Welcome to the Math PhD program at Harvard University and the Graduate School of Arts and Sciences.
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For information on admissions and financial support, please visit the Harvard Graduate School of Arts and Sciences.
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Demand for mathematicians is projected to grow 21% during the next 10 years. Whether you are interested in becoming a college professor, working in industry or government, or collaborating with top businesses around the globe, this degree provides you a solid foundation from which to pursue your mathematics career.
Degree Awarded: PHD Mathematics
The PhD program in mathematics is intended for students with superior mathematical ability. The program emphasizes a solid mathematical foundation and promotes creative scholarship in mathematics and its many related disciplines. The School of Mathematical and Statistical Sciences has very active research groups in analysis, number theory, geometry and discrete mathematics.
At a Glance: program details
- Location: Tempe campus
- Additional Program Fee: No
- Second Language Requirement: No
Required Core (3 credit hours) MAT 501 Geometry and Topology of Manifolds I (3) or MAT 516 Graph Theory I (3) or MAT 543 Abstract Algebra I (3) or MAT 570 Real Analysis I (3)
Other Requirements (3 credit hours) MAT 591 Seminar (3)
Electives (24-39 credit hours)
Research (27-42 credit hours) MAT 792 Research
Culminating Experience (12 credit hours) MAT 799 Dissertation (12)
Additional Curriculum Information Electives are to be chosen from math or related area courses approved by the student's supervisory committee.
Students must pass:
- two qualifying examinations
- a written comprehensive examination
- an oral dissertation prospectus defense
Students should see the department website for examination information.
Each student must write a dissertation and defend it orally in front of five dissertation committee members.
Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.
Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics, or a closely related area from a regionally accredited institution.
Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program, or applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.
All applicants must submit:
- graduate admission application and application fee
- official transcripts
- statement of education and career goals
- three letters of recommendation
- proof of English proficiency
Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of current residency.
Additional eligibility requirements include competitiveness in an applicant pool as evidenced by coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371). It is expected that applicants will have a B or better in these pre-requisite classes. It is also desirable that applicants have scientific programming skills.
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Program Contact Information
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COLLEGE OF LIBERAL ARTS AND SCIENCES
Department of Mathematics
- Ph.D. in Mathematics
- General Information
- Financial Aid
- Answers to FAQ
- M.S. in Mathematics
- Ph.D. in Mathematics with Thesis in Actuarial Science
- M.S. in Actuarial Science
- M.S. in Applied Financial Mathematics
- M.S. in Applied Financial Mathematics – Actuarial Science
- Course Schedule
- Preliminary Exams
Overview of Requirements
The Ph.D. program is open to students who have a broad mathematical background and who have demonstrated ability and evidence of special aptitude for research in mathematics in their prior work. Students with B.S. or B.A. degrees can apply directly to the Ph.D. program.
To graduate with a Ph.D. in Mathematics, a student must satisfy all of the following requirements:
- Course Credits: 45 credits are required (including 15 doctoral dissertation research credits/GRAD 6950) or if you get a master’s degree in mathematics at UConn then 30 credits are required (including 15 doctoral dissertation research credits) beyond the master’s degree.
- Pass three preliminary exams and two core courses (described below). The exams are officially referred to as “the written part of the qualifying exam”.
- Student is expected to choose a thesis advisor no later than the end of the fall semester in their third year.
- Submit Plan of Study prior to the oral (general) exam.
- Pass an oral exam (called general exam). This exam is the “oral part of the qualifying exam”, and is meant to further the student’s education, scholarship and professional development. It should cover material in the broad area in which the student intends to write a dissertation, but should not focus on the actual thesis research. The student is expected to present the material he/she has studied, and to answer questions about that material. The exam is prepared by the student’s advisory committee, and is normally taken at the end of the third year or beginning of the fourth year.
- Write a dissertation under the direction of a member of the Graduate Faculty . The Graduate School has required specifications for the dissertation here . (More generally, this page has useful information.)
- Thesis templates (for LaTex) are available on our Thesis Formatting page.
Preliminary Examination and Core Course Requirements
Students entering the Mathematics PhD program in Fall 2017 or later need to pass three prelim exams and get a grade B or better in two core courses, chosen from one of the options below (prelims are indicated by the course for that prelim).
- Three prelims from Math 5111 , Math 5120 , Math 5210 and Math 5310 .
- Two core courses from Math 5111 , Math 5120 , Math 5160 , Math 5210 , Math 5211 , Math 5260 , Math 5310 , and Math 5360 .
- Three prelims from Math 5111 , Math 5120 , Math 5310 , Math 5410 and Math 5510 .
- Two core courses from Math 5111 , Math 5120 , Math 5310 , Math 5160 , Math 5410 , Math 5440 , Math 5510 , and Math 5520 .
In this requirement, a core course is any course in the list except for prelim courses that correspond to prelims the student passed. For example, if a student completes the Pure Math prelim requirements by passing the Real Analysis ( Math 5111 ), Algebra ( Math 5210 ), and Topology ( Math 5310 ) prelims, then Complex Analysis ( Math 5120 ) or Abstract Algebra II ( Math 5211 ) may count as a core course, but Abstract Algebra I ( Math 5210 ) may not. Students cannot mix and match different requirements from the different tracks, so they should consult with their advisors upon arrival about which track to take. A student is considered to have passed this requirement after doing so from either track.
For core courses, instructors are expected to have a rigorous assessment of students’ performance.
Time Frame and Progress Requirements
Students are expected to pass at least one prelim exam after each semester for the first three semesters of their graduate study and finish all prelim exam requirements by the beginning of the spring semester in their second year and finish all core course requirements by the end of their second year of graduate study. Failure to meet the requirements could result in a loss of funding for the following year.
The Philosophy behind Prelims
The role of prelims is distinct from the role of final examinations in undergraduate courses. Prelims, as comprehensive examinations, require the student to gather together knowledge, skills and insights from diverse mathematical areas. Traditionally, exams in different areas are given during short time periods, which forces the students to study different areas concurrently. The desired effect, proved over the years, is for students to develop a sense of mathematical ideas that span the discipline and, thereby, to prepare the student for independent research. Contrast is drawn here with the passive activity of taking courses.
Well-prepared entering students are encouraged to take prelim exams before the first semester, but students are not encouraged to take prelim exams without proper preparation. While a student is allowed to take a prelim exam without taking the corresponding prelim course, this should be discussed with an advisor if it is after the first semester.
Graduate study in mathematics is a rigorous enterprise and requires a sincere commitment. The faculty has a responsibility both to the student body at large and to the profession to maintain adequate standards for the Ph.D. degree. Piecemeal passing of prelims over an extended number of years is not, in the opinion of this committee, generally compatible with the goals of a mathematics Ph.D. program.
At times, individual circumstances will dictate that the pace and/or content of the doctoral program should be altered. The Graduate Program Committee welcomes petitions from students and/or their advisors which will recognize the students’ individual interests, backgrounds and goals, within the constraints established by the Graduate School, the College and the Department.
Past Exams- The Department maintains a collection of past prelims going back to August 2000. Students are encouraged to peruse these in preparation for their own examinations.
Plan of Study
Your committee must be established prior to completing the Plan of Study for PhD . This should be done before or at the same time as the general (oral) exam, but at the very latest, in the first four weeks of your final semester.
Language Requirement – The Language Requirement was waived by the Graduate Program committee starting in Fall 2020.
General/Oral Exams, Dissertation Proposal and Applying for Graduation
The doctoral General Exam usually is undertaken when the student has completed at least 75% of the content coursework listed on the approved Plan of Study. The student should not take the General Exam before the Plan of Study has been approved.
The Dissertation Proposal for Doctoral Degree must be submitted at least six months before your expected date of degree completion and must be fully approved at least three months prior to the date of your defense. After approval of the committee, the form must be submitted to the Director of Graduate Studies, who must find two reviewers to read the proposal and provide feedback prior to signing the form.
Instructions for applying for graduation in Student Admin. Applications must be made during the first four weeks of your last semester.
Update your diploma (DIPL) address in Student Admin: In Self-Service/Student Center/Personal Information choose “addresses” in the pull down menu. Select “ADD A NEW ADDRESS” and fill in the form. Once you hit, OK, check “DIPL addr” and SAVE. Also update any other address, if applicable. NOTE: It takes approximately three months to get your diploma.
Doctor of Philosophy
Prepare for a career as a mathematics research scholar.
If you're interested in advancing your math education to enter the ranks of college professorship or high-level research, a PhD in mathematics can help you achieve these goals.
You will learn from the highly qualified and dedicated faculty in the University at Albany's mathematics program as you to delve deeper into your chosen area of specialty on the way to your independent research and doctoral thesis defense.
As your dissertation advisor and other faculty members guide you through a personalized course of study, you'll acquire the knowledge and experience you need to become a contributor to the field of mathematics, whether it’s in research, industry or a classroom.
Program of Study
- Career Outcomes
- Admissions Requirements
You will begin specializing and focusing your educational path soon after entering the PhD program. Your dissertation advisor will help you set and reach applied research goals throughout your time in the courses. The university has research groups in algebra, combinatorics and number theory, analysis, geometry and topology, and probability and statistics.
