Introduction |
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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 is assigned a Faculty Oversight Committee consisting of two to four faculy members. The committee advises the student on course selection and program milestones in addition to assessing the student's progress. In the second or third year, Ph.D. students and MS students who choose to write a thesis pick a thesis advisor based on their research interests. From that time forward, the thesis advisor takes on primary responsibility for student mentoring.
Degree Requirements |
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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:
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.
The credits must be equivalent to coursework offered at Temple with a grade of "B" or better.
No course that was completed more than five years before the date of the application will be awarded credit.
Credit for courses substantially similar to courses taken since matriculation will not be awarded.
Advanced standing credit is only available for graduate level courses in mathematics, or courses in related fields that have a substantial mathematical content.
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.
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.
Full time students must complete the requirements for the M.S. degree within three years.
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 courses for the topics in which the student plans to take the Ph.D. Qualifying 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. Qualifying Examination, pass the Ph.D. Preliminary (Oral) Examination, and write and successfully defend a research dissertation.
To be eligible for the Ph.D. Preliminary Examination, students must have obtained a Ph.D. pass on the Qualifying Examination.
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.
All requirements for the Ph.D. degree should be fulfilled within six years of graduate study at Temple University, and must be fulfilled within seven years.
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.
Please consult the Graduate Academic Policies and Regulations Section of the Graduate Bulletin for further details.
Graduate Courses in Mathematics |
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A current listing of graduate courses in mathematics can be found here.
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:
The Master's Qualifying Examination |
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For students selecting this option, a Master's Qualifying Examination 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 Qualifying 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. qualifying examinations.
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 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.
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.
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.
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 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.
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.
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. Qualifying Examination |
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The Ph.D. Qualifying Examination is a written exam comprised of three separate sections selected from the following areas:
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 Qualifying Exam as soon as possible after finishing the corresponding coursework. Students are expected to complete and pass the Qualifying 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 Qualifying 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 Qualifying 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 Qualifying 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.
All syllabi include the topics listed in the corresponding syllabi for the Master's Qualifying Examination in Pure mathematics. In addition, the student should be familiar with the following:
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:
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.
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:
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:
Floating point arithmetic and error analysis. Approximation and interpolation of functions. Numerical integration. Solution of nonlinear equations. Runge-Kutta, multistep, and Taylor series methods. Deferred correction. Convergence and stability. Error analysis. Stiff problems. Boundary value problems. Finite differences. Stochastic ODEs.
REFERENCES:
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:
Language Examinations |
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Beginning in Fall 2016, a language requirement is no longer in effect.
Ph.D. Preliminary (Oral) Examination |
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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 (typically the prospective thesis advisor). Consenting to be the Chief Examiner is tantamount to accepting 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 Qualifying 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 syllabus 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 and the examiners for the other topics. The student will be considered to have passed if the Chief Examiner and at least half of the other examiners 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 must be registered for 9994, "Preliminary Examination Preparation:," in the semester in which the examination is taken, including the Summer session. Note that one semester hour of course number 9994 is sufficient for full time status.
Further regulations concerning the preliminary examination may be found in the Temple University Graduate Academic Policies and Regulations Section of the Graduate Bulletin.
Repetition of examinations: Students seeking to pass the Ph.D. Qualifying 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 Qualifying Examination.
The Dissertation |
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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 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. 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.
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.
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 |
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Qualifying Examination: The qualifying 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 fourth year, but must do so before the end of their ninth semseter of studies.
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.
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.