks whose keys are expensive to replace.