Emerge from the PhD program with in-depth knowledge of your specified topic of interest, the communication skills to express that information and the ability to comprehend a second language in writing.
A minimum of 60 graduate credits in mathematics as approved by the advisor.
Show proficiency in four of five areas: algebra, topology, complex analysis, real analysis, or probability and statistics. You are required to successfully pass written preliminary examinations in three areas and satisfactorily finish one breadth sequence in an additional fourth area. Breadth Sequence: Two semester sequence in an area complementary to your choice of preliminary exams areas.
- Foundations of Analysis & Real Analysis
- Advanced Linear Algebra & Algebra I
- Topology I & Topology II
- Complex Analysis I & Complex Analysis II
- Introduction to Theory of Statistics & Introduction to Theory of Statistics II
Successful Completion of the Oral Qualifying Examination
Within a month of completion of all the breadth requirements, you will select a faculty member as your dissertation advisor and the area of specialty for your dissertation. Within two semesters you will take a qualifying oral exam to test your readiness to begin work on a dissertation.
Completion of a Research Tool
Display a reading knowledge of French, German, Russian, or another foreign language appropriate to your area of specialization and approved by the department. You will take a departmental examination before the oral qualifying examination.
Full Time Study in Residence
You will be engaged in full-time study beyond the master's degree or equivalent at the University in at least two sessions after admission to the advanced program. You must: a. be registered for 12 credits per semester, or b. hold a full assistantship and (i) complete one academic year in such a position, (ii) be registered for a minimum of 9 credits per term, and (iii) satisfactory completion of assigned duties.
Continuous Registration of Doctoral Students
You must maintain continuous registration for each fall and spring session until you have completed all program requirements. Minimum registration consists of 3 credits of approved course work or registration for dissertation load.
You must be registered for Doctoral Dissertation during the semester in which the degree is conferred. The dissertation is based on your independent research and should constitute a significant original contribution to mathematics or statistics.
In addition to the completion of course requirements, you are required to engage in some ancillary teaching, research, or practicum duties contributing to the academic development.
See the Graduate Bulletin for details .
For more information, contact Marco Varisco at [email protected] .
The doctoral program in mathematics at UAlbany will prepare you for analysis and problem-solving in business or government, or for research and teaching at the university level.
Whether you choose to share your knowledge with the next generation of advanced mathematics students or commit to further developing your own theorems, career outcomes uniquely suited to math PhD graduates include:
- College mathematics professor
- Research mathematician
- AI and machine learning consultant
- Investment researcher
- Insurance actuary
- Software engineer
This degree is designated as a STEM program. International students maintaining F-1 status are allowed to apply for up to 12 months of post-completion Optional Practical Training (OPT) following completion/graduation from their degree program. Currently, this degree program is also designated by the Department of Homeland Security (DHS) as an eligible degree for the F-1 STEM OPT work authorization extension; students who secure qualifying employment may be eligible to apply for the STEM OPT extension for a cumulative total of up to 36 months of F-1 OPT work authorization.
Departmental Assistantship Consideration
- Fall: February 1
- Spring: January 1
- Summer: Not Available
No Departmental Assistantship Consideration
- Fall: August 1
- Summer: May 1
- Transcripts all schools attended
- Three letters of recommendation
- Official GRE test scores*
- Statement of goals
*The GRE will be waived as a requirement for admission through Fall 2023.
The statement is generally one to two pages discussing what you have to offer the program and what you wish to get out of the program. It should include a brief description of the applicant's field of interest, related background, desired area of study and research emphasis or career goals.
Available information for International Applicants .
Learning objectives that UAlbany students are expected to attain through their course of study within their academic program.
- Develop a deep understanding of proposed area of dissertation research.
- Complete a doctoral dissertation.
- Gain a solid understanding of three different areas of mathematics and statistics, namely, one-year sequence in each of the three areas chosen from the following fields: algebra, real analysis, complex analysis, topology, mathematical statistics, and probability.
- Dissertation Filing
The final requirement in the Math Department’s PhD program is filing the dissertation. The dissertation committee has the responsibility for determining whether a submitted dissertation draft is acceptable for the PhD. It is the responsibility of the student to keep in touch with all members of the committee and communicate their expected timeline to completion with adequate time prior to the filing date for all members of the committee to read and provide comments on the dissertation draft. It is also the responsibility of the student to stay informed about all deadlines associated with dissertation filing.
There are several steps, outlined below, that must be completed by the student during the term in which they plan to file. PhD candidates should review the checklist carefully and in full no later than the start of the semester in which they intend to file, and follow up with the GSAO with any unanswered questions. Please note that the checklist items detailed below are for Math and Applied Math PhD candidates; Logic students will find a separate checklist linked below under “Dissertation Filing Resources”.
Steps to Filing Your Dissertation (Math & Applied Math)
- Review Graduate Division dissertation filing protocol (beginning of term): Review the guidelines here . Please read through carefully.
- Check Dissertation Committee on CalCentral (by beginning of term): Ensure that all members listed are correct and show as “active”. If you have a committee member outside of the university, double-check that their email address is correct as well, as the final signature eForm will be routed to this email address for their approval. If your committee is not up to date, you can change it at any time by going to CalCentral > My Dashboard > Submit a Form > Higher Degrees Committee eForm > Committee type “Doctoral”.
- Update Expected Graduation Term to current term (by beginning of term): Expected Graduation Term (EGT) must be set to the term in which you plan to graduate in order for you to be added to the campus degree list. Students can update their EGT by going to CalCentral > My Dashboard > Submit a Form > Change of Expected Graduation Term Request.
- Check Academic Progress Report for incomplete milestones (by beginning of term): The Academic Progress Report (APR) shows all milestones required to complete a degree. Check your APR to ensure that only the dissertation filing milestone is incomplete. If any other milestones show as incomplete, please contact your GSAO. You can view your APR at CalCentral > My Academics > Degree Progress > View Academic Progress Report.
- Inform your committee members of your intent to file (by beginning of term): Check in with all of your committee members (not just your advisor!) and make sure they are aware of your planned timeline to completion. Ensure that all members will be available to read and approve your thesis by the filing date.
- Submit Departmental Intent to Graduate Form (by departmental deadline): Advising staff will send this prior to the deadline each semester, generally in February and October for spring and fall respectively, but you are welcome to submit as soon as the semester in which you intend to graduate begins.
- Submit Research Presentation Form to GSAO (at least three weeks prior to filing): Students must give a research presentation, at least 50 minutes in duration, after advancement to candidacy and during a seminar offered by the Math Department; research presentations held at MSRI or LBNL are also acceptable. After your presentation, please download the form template (found below), to be signed by the faculty in whose seminar you presented, and then return it via email to the GSAO once completed.
- Get for any previously published or co-authored material (at least three weeks prior to filing): Please review Graduate Division guidelines on previously published or co-authored material here . Note that this process is governed by Graduate Division, not the department, so all questions should be directed to degrees [at] berkeley [dot] edu .
- Review CalCentral incomplete tasks and complete any outstanding forms or surveys (at least three weeks prior to filing): A list of incomplete tasks can be found on your CalCentral dashboard.
- Submit your final thesis to ProQuest (by the semester filing deadline ): Upload a PDF of your dissertation to ProQuest as outlined in the procedure for filing your dissertation on Graduate Division's website . You must review guidelines for submission carefully! Graduate Division has very specific formatting guidelines. You cannot make edits to your dissertation after it has been accepted.
- Submit the Final Signature eForm (by the semester filing deadline ): The final signature page is now an eForm accessible via CalCentral. If your EGT is correct, the eForm will show as an incomplete task on your dashboard. Inform your committee prior to submitting the form for their approval so that they can look out for the email notification. Make sure to follow up with all committee members before the filing deadline to ensure that everyone has signed. You will also use this form to select an embargo period for your manuscript, if applicable.
- Complete the AMS Survey of Doctoral Recipients: The Survey of Doctoral Recipients is now housed on AMS' website and completed online. We will email the survey to all graduating students.
Once you have completed all steps of the filing process (including surveys) and have a satisfied APR your certification of conferral will be emailed to you within 2-3 business days
Dissertation Filing Resources
Departmental Intent to Graduate Form
MATH Dissertation Filing Checklist
LOGIC Dissertation Filing Checklist
Research Presentation Form
Current AMS Survey of Doctoral Recipients
Graduate Division Filing Guidelines
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- Degree Programs
- Dissertation & Exams
When planning a thesis defense, keep in mind university submission deadlines , the three-week minimum that the examining committee requires to read the thesis as well as time needed after the defense for revisions. Also review the GW Libraries website for guidance on dissertation format and submission.
Note: Schedules for the next exam period will be posted when available.
The thesis advisor, in consultation with the graduate committee, assembles the final examination committee. This committee is made up of the thesis advisor (director), the co-director (if applicable), two readers and two additional examiners; at least one examiner must be from outside the department. Also, another person is selected to serve as chair of the defense.
The PhD candidate, with the assistance of the advisor, must make every effort to ensure that the thesis is free of errors before it is submitted to the final examination committee. The thesis must be submitted to the committee at least one month before the defense. Thesis defenses are open to the public; the advisor should ensure that an announcement is sent to the entire department at least one month in advance.
The defense consists of an hour-long talk by the candidate on his or her research followed by several rounds of questions by the final examination committee, excluding the director (and co-director, if applicable). Once the question session ends, the defense is over and the chair dismisses the candidate and the audience.
The committee then meets to determine the outcome, which can be:
- The dissertation is acceptable as is,
- the dissertation is acceptable subject to requested revisions (in this case, the requested revisions are conveyed to the thesis advisor, a subset of the committee is appointed to check and ultimately sign-off on the revised thesis, and a date is chosen by which revisions should be completed) or
- the dissertation is unacceptable (in this case, the student may write and submit a new thesis).
The chair facilitates this discussion and, at its conclusion, sees that the Final Examination Committee Sign-Off Form is completed.
Following a thesis defense at which the thesis was accepted as is, or following the completion of the required revisions and the subsequent filing of the Final Dissertation Approval Form, the successful PhD candidate submits his or her dissertation electronically, following the instructions on the Electronic Theses and Dissertations (ETD) website . This website has links to the ProQuest/University Microfilm (UM) Form, which must also be submitted electronically. (See the graduate committee chair for the username and password needed for the UM form.)
Paper copies of two additional forms must be submitted to CCAS (Phillips 107): the ETD Access Approval Form (signed by the student and the thesis advisor) and the Survey of Earned Doctorates . It is highly recommended that all of these items be submitted well in advance of the CCAS deadline. After this material has been submitted, the graduate committee chair approves the candidate for graduation .
The qualifying exams are offered twice a year: once in August, before the beginning of the fall semester, and once in January, before the beginning of the spring semester. A student may take each exam at most twice. Students have four total opportunities to pass the exams over the course of their first two years (first year August, first year January, second year August and second year January).
Questions about the qualifying exams should be directed to Joel Lewis , the exam coordinator.
In consultation with their advisors, students select exams from the four designated options:
- Applied Mathematics
- Groups, subgroups, normal subgroups, quotient groups, homomorphisms
- Isomorphism theorems, Cayley’s theorem
- Normal and composition series, Jordan-Holder theorem, solvable groups
- Simple groups
- Direct products, structure theorem for finitely generated abelian groups
- Groups acting on sets, class equation, Sylow’s theorems
- Free groups, group presentations
- Elementary properties of rings, ideals, Zorn’s lemma, quotient rings
- Homomorphism and isomorphism theorems, field of fractions of an integral domain, polynomial rings, Noetherian rings, algebras
- Unique factorization domains, principal ideal domains, Euclidean domains, prime ideals
- Larry Grove, Algebra, Dover Publications, 2004.
- Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics Vol. 73, Springer Verlag, 2003.
- Nathan Jacobson, Basic Algebra I and Basic Algebra II, Dover Publications, 2009.
The background for this exam consists of a good knowledge of advanced calculus, such as one might acquire in MATH 4239 & 4240 or MATH 6201 & 6202. This includes:
- Continuity, uniform continuity and differentiability of real valued functions of a real variable;
- Completeness of R and compactness of [0, 1];
- Complete, connected and compact metric spaces, including basic topological concepts like open sets, limit points, etc.;
- The continuity and derivatives of functions f : Rn → Rm;
- The implicit and inverse function theorems.
The remaining topics are mostly covered in MATH 6214. Any topics not covered in that class are the student’s responsibility.
- Lebesgue measure and integration in abstract spaces
- Fatou’s lemma; the dominated and monotone convergence theorems
- Constructing Lebesgue measure on [0, 1] and on Rn
- Functions of bounded variation on [0, 1], their differentiability (Lebesgue almost everywhere), and absolute continuity for real valued functions on [0, 1]
- The Radon-Nikodym and Lebesgue decomposition theorems
- Measures on product spaces, and the Fubini theorem
- Hölder’s inequality. The L p spaces and their properties, including completeness
- Relations between different modes of convergence: uniform, pointwise almost everywhere, L p and in measure
Applied mathematics focuses on mathematical techniques that yield practical information on the problems of the natural world. The fundamental processes in the natural world are to a large extent described by partial differential equations.
This syllabus focuses on a number of basic partial differential equations and some standard techniques used to analyze these equations. We see how some first order equations can be solved along characteristic curves, how explicit solutions may be found for the wave equation in the entire space, and how the Fourier transform is used to solve linear evolution equations. We learn to analyze the Laplacian operator on a bounded domain.
Afterwards we will solve the linear wave equation, the heat equations and Schrödinger equation via various methods. Calculus of variations is a rich source of problems from physical sciences and other branches of mathematics and we learn how this approach leads to differential equations.
- Applied Mathematics by David Logan
- Partial Differential Equations (Graduate Studies in Mathematics) by Lawrence C. Evans.
1. One-dimensional boundary value problems
- Eigenvalues and eigenfunctions
- Sturm-Liouville theory
- Green’s functions, integral equations, Arzel`a-Ascoli theorem
2. First order equations, characteristics and shock waves
- First order equations, the method of characteristics
- Failure of the characteristic method, weak solutions
- Shock waves
3. Linear elliptic and evolution equation
- Fourier Transforms
- The Heat equation, Schrödinger equation and kernels of linear operators
- Separation of variables
- The Laplace operator on a bounded region
- Green’s functions
4. Calculus of variations
- Variational problems, Euler-Lagrange equations
- The classical harmonic oscillator, the pendulum, minimal surfaces
- The Dirichlet boundary condition, the Neumann boundary condition
- Variational problems with constraints
- Topological spaces, basis, subbasis, open sets and closed sets, quotient topology, product topology, continuous maps and homeomorphisms, finite topological spaces.
- Connectedness, path connectedness, compactness, compactness in metric spaces, local compactness, paracompactness.
- Metric topology, equivalent metrics, complete metric spaces.
- Product topology, Tychonoff Theorem, Alexander subbase Theorem.
- Separation axioms and countability axioms, Urysohn lemma, Tietze extension theorem.
- Brouwer fixed-point theorem via Sperner Lemma.
- Fundamental groups, homotopic maps, strong deformation retracts, Fundamental groups for Rn, Sn, graphs, torus and link exteriors.
- Covering spaces, universal covering, classifications of covering spaces.
- The first homology group as the abelianization of a fundamental group.
- A. Hatcher, Algebraic Topology . Cambridge University Press, Cambridge, 2002. xii+544
- J. Munkres, Topology: A First Course , Prentice-Hall.
- O. Ya. Viro, O. Ya. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary Topology Problem Textbook , AMS, 2008
Details and Scheduling
Each exam is graded out of 30 points. The student’s subject score in an area is the highest qualifying exam score received in that area. The student passes the qualifying exams if the total of any three subject scores equals 50 or higher.
For a full-time PhD student, failure to pass the qualifying exams by the start of the fourth semester in the PhD program results in the termination of his or her PhD program. Students who are receiving funding as graduate teaching assistants must pass the qualifying exams by January of their second year to be eligible for funding in their third year.
The qualifying exams must be completed successfully before the specialty exam may be attempted. Students should let the graduate committee know which exams they intend to take by the middle of the semester preceding the desired exam session.
The qualifying exam in each area is
- Two hours long
- Scored out of 30 points
- Based on the corresponding graduate core course
The exams are given on two days during the week preceding the first week of classes each semester. On each of these two days two exams are given: one in the morning and one in the afternoon. It is not required that all the exams be taken for the first time during the same exam session. Students should consult their advisors to develop a sound plan for when to take the exams.
While the graded exams remain in the department files, students can ask the department for copies of their own graded exams. Obtaining copies of one's exams and discussing them with the graders is especially important if an exam has to be retaken.
Copies of past exams are also available for download. Note that some older exams may contain problems on subjects no longer required by the current qualifying exam syllabi.
Download Past Exams on Box
- Applied Math
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If you require these documents in an alternate format, please contact the Department of Mathematics at [email protected] .
- Analysis: Tuesday, January 10, 10 a.m. – 12 p.m.
- Topology: Tuesday, January 10, 1 – 3 p.m.
- Algebra: Thursday, January 12, 10 a.m. – 12 p.m.
- Applied math: Thursday, January 12, 1 – 3 p.m
The specialty exam is an oral exam in the student's intended research area. It is conducted by a specialty exam committee made up of three faculty members (usually all in the department), including a prospective dissertation advisor. Starting soon after the completion of the qualifying exams, students typically spend the better part of a year preparing for and then taking the specialty exam.
The prospective advisor prepares a syllabus; in addition to listing the exam topics and recommended sources, this document sets a target completion date and names the exam committee. Early in the exam preparation process, the graduate committee is informed of the arrangements and the syllabus is filed in the student's departmental file. The exam committee (especially the advisor) assists the student in preparing for the exam.
The advisor chairs the exam, which is conducted by the exam committee and is open to attendance by all mathematics faculty members and graduate students. The advisor should ensure that an announcement is sent to the entire department at least a week in advance.
The exam committee meets privately immediately after the exam to determine the outcome, which can be Pass with Distinction, Pass, Low Pass or Fail. A student may not take the specialty exam more than twice. The second attempt, if needed, must be scheduled within one semester of the first.
Soon after passing the specialty exam, if not earlier, the student starts working with a thesis advisor. A student who wishes to change thesis advisors may do so at any time; all that is required is the consent of the new advisor. The new advisor may require the student to retake the specialty exam.
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Home > Physical and Mathematical Sciences > Mathematics Education > Theses and Dissertations
Mathematics Education Theses and Dissertations
Theses/dissertations from 2022 2022.
Understanding College Students' Use of Written Feedback in Mathematics , Erin Loraine Carroll
Identity Work to Teach Mathematics for Social Justice , Navy B. Dixon
Developing a Quantitative Understanding of U-Substitution in First-Semester Calculus , Leilani Camille Heaton Fonbuena
The Perception of At-Risk Students on Caring Student-Teacher Relationships and Its Impact on Their Productive Disposition , Brittany Hopper
Structural Reasoning with Rational Expressions , Dana Steinhorst
Student-Created Learning Objects for Mathematics Renewable Assignments: The Potential Value They Bring to the Broader Community , Webster Wong
Theses/Dissertations from 2021 2021
Emotional Geographies of Beginning and Veteran Reformed Teachers in Mentor/Mentee Relationships , Emily Joan Adams
You Do Math Like a Girl: How Women Reason Mathematically Outside of Formal and School Mathematics Contexts , Katelyn C. Pyfer
Developing the Definite Integral and Accumulation Function Through Adding Up Pieces: A Hypothetical Learning Trajectory , Brinley Nichole Stevens
Theses/Dissertations from 2020 2020
Mathematical Identities of Students with Mathematics Learning Dis/abilities , Emma Lynn Holdaway
Teachers' Mathematical Meanings: Decisions for Teaching Geometric Reflections and Orientation of Figures , Porter Peterson Nielsen
Student Use of Mathematical Content Knowledge During Proof Production , Chelsey Lynn Van de Merwe
Theses/Dissertations from 2019 2019
Making Sense of the Equal Sign in Middle School Mathematics , Chelsea Lynn Dickson
Developing Understanding of the Chain Rule, Implicit Differentiation, and Related Rates: Towards a Hypothetical Learning Trajectory Rooted in Nested Multivariation , Haley Paige Jeppson
Secondary Preservice Mathematics Teachers' Curricular Reasoning , Kimber Anne Mathis
“Don’t Say Gay. We Say Dumb or Stupid”: Queering ProspectiveMathematics Teachers’ Discussions , Amy Saunders Ross
Aspects of Engaging Problem Contexts From Students' Perspectives , Tamara Kay Stark
Theses/Dissertations from 2018 2018
Addressing Pre-Service Teachers' Misconceptions About Confidence Intervals , Kiya Lynn Eliason
How Teacher Questions Affect the Development of a Potential Hybrid Space in a Classroom with Latina/o Students , Casandra Helen Job
Teacher Graphing Practices for Linear Functions in a Covariation-Based College Algebra Classroom , Konda Jo Luckau
Principles of Productivity Revealed from Secondary Mathematics Teachers' Discussions Around the Productiveness of Teacher Moves in Response to Teachable Moments , Kylie Victoria Palsky
Theses/Dissertations from 2017 2017
Curriculum Decisions and Reasoning of Middle School Teachers , Anand Mikel Bernard
Teacher Response to Instances of Student Thinking During Whole Class Discussion , Rachel Marie Bernard
Kyozaikenkyu: An In-Depth Look into Japanese Educators' Daily Planning Practices , Matthew David Melville
Analysis of Differential Equations Applications from the Coordination Class Perspective , Omar Antonio Naranjo Mayorga
Theses/Dissertations from 2016 2016
The Principles of Effective Teaching Student Teachershave the Opportunity to Learn in an AlternativeStudent Teaching Structure , Danielle Rose Divis
Insight into Student Conceptions of Proof , Steven Daniel Lauzon
Theses/Dissertations from 2015 2015
Teacher Participation and Motivation inProfessional Development , Krystal A. Hill
Student Evaluation of Mathematical Explanations in anInquiry-Based Mathematics Classroom , Ashley Burgess Hulet
English Learners' Participation in Mathematical Discourse , Lindsay Marie Merrill
Mathematical Interactions between Teachers and Students in the Finnish Mathematics Classroom , Paula Jeffery Prestwich
Parents and the Common Core State Standards for Mathematics , Rebecca Anne Roberts
Examining the Effects of College Algebra on Students' Mathematical Dispositions , Kevin Lee Watson
Problems Faced by Reform Oriented Novice Mathematics Teachers Utilizing a Traditional Curriculum , Tyler Joseph Winiecke
Academic and Peer Status in the Mathematical Life Stories of Students , Carol Ann Wise
Theses/Dissertations from 2014 2014
The Effect of Students' Mathematical Beliefs on Knowledge Transfer , Kristen Adams
Language Use in Mathematics Textbooks Written in English and Spanish , Kailie Ann Bertoch
Teachers' Curricular Reasoning and MKT in the Context of Algebra and Statistics , Kolby J. Gadd
Mathematical Telling in the Context of Teacher Interventions with Collaborative Groups , Brandon Kyle Singleton
An Investigation of How Preservice Teachers Design Mathematical Tasks , Elizabeth Karen Zwahlen
Theses/Dissertations from 2013 2013
Student Understanding of Limit and Continuity at a Point: A Look into Four Potentially Problematic Conceptions , Miriam Lynne Amatangelo
Exploring the Mathematical Knowledge for Teaching of Japanese Teachers , Ratu Jared R. T. Bukarau
Comparing Two Different Student Teaching Structures by Analyzing Conversations Between Student Teachers and Their Cooperating Teachers , Niccole Suzette Franc
Professional Development as a Community of Practice and Its Associated Influence on the Induction of a Beginning Mathematics Teacher , Savannah O. Steele
Types of Questions that Comprise a Teacher's Questioning Discourse in a Conceptually-Oriented Classroom , Keilani Stolk
Theses/Dissertations from 2012 2012
Student Teachers' Interactive Decisions with Respect to Student Mathematics Thinking , Jonathan J. Call
Manipulatives and the Growth of Mathematical Understanding , Stacie Joyce Gibbons
Learning Within a Computer-Assisted Instructional Environment: Effects on Multiplication Math Fact Mastery and Self-Efficacy in Elementary-Age Students , Loraine Jones Hanson
Mathematics Teacher Time Allocation , Ashley Martin Jones
Theses/Dissertations from 2011 2011
How Student Positioning Can Lead to Failure in Inquiry-based Classrooms , Kelly Beatrice Campbell
Teachers' Decisions to Use Student Input During Class Discussion , Heather Taylor Toponce
A Conceptual Framework for Student Understanding of Logarithms , Heather Rebecca Ambler Williams
Theses/Dissertations from 2010 2010
Growth in Students' Conceptions of Mathematical Induction , John David Gruver
Contextualized Motivation Theory (CMT): Intellectual Passion, Mathematical Need, Social Responsibility, and Personal Agency in Learning Mathematics , Janelle Marie Hart
Thinking on the Brink: Facilitating Student Teachers' Learning Through In-the-Moment Interjections , Travis L. Lemon
Understanding Teachers' Change Towards a Reform-Oriented Mathematics Classroom , Linnae Denise Williams
Theses/Dissertations from 2009 2009
A Comparison of Mathematical Discourse in Online and Face-to-Face Environments , Shawn D. Broderick
The Influence of Risk Taking on Student Creation of Mathematical Meaning: Contextual Risk Theory , Erin Nicole Houghtaling
Uncovering Transformative Experiences: A Case Study of the Transformations Made by one Teacher in a Mathematics Professional Development Program , Rachelle Myler Orsak
Theses/Dissertations from 2008 2008
Student Teacher Knowledge and Its Impact on Task Design , Tenille Cannon
How Eighth-Grade Students Estimate with Fractions , Audrey Linford Hanks
Similar but Different: The Complexities of Students' Mathematical Identities , Diane Skillicorn Hill
Choose Your Words: Refining What Counts as Mathematical Discourse in Students' Negotiation of Meaning for Rate of Change of Volume , Christine Johnson
Mathematics Student Teaching in Japan: A Multi-Case Study , Allison Turley Shwalb
Theses/Dissertations from 2007 2007
Applying Toulmin's Argumentation Framework to Explanations in a Reform Oriented Mathematics Class , Jennifer Alder Brinkerhoff
What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof? , Karen Malina Duff
Probing for Reasons: Presentations, Questions, Phases , Kellyn Nicole Farlow
One Problem, Two Contexts , Danielle L. Gigger
The Main Challenges that a Teacher-in-Transition Faces When Teaching a High School Geometry Class , Greg Brough Henry
Discovering the Derivative Can Be "Invigorating:" Mark's Journey to Understanding Instantaneous Velocity , Charity Ann Gardner Hyer
Theses/Dissertations from 2006 2006
How a Master Teacher Uses Questioning Within a Mathematical Discourse Community , Omel Angel Contreras
Determining High School Geometry Students' Geometric Understanding Using van Hiele Levels: Is There a Difference Between Standards-based Curriculum Students and NonStandards-based Curriculum Students? , Rebekah Loraine Genz
The Nature and Frequency of Mathematical Discussion During Lesson Study That Implemented the CMI Framework , Andrew Ray Glaze
Second Graders' Solution Strategies and Understanding of a Combination Problem , Tiffany Marie Hessing
What Does It Mean To Preservice Mathematics Teachers To Anticipate Student Responses? , Matthew M. Webb
Theses/Dissertations from 2005 2005
Fraction Multiplication and Division Image Change in Pre-Service Elementary Teachers , Jennifer J. Cluff
An Examination of the Role of Writing in Mathematics Instruction , Amy Jeppsen
Theses/Dissertations from 2004 2004
Reasoning About Motion: A Case Study , Tiffini Lynn Glaze
Theses/Dissertations from 2003 2003
An Analysis of the Influence of Lesson Study on Preservice Secondary Mathematics Teachers' View of Self-As Mathematics Expert , Julie Stafford
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Navigate ohio, connect with us, ph.d. in mathematics.
Program Name and Number: Mathematics - PH3101
- Admissions (Note: Students without a master's degree or similar post-baccalaureate experience will not be admitted directly to the doctoral program; instead, they should apply to the master's program doctoral preparation track .)
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- Program Mission and Learning Objectives
The Ph.D. in Mathematics is intended for students who wish to advance mathematical knowledge itself, apply such knowledge to problems confronting society and science, and educate others in mathematical methods and ways of thinking.
The Mathematics Department offers students the possibility of designing study plans to meet their individual goals and interests. In particular, it offers a broad spectrum of possible research areas for Ph.D. students, including algebra, analysis, coding theory, computational harmonic analysis, partial and ordinary differential equations, dynamical systems, financial mathematics, mathematical biology, numerical analysis, optimal control theory, set theory, statistics, stochastic processes, and topology.
The first phase in doctoral education in Mathematics is to understand a few subjects deeply and a range of subjects in less detail. OHIO's program accomplishes this through a system of courses and written examinations. An exceptionally well-prepared student can attempt the examinations early and spend relatively little time doing coursework.
The second phase is to become the expert on a specific problem and produce new mathematical results on it suitable for a dissertation. In the program, this phase is done one-on-one with a faculty adviser or in a small research group. The dissertation is a scholarly work demonstrating the ability to understand, organize, improve, and present mathematical ideas of outstanding importance, depth, or interest. It must include original mathematical research and be worthy of publication.
Most doctoral students are trained and financially supported as teaching assistants and have the opportunity to teach classes as the primary instructor.
Most graduates work in academia, teaching and/or doing research in Mathematics.
To train students to create, apply, and disseminate mathematical knowledge and understanding.
Program Learning Objectives
- Graduates will be able to extend the frontier of mathematical knowledge by producing quality research with original results.
- Graduates will be able to apply a range of mathematical tools to problems within mathematics and in other disciplines.
- Graduates will be able to effectively disseminate mathematical knowledge and understanding through publications, seminars, classroom teaching, or other means.
Department of Mathematics
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Graduate Degree Programs
Thank you for your interest in the graduate program in Mathematics at UTA! We look forward to your application to one of our degree programs. The objectives of our graduate program are to develop the students’ ability to do independent research and prepare them for more advanced studies in mathematics, and to provide advanced training and preparation for professional careers as mathematicians, mathematics teachers, and those employed in engineering, scientific and business fields.
Master of Science in Mathematics
The M.S. in Mathematics degree option provides opportunities to prepare students for more advanced study in mathematics and offers specializations in mathematics (Algebra, Geometry, Applied Mathematics, Computational Mathematics, Data Science, Mathematical Biology, Mathematics Education, and Probability) and in Statistics. Students may choose the Thesis Plan, Thesis-Substitute Plan, or Non-Thesis Plan. Program Details
Master of Arts in Mathematics
The M.A. in Mathematics degree option provides a unique learning experience for those who are interested in strengthening their understanding of mathematics and enriching their understanding of mathematics teaching and learning. Program Details
Master of Science in Applied Statistics and Data Science
The M.S. in Applied Statistics and Data Science (ASDS) degree option provides a unique learning experience for those who are interested in statistical methodologies, data science, big data analytics, and machine learning. The program prepares work-ready students for statistics/data science positions in multiple disciplines and industries. Program Details
Certificate in Applied Statistics
The Certificate in Applied Statistics offers individuals with an undergraduate degree an opportunity to receive graduate instruction in applied statistics as a means of maintaining and enhancing their professional development. The certificate program will provide coursework in statistics to an individual whose undergraduate major was outside the area of statistics. Since the requirements for the certificate are substantially less than those for the M.S. in Mathematics with a concentration in Statistics, the certificate can be earned in a much shorter time span. When applicable, the courses taken in the certificate program can also be used to satisfy an M.S. in Mathematics degree requirement at UTA and other institutions. Consult with the graduate advisor for advice on graduate course transfer. Certificate Requirements
Ph.D. in Mathematics
The Ph.D. in Mathematics program prepares students as researchers to enter a wide range of professional roles including academia, industry, and business. The program is offered in Mathematics (Algebra, Geometry, Applied Mathematics, Computational Mathematics, Data Science, Mathematical Biology, Mathematics Education, and Probability) and in Statistics. There are two different routes to obtain the Ph.D. degree: one is to obtain the M.S. degree first and then enter Ph.D. program, while the other is to enter the Ph.D. program directly with a B.S. degree (i.e., the B.S. to Ph.D. track). Program Details
How to Apply
Application to our graduate programs in Mathematics must be completed through the UTA Admission Page . In addition, please submit three letters of recommendation directly to the Department of Mathematics via the upload page . If for some reason the upload page is not working, please feel free to email or send by mail:
Department of Mathematics Graduate Program The University of Texas at Arlington P.O. Box 19408 Arlington, TX 76019-0408 E-mail: [email protected] UTA Recommendation Form .-->
Application Priority Dates
1. UTA Priority Dates: The Graduate General Application Priority Dates are posted at: UTA Priority Dates .
2. GTA Support Priority Dates: In order to receive full consideration for Graduate Teaching Assistant (GTA) support starting in Fall, applicants need to complete their admission application by February 1 of that year. Admission application also serves as GTA application, which will continue to be reviewed until all positions are filled.
The Math Graduate Affairs committee carefully evaluates and ranks all complete doctoral applications and then makes the recommendations for funding based on available resources. Preference for financial support is given only to Ph.D.-bound graduate students admitted unconditionally into one of our doctoral programs. Due to limited financial resources, not all unconditionally admitted graduate students are offered support. The number of newly supported graduate students depends on the available financial resources at that time, which is based on how many supported students have graduated, the new external grants that the department secures, and other factors.
To check the status of your application please visit the UTA: Apply website or contact the UTA Admission Office .
*Note that admission decisions cannot be made unless the UTA Admissions Office has received all the required application documents and has determined that your application is complete and ready for an evaluation! Once an admission decision is made on your application, then you will be immediately notified by the UTA Admissions Office.
To request information about our graduate program, call or send an email to the Mathematics Graduate Program Advisor s .
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Location : Pickard Hall, Room 478, 411 S. Nedderman Dr., Arlington, TX 76019 Phone : 817-272-3261 Fax : 817-272-5802 Email : [email protected]
- Dissertation & Defense
What follows is a summary of the many interconnected tasks and deadlines necessary to graduate with a PhD. More information may be found in the Bulletin , the official policy document of the Graduate School.
Student To Do List
- Familiarize yourself with the various deadlines and required paper work .
- Ensure that you have satisfied the Responsible Conduct of Research (RCR) requirement of 4 forums beyond the entering orientation.
- Your advisor must must email The Graduate School a letter confirming that the thesis is complete and ready to defend ( [email protected] ). This letter must be in the form of PDF attachment (email text is no longer accepted). Contact the DGSA for an easy-to-fill-in template. The letter must be received 30 days before the scheduled defense . Please cc the DGS and DGSA ( [email protected] ).
- Prepare Dissertation utilizing Duke Graduate School Thesis and Dissertations formatting requirements . ( Link to templates including LaTex )
- Confirm your committee with the DGSA at least 30 days prior to the dissertation defense . If your committee has not changed, you must send an email to [email protected] notifying them that your thesis committee remains unchanged. If your committee will change, notify the DGSA at least 30 days before the dissertation defense .
- Select a date for the dissertation defense . It is strongly recommended that you schedule this with your committee at least two months in advance. Zoom defenses and remote participation is allowed but please confirm with DGSA. Inform the DGSA of date, time, and location of defense .
- Submit your dissertation for format checking . Again visit the Graduate School website for the exact deadline dates. The initial submission must take place at least two weeks before your defense and no later than 5:00 p.m. on the deadline date. The Graduate School website says that "Initial submission of the dissertation or thesis means that the document is complete.''
- Copies of the most up-to-date version of the dissertation must be given to all committee members for them to read. This must be done at least two weeks before the defense. Some committee members may request more time, so consult with them early to learn their requirements.
- A public copy of dissertation should be available. Please send DGSA a pdf copy to upload to the Intranet.
- Confirm with your committee that all exam form have been received and filled. All forms are due before the deadline for final submission.
- Complete departure checklist and send in thesis poster materials.
- Submit the final version of your dissertation to the Graduate School. This must be done within 30 days of the final defense or by the submission deadline date . Small changes in the content can (and probably will) be made after submission but all of the pieces must be there.
- Please do all the above steps before leaving campus.
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Mathematics Education PhD
Doctor of philosophy.
The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.
- Points/Credits: 75
- Entry Terms: Spring/Summer/Fall
- Spring: November 1
- Summer/Fall (Priority): December 1
- Summer/Fall (Final): Rolling
* For details about rolling deadlines , visit our admission deadlines page.
Supplemental Application Requirements/Comments
- Online Degree Application , including Statement of Purpose and Resume
- Transcripts and/or Course-by-Course Evaluations for all Undergraduate/Graduate Coursework Completed
- Results from an accepted English Proficiency Exam (if applicable)
- $65 Application Fee
- Two (2) Letters of Recommendation
Additional Degree Information
Program Study Guide
Requirements from the TC Catalog
View Full Catalog Listing
All candidates for the Ed.D., Ed.D.C.T., or Ph.D. degrees are expected to demonstrate both mathematics and mathematics education competencies through a series of certification examinations taken upon the completion of 60 graduate points. Certification examinations test the student’s knowledge of current research and theory in mathematics education and mathematics content. Examinations are offered once in the fall, spring, and summer terms. Courses recommended as preparation for the examinations include MSTM 4019/4020, MSTM 6037, and 6000- level mathematics content courses.
Students must demonstrate acceptable proficiency in at least three of the following six mathematics content areas: algebra, analysis, discrete mathematics, foundations of mathematics, geometry and topology, and probability and statistics. Students may sit for the examination in mathematics content during the regular certification examination times. Alternatively, they may register for advanced content courses and, with permission of the program, sit for the content area certification examination upon completion of the course. Incoming doctoral candidates should register for Professional Seminar in Mathematics during the first year of doctoral studies.
Doctoral students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.
Doctor of Philosophy in Mathematics Education
Candidates are encouraged to develop an association with a faculty member early in their studies to identify a problem area of mutual interest to plan a course of studies that leads to the competencies needed to complete dissertation research and prepare for a professional role. Further details are available in the brochures on doctoral studies and in the general descriptions of doctoral programs available from the Office of Doctoral Studies.
A program of study for the Doctor of Philosophy degree must include at least 45 points taken under Teachers College registration. In order to permit the acquisition of broad and basic scholarship, each program of study should include at least 60 points in mathematics, mathematics education, statistics, and computing. At least 35 points should be in advanced courses – including research courses (MSTM 6500 or 6501 and MSTM 7500). (Any Teachers College course at the 6000 level or above, any Columbia University Graduate School of Arts and Sciences course with a “G” prefix, any “W” course numbered above 4000, or any transferred course with a graduate-level prerequisite will be considered an advanced course.) Further, 15 points in the philosophical, psychological, and curricular foundations of education must be included in every Ph.D. program. Students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.
Candidates for the Ph.D. degree are required to demonstrate competency in two languages chosen from among French, German, and Russian. Students who require other languages for the preparation of their dissertation may petition the program to request one substitution. Students in mathematics may not use computer languages or statistics to satisfy the language requirement.
The Ph.D. dissertation is a scholarly study contributing new theoretical knowledge to the field and should be planned early in the program when sufficient advanced courses have been completed to permit the candidate to enroll in relevant research courses. Ph.D. dissertations in mathematics education should be (1) experimental studies in learning, (2) analytical studies in policy theory in mathematics education, or (3) other scholarly investigations of problems and issues of broad significance in the field.
The site of the Program offers a list of Topic study groups which doctoral students are recommended to join.
- View Other Degrees
Program Director : Professor Alexander Karp
Teachers College, Columbia University 323 Thompson
Phone: (212) 678-3381 Fax: (212) 678-8319
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How to Write a Dissertation in Mathematics: Step-by-step Guide
A dissertation in mathematics is a significant component of postgraduate education where students have to work independently and produce a coherent piece of text in which they need to describe the results of their investigation. It’s a challenging and time-consuming project which requires thorough understanding of complex mathematical concepts and producing coherent research outcome.
To help you cope with this complex academic paper, we’ve created this guide which covers the essential steps in writing a dissertation in mathematics and addresses the most common difficulties that students face during the writing process. You will learn how to structure your dissertation and write its chapters. We will provide you with editing and proofreading tips and give you a list of 15 interesting dissertation topics. If you struggle with writing any other academic project, on our website, you will find full guides to all writing assignments out there.
What Is a Dissertation in Mathematics? Basic Requirements
Dissertation in mathematics is an individual project where students investigate and study a specific area of mathematical research or an application of advanced mathematical techniques and write a report on their findings. When writing a dissertation in mathematics, you will learn to understand complex mathematical texts, work on open-ended problems, and communicate mathematical reasoning and ideas clearly.
In your dissertation, you need to report what was previously known on a particular topic and contribute to some field of mathematics. A Masters dissertation can be either expository that explains some results that are already known or research-based that includes a new theorem. A PhD dissertation in mathematics is based on original mathematical research and includes at least one previously unknown substantial theorem.
The length of a dissertation varies with the topic and typically is between 50 to 80 pages for Masters’ dissertation. Keep in mind that quality is more important than quantity. Mathematical sentences contain notations, figures, and equations that are difficult to type if you use a typical word-processing program. That’s why dissertations in mathematics should normally be produced in TEX or LaTEX which were specifically designed for the production of technical and scientific documentation. If you are going to type your dissertation in mathematics yourself, you should dedicate a certain amount of time to become familiar with this software.
And keep in mind that you will need several drafts because it’s practically impossible to concentrate on getting all aspects of your paper (layout, mathematics, grammar, spelling etc.) correct at once. Writing is a process and no one is capable of producing a well-organized, clear, and error-free exposition the first time he/she sits down to write. Typically, good papers are thoroughly rewritten at least once or twice and the key sections often require 3 to 5 major revisions.
Try a quicker way
A dissertation in mathematics reports results of your investigation and should meet the following standards:
- It should follow a standard format for scientific communication and include a title, introduction, results, discussion, conclusions, and bibliography. You need to follow conventions of mathematical writing and citations.
- Your final paper should be correct and logically coherent. You should clearly explain which contributions are your own and which you have taken from other scholarly sources. You also have to report negative results.
- The text of your dissertation should be clear, interesting, and readable. You should write it with your readers in mind and ensure that even non-experts could be able to make sense of what you write.
- Precision is crucial. Every mathematical statement in your dissertation must have a precise mathematical meaning. You must define every term and every symbol and use them according to their definition. You must also justify every mathematical conclusion that you reach in your paper.
- If you write something (ideas, facts, proofs, mathematical results) which you obtained from a published source, you must refer to such a source in the text and list it in your bibliography to avoid plagiarism.
Choosing a Good Topic for a Dissertation in Mathematics
The first step in writing a dissertation in mathematics is choosing a topic. The dissertation project typically relates to the research interests of your supervisor so you should speak to members of the staff about possible dissertation topics and find out who works in the areas that you are interested in.
If you can’t think of something to write about, here is a list of topic suggestions for your inspiration.
- Representations of Finite Hecke Algebras for Complex Reflection Groups;
- Homotopy Type Theory: Computing Homotopy Groups of Spheres;
- Geometric Quantisation of Symplectic Manifolds;
- Cohomology of Abelian Varieties;
- Geometric Class Field Theory;
- Application of Theorems of Grushko and Stallings for Classification of Groups up to Quasi-isometry;
- Evolution Equations: Problems Arising in Population Dynamics;
- Low-rank Approximations in Multiple Variables;
- Stochastic Gradient Descent Methods;
- Application of Discrete Mathematical Techniques in Cellular Automation Models in Biology;
- Fluctuation Theory of Levy Processes;
- Random Matrices and Free Probability;
- Intersection Theory of Singular Varieties;
- Linear Programming Methods in Combinatorics;
- Mathematical Modelling in Metal Forming Processes.
How to Structure Your Dissertation in Mathematics
Dissertations in mathematics typically give an extended analysis of a particular topic and report on a research project or study. A dissertation commonly consists of multiple chapters:
- Abstract of approximately 300 words summarizes the aims, scope, and the conclusions of a dissertation.
- Introduction introduces a specific research topic, its scope, and significance, gives a review of relevant literature, and briefly outlines the logical structure of your dissertation.
- Body of the paper should be structured into several chapters that contain proofs and details of numerical implementations.
- Conclusion restates the main arguments, tells about the consequences, and provides suggestions for future work.
You may structure the body of the work like this:
- Proof of Theorem 1
- Proof of Theorem 2
- Numerical Examples
Writing a Dissertation in Mathematics
A dissertation in mathematics is based on independent mathematical research which differs from most researches in other fields of knowledge. To get your PhD in math, you need to create at least one new theorem and become a world’s expert in your niche. That’s not easy. Typically, it takes about a year and a half of hard work reading papers and attacking different math problems before you find one that you can prove.
So first, you need to choose an advisor, select a topic, and work hard on your research reading papers, making conjectures, and proving lemmas. While reading, you should analyze examples, and memorize important vocabulary. You should understand that you won’t be able to prove your theorem by making one big step. You’ll need work hard and add lemma after lemma, write down your research results, and keep them organized in your notebook. You may need a couple of years to prove your theorem and you never know if you are following the right or the wrong path for a proof until you actually find it. Research will take up most of your time but you’ll also need to meet regularly with your supervisor and attend all relevant seminars.
When you start writing your dissertation, you should provide definitions for terms and notations that you will use throughout the paper so that your readers can understand your work. The best way to do it is to provide explanations of the key terms in the introduction section. Your introduction should also clearly describe the problem and put it into the context.
When writing the main chapters of your dissertation, you should split lengthy proofs in several steps or several lemmas, taking care of the logical flow. When writing a proof, you must start with the hypothesis and use other mathematical truths such as definitions, axioms, computations or theorems to arrive at the desired conclusion. Write your proof like a manual in a natural step-by-step order. You need to provide all logical steps but there is no need to explain obvious arguments. It’s important to provide clear arguments and justify every step. Don’t just add a parenthetical remark, telling readers that a certain equation was used. You should provide step-by-step explanations in full sentences. When you finish a long argument, you should summarize it. You should also mark the end of your proof.
Don’t forget to link ideas in different chapters using transition words and phrases. This way, you will integrate your work and see your dissertation as one whole in your mind. It’s important to number equations, as well as theorems, lemmas, etc. by section to help your readers follow your argument. You should also include numerical results, examples, computations, comparisons, and interpretations.
Don’t try to write an abstract of your dissertation in mathematics first because you’ll only waste your time. It’s the most difficult part of any academic paper and it should be written last in spite of the fact that it appears at the very beginning of a document. You should try to condense the content of your dissertation into one paragraph and ensure that your readers will be able to understand your message. Abstract provides readers with a quick overview of your dissertation and its organization. It should express your hypothesis and your key points in the proof. Besides, your abstract should suggest applications o implications of the research you discuss in the paper.
How to Cite Sources in Mathematical Writing
You must properly cite all sources of the results used in your dissertation that are not your own because if you state a definition, theorem or an idea and don’t give reference to it, you tell your audience that it is something absolutely new and make a claim that you have created it. And if it’s not true, you commit plagiarism. Every time you use a person’s words or even ideas, you need to give them credit. This includes direct citing and paraphrasing. All major citation styles require using in-text citations and an entry in a reference or bibliography list.
When referring to your bibliography in text of your dissertation in mathematics, you need to include the item into the square brackets e.g.  and then provide more specific information such as theorem number and page number.
e.g. E. Snow [46, Proposition 4.1, p.3] gave a good example of …
When you present a technical material word for word, such as a definition or a statement of theorem, you don’t need to use quotation marks if you provide an appropriate reference.
Revising and Polishing Your Dissertation in Mathematics
As you write each chapter of your dissertation in math and finish it, you should read it for understanding and pay attention to the logical flow and sense of continuity. Then, you should read it again and this time, you need to pay attention to how comprehensible it is to ensure that your audience will understand your arguments. Finally, you should read it once more and pay attention to language issues like spelling, grammar, and punctuation. Following these three stages, you will be able to evaluate each chapter for its structure, content, and style.
When you complete all chapters, you should read the dissertation again and this time, check overall understanding, coherence, clarity, precision, and presentation. Ask your supervisor to read and criticize your early draft of your dissertation. Keep in mind that the more you polish up your math dissertation, the better your chances of getting good mark for it. A good idea is to do a final revision of the text of your dissertation in math on a printed copy instead of doing it on the computer. Printing your math dissertation out and revising it on hard copy can help you feel as if you are reviewing someone else’s paper. It’s a useful technique because it will help you overcome the writers’ blindness and be more objective so that you will be able to spot more mistakes, typos or inconsistencies.
If you take the right approach and take full responsibility for your project from the very beginning, you will be able to write a competent dissertation in mathematics and make a significant contribution to this field of study.
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Degree Programs & Requirements
Note: The information and policies contained on this website is for informational purposes only. The Undergraduate Record and Graduate Record represent the official repository for academic program requirements. These publications may be found at http://records.ureg.virginia.edu/index.php .
Official requirements for our graduate degrees are listed in the Mathematics section of the Graduate Record .
The general rules and requirements governed by the Graduate School of Arts and Sciences can also be found in the Graduate Record under Policies .
Courses and research amounting to 72 credit hours. Courses must include satisfactory work in two semesters of analysis (MATH 7310, 7340), algebra (MATH 7751, 7752), and topology (MATH 7820, 7800), or the equivalent. Students may take approved courses from other departments. Students entering our program with a masters degree from another institution may get credit for up to 24 hours of coursework.
Passing grades on two General Examinations, chosen from Algebra, Analysis, and Topology. These written exams cover the following material: Algebra (MATH 7751, 7752), Analysis (MATH 7310, 7340), and Topology (MATH 7820, 7800). The Analysis exam is split into two parts: Complex and Real, which are taken and graded independently.
These exams are typically offered twice a year, January and August. Students are expected to attempt two different general examinations by August of their second year, to pass at least one general examination by January of their second year, and to pass both exams by January of the student's third year. A student may repeat the examinations as many times as needed in order to satisfy these requirements.
Current syllabi for the Algebra, Analysis, and Topology General Examinations:
- Syllabus for Algebra General Exam
- Syllabus for Analysis General Exam
- Syllabus for Topology General Exam
Examples of past general exams can be found here .
Second-Year Proficiency Examination
The purpose of the oral exam is to assess whether a student is prepared to begin doing research in their chosen area with a thesis advisor. The exam format is flexible and can be tailored to the students’ and examiners’ preferences. For example, it can consist of the student giving a presentation, followed by a question-and-answer period with the examiners, but it could also simply take the form of a question-and-answer exam. Typically these exams should be expected to take 1.5-2 hours.
A student should develop an exam syllabus together with the intended thesis advisor and one additional permanent faculty member. The nature of the syllabus can be flexible, depending on the research area and preferences of the student and examiners. It could still take the form of a list of topics in a given area, drawn from a course or from standard textbooks, but could also take the form of a review of relevant papers, or a suitable combination. The quantity of material should be roughly comparable to the content of two second-year courses. While the exam format can be flexible, in the ways discussed above, it should be specified in the syllabus for approval.
The syllabus should be submitted to the Graduate Committee at least four weeks before the intended exam to allow for adequate time for review and revision, if necessary. After the exam, regardless of the outcome, a form should be submitted to the department, including signatures of the two faculty examiners, indicating which of the two is the intended advisor, as well as the Director of Graduate Studies, to confirm that the syllabus was approved by the Graduate Committee.
Students already enrolled in Fall 2022 or earlier, before this format was approved, may opt to take the oral exam in the previous format of two faculty examining a student on the material from two semester courses on material beyond the first-year courses.
Successfully teaching and directing discussion sections of undergraduate courses, and participating in the professional training seminars MATH 7000 and MATH 7010. Teaching is evaluated, and a good record is an asset to graduates seeking academic employment.
Dissertation and Defense
Written under the supervision of the major advisor, the PhD dissertation must contain original contributions to the field of mathematics. The main results of the dissertation are presented at a public oral defense. A committee consisting of the major advisor and three other faculty members (two from within the Department and one from outside) must approve the dissertation and defense in order for the dissertation to be accepted by the faculty.
Masters Degree Requirements
Please note that Department of Mathematics does not normally admit students seeking only a Masters degree; we are primarily a PhD program.
Information on the The 4+1 BA/MA Program for current UVA undergraduate students
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The Department of Mathematics offers graduate programs leading to the Master of Science (M.S.) and Doctor of Philosophy (Ph.D.) degrees. The Ph.D. program provides for a common core of fundamental mathematics followed by specialized studies culminating in an original research dissertation directed by a faculty advisor. Depending on the student's program and interests, there are diverse career opportunities available in education, government, and industry.
Each graduate program sets their own terms of admission and application deadlines. Applicants can only apply for admission for the terms displayed below. Any questions regarding the application deadline should be directed to the graduate program representative. Fall: January 15 Spring: October 15
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To be eligible for admission into a graduate program at WVU an applicant must submit official, bachelors degree transcripts from a regionally accredited institution and hold a GPA of at least 2.75.
WVU operates decentralized admissions. Decentralized admissions allows each graduate program to set its own application requirements in addition to the University requirements.
To be eligible for admission into the Mathematics graduate program an applicant must submit the following documentation:
- Masters Degree Required
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Here is the complete list of all doctoral dissertations granted by the School of Math, which dates back to 1965. Included below are also all masters theses produced by our students since 2002. A combined listing of all dissertations and theses , going back to 1934, is available at Georgia Tech's library archive. For the post PhD employment of our graduates see our Alumni Page .
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Three Minute Thesis/Dissertation Competition
On this page:
About the Event
Eligibility, application submission, rules for final, in-person competition, evaluation and winners, frequently asked questions.
- Midwestern Association of Graduate Schools (MAGS) Conference
Tuesday, February 21, 2023
If you are working on a thesis or dissertation, you may be wondering how to explain your research ideas easily and engage your audience. The College of Graduate Programs & Honors Studies will host our inaugural Three Minute Thesis/Dissertation Competition on Tuesday, February 21, 2023 . By participating in this competition, you will develop the best “elevator speech” for effective research communication. You will become confident at job interviews and talking to an intelligent, broad audience.
- Friday, January 27, 2023: Deadline for submissions. View application submission requirements .
- Friday, February 10, 2023: The reviewing committee will select a group of contenders to compete at the final, in-person event.
- Tuesday, February 21, 2023, 5 to 7 p.m.: Final, in-person competition at the Student Union Endeavour Room.
- Participating students must be enrolled in either master's or doctoral programs in Spring 2023.
- A student’s program of study must contain an original research project.
- If the degree program does not formally require a thesis or dissertation, the research topic must be an original research project.
- Students must compete as an individual. No team competition is allowed.
- One-page Research Statement is required (max 400 words). The Research Statement should include the originality of the research, the research question(s), the research methodology, data analysis, findings, and (expected) impact.
- Application closes on January 27, 2023.
- The final contenders must attend a coaching session before the final competition.
- The final competition will allow one single PDF slide. No animations or transitions are allowed.
- No additional electronic media (e.g., sound and video files) are permitted.
- No additional props (e.g., notecards, scripts, pointers, costumes, musical instruments, or laboratory equipment) are permitted.
- Each presentation is limited to 3 minutes maximum talking on stage. Competitors exceeding 3 minutes are disqualified.
- Presentations are to be spoken word (e.g., no poems, raps, or songs).
- Presentations are considered to have commenced when a presenter begins through movement or speech.
- A panel of reviewers will use the rubric (PDF) for evaluation and select participants to compete in the final presentation.
- 1 First Place Winner wins a $400 award. The winner will also compete in Chicago in March 2023 at the MAGS conference. The college will cover the travel expenses (more details are below).
- 2 Second Place Winners win a $300 award each.
- 3 Third Place Winners win a $200 award each.
- 1 People’s Choice Winner wins a $150 award. The audience at the event (in-person) will be able to vote for their People’s Choice winner.
- The decision of the judging panel is final.
- The award will be processed through WrightBuy after the event. The award is subject to income tax.
You are eligible to compete if you are enrolled in a master’s or doctoral program in Spring 2023. Your degree program must involve a thesis, dissertation, or original research, such as a master’s research project.
It is up to each competitor to determine whether they have enough context and results to present their research in a compelling 3MT. If your research does not have results yet, you may focus on what you hope to achieve or the expected outcomes. Please seek your supervisor’s advice on how to best present your research.
A countdown timer will be visible to the speaker at the 3MT competition. Presenters who go over 3 minutes will become disqualified . However, we will not interrupt; if you go over, keep talking and finish your presentation.
For the final in-person competition, one PDF slide is allowed. No animations or transitions are allowed. The final contenders will receive more information on preparing for the final competition.
Yes. All Wright State students will receive the invitation to the final competition to be held on February 21. The audience present will be able to vote for a People’s Choice Award. Final contenders are encouraged to invite their fellow students and supporters to the event.
For the final competition, please dress business casual or business. You will be giving a public presentation and will be evaluated by a panel of judges. The dress code should be similar to attending a conference presentation.
Notecards and/or scripts are NOT allowed during the presentation.
No remotion option is allowed. All final contenders will compete in person and be evaluated by a panel of judges in person.
Judges will provide scores on the grading sheets. Judges may provide a brief written comment, if any, on the grading sheets. We will share the brief written feedback with the final contenders after the competition. The scores will not be shared.
No, the judges’ scores are confidential. We will announce the winners but will not share the actual scores.
3-Minute Thesis® Competition at the Midwestern Association of Graduate Schools (MAGS) Conference 2023
MAGS will host the 2023 CGS-affiliate midwestern region competition in March 2023 in Chicago, IL. Wright State University will send our first-place winner to compete at the MAGS Annual Conference. Participants will compete on the final day of the meeting on March 31, 2023. Awardees will be selected and announced at that time. More details can be found on the Competition website .
To take it to another level, the winner of the MAGS competition (midwestern region) will enter the national competition at the Council of Graduate Schools (CGS)’s national conference in Washington, D.C., in December 2023.
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Welcome to the Math PhD program at Harvard University and the Graduate School of Arts and Sciences. Learn more about Harvard's Math communityand our statement on diversity and inclusion. The GSAS Office of Diversity and Minority Affairs offers student affinity groups for graduate students and many other resources.
The doctoral program in the Department of Mathematics offers the personalized attention of a small department while also providing a wide range of faculty who offer expertise to support dissertation research. The graduate student environment in the department is collaborative and rigorous, with many opportunities for mentorship, peer interaction and interdisciplinary
The PhD program in mathematics is intended for students with superior mathematical ability. The program emphasizes a solid mathematical foundation and promotes creative scholarship in mathematics and its many related disciplines. The School of Mathematical and Statistical Sciences has very active research groups in analysis, number theory ...
The Ph.D. program is open to students who have a broad mathematical background and who have demonstrated ability and evidence of special aptitude for research in mathematics in their prior work. Students with B.S. or B.A. degrees can apply directly to the Ph.D. program.
To get your doctoral dissertation done in 20 days, you will have to pay around 20 dollars per page. However, if you are a Master's student and need to get your thesis done, the price would be 18 dollars for each page. Just like any other dissertation writing service, you need to fill in the necessary information, pay, and you are good to go.
The doctoral program in mathematics at UAlbany will prepare you for analysis and problem-solving in business or government, or for research and teaching at the university level. Whether you choose to share your knowledge with the next generation of advanced mathematics students or commit to further developing your own theorems, career outcomes ...
Steps to Filing Your Dissertation (Math & Applied Math) Review Graduate Division dissertation filing protocol (beginning of term): Review the guidelines here. Please read through carefully. Check Dissertation Committee on CalCentral (by beginning of term): Ensure that all members listed are correct and show as "active".
The specialty exam is an oral exam in the student's intended research area. It is conducted by a specialty exam committee made up of three faculty members (usually all in the department), including a prospective dissertation advisor. Starting soon after the completion of the qualifying exams, students typically spend the better part of a year ...
Mathematics Education Theses and Dissertations Theses/Dissertations from 2022 PDF Understanding College Students' Use of Written Feedback in Mathematics, Erin Loraine Carroll PDF Identity Work to Teach Mathematics for Social Justice, Navy B. Dixon PDF
The dissertation is a scholarly work demonstrating the ability to understand, organize, improve, and present mathematical ideas of outstanding importance, depth, or interest. It must include original mathematical research and be worthy of publication.
The Ph.D. in Mathematics program prepares students as researchers to enter a wide range of professional roles including academia, industry, and business. The program is offered in Mathematics (Algebra, Geometry, Applied Mathematics, Computational Mathematics, Data Science, Mathematical Biology, Mathematics Education, and Probability) and in ...
Please cc the DGS and DGSA ([email protected]). Prepare Dissertation utilizing Duke Graduate School Thesis and Dissertations formatting requirements. ... [email protected] Mathematics. 120 Science Drive 117 Physics Building Campus Box 90320 Durham, NC 27708-0320 p: 919.660.2800 f: 919.660.2821
Ph.D. dissertations in mathematics education should be (1) experimental studies in learning, (2) analytical studies in policy theory in mathematics education, or (3) other scholarly investigations of problems and issues of broad significance in the field.
How to Structure Your Dissertation in Mathematics. Dissertations in mathematics typically give an extended analysis of a particular topic and report on a research project or study. A dissertation commonly consists of multiple chapters: Abstract of approximately 300 words summarizes the aims, scope, and the conclusions of a dissertation.
The doctoral dissertation (or thesis) is completed under the direction of a thesis advisor, and must be approved by a faculty committee of (at least) four members, one of which is an ''outside member". Additional information about the rules for forming the thesis committee can be found here.
Courses and research amounting to 72 credit hours. Courses must include satisfactory work in two semesters of analysis (MATH 7310, 7340), algebra (MATH 7751, 7752), and topology (MATH 7820, 7800), or the equivalent. Students may take approved courses from other departments. Students entering our program with a masters degree from another ...
The Ph.D. program provides for a common core of fundamental mathematics followed by specialized studies culminating in an original research dissertation directed by a faculty advisor. Depending on the student's program and interests, there are diverse career opportunities available in education, government, and industry. Application Deadlines
Here is the complete list of all doctoral dissertations granted by the School of Math, which dates back to 1965. Included below are also all masters theses produced by our students since 2002. A combined listing of all dissertations and theses, going back to 1934, is available at Georgia Tech's library archive.
Participating students must be enrolled in either master's or doctoral programs in Spring 2023. A student's program of study must contain an original research project. If the degree program does not formally require a thesis or dissertation, the research topic must be an original research project. Students must compete as an individual.
DEPARTMENT OF MATHEMATICS. The graduate faculty in the Department of Mathematics agrees with the principles in the newly adopted Graduate School-New Brunswick's GUIDELINES ON TIME FOR REVIEW AND ASSESSMENT OF QUALIFYING PAPERS, THESES AND DISSERTATIONS": to maintain a culture of mutual respect between students and faculty members and that this include excellent communication among them.