The Doctoral Program in Mathematics
Students working toward a Ph.D. degree usually require from four to six years to complete the requirements for the doctoral degree in mathematics. Ninety-six hours of credit and a Ph.D. thesis are required. At least sixty-four hours must be earned in residence. In addition, the student must meet the Departmental Comprehensive Requirements and pass the Preliminary and Final Examinations for the Ph.D. degree.See the Ph.D. in Mathematics with Computational Science and Engineering Option for additional requirements for this option. The principal requirements in the order that most students complete them are:
- The Comprehensive Requirements
- The Foreign Language Requirement
- The Preliminary Examination
- The Dissertation Committee
- The Doctoral Thesis
- The Final Examination
Ph.D. students who do not already hold an M.S. are encouraged to earn an M.S. here on the way to the Ph.D. See Karen Mortensen for procedures.
The Comprehensive Requirements
The purpose of the Comprehensive Requirements is to ensure that graduate students have acquired a suitable mathematical foundation for undertaking high level research. These cover a basic level of competency in core courses, and beginning students are encouraged to complete them in a timely fashion. These can be met by coursework or exams. Previously given exams can be found on the web at www.math.illinois.edu/GraduateProgram/Study/study.html.
Students must demonstrate competence in five core courses. Two of these are required to be Math 500 (basic algebra) and Math 540 (basic analysis).
For any of the five courses, competence can be demonstrated by either passing the Comprehensive Examination, or by receiving a grade of A- or better in the corresponding class. For at most two of the courses, a grade of B+ in the class is also sufficient. In very special circumstances the Director of Graduate Studies may approve substituting up to two courses with final exams for listed courses.
In addition, students must demonstrate proficiency in undergraduate complex analysis. This can be done in one of two ways:
- By passing a proficiency exam in the subject. This exam will be offered at the same times as the exams in the core courses. Passing the core exam in Math 542 also counts as passing the proficiency exam in complex analysis.
- By getting a grade of B or better in Math 448 or Math 542 for complex analysis.
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Current
Core Courses |
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| Course | Area | Description |
| Math 500 | Algebra I | Isomorphism theorems for groups, centers of p-groups, simplicity of A n, Jordan-Holder Theorem; Commutative Rings and Fields, PIDs, UFDs, Gauss's Lemma, splitting fields, Hilbert Basis Theorem, Zariski topology; Modules over Commutative Rings, structure theorem for finitely generated modules over PIDs, with applications to abelian groups and canonical forms for matrices; Zorn's lemma and applications, existence and uniqueness of algebraic closures; Categories and Functors, universal mapping properties, natural transformations, limits and colimits. |
| Math 501 | Algebra II | Solvable groups, finite p-groups, semidirect products, Sylow's theorem; Galois Theory, transcendental extensions, separable and normal extensions, finite Galois groups, Theorem of the Primitive Element, Fundamental Theorem of Galois Theory, symmetric Function Theorem, examples, cyclotomic, cyclic and radical extentions; Modules over Arbitrary Rings, exact sequences, projective and injective modules, Tensor products, Matrix rings, Schur's lemma, Wedderburn's theorem on semisimple rings, group algebras, Maschke's theorem; Algebraic Geometry, varieties, morphisms of varieties, Noetherian properties, Irreducible varieties and prime ideals. |
| Math 530 | Algebraic Number Theory | Further development of the theory of fields covering topics from valuation theory, ideal theory, units in algebraic number fields, ramification, function fields, and local class field theory. |
| Math 570 | Logic | Development of first order predicate logic; completeness theorem; formalized number theory and the Godel incompleteness theorem. |
| Math 518 | Differentiable Manifolds I | Definitions and properties of differentiable manifolds and maps, (co)tangent bundles, vector fields and flows, Frobenius theorem, differential forms, exterior derivatives, integration and Stokes' theorem, DeRham cohomology, inverse function theorem, Sard's theorem, transversality and intersection theory. |
| Math 525 | Topology | Winding numbers, fixed point theorems, covering spaces, fundamental groups, classification of surfaces, van Kampen Theorem, singular homology, Eilenberg-Steenrod axioms, homology groups of surfaces. |
| Math 542 | Complex Analysis | Topics include the Cauchy theory, harmonic functions, entire and meromorphic functions, and the Riemann mapping theorem. |
| Math 540 | Real Analysis | Lebesgue measure on the real line; integration and differentiation of real valued functions of a real variable; and additional topics at discretion of instructor. |
| Math 550 | Dynamical Systems I | An introduction to the study of dynamical systems. Considers continuous and discrete dynamical systems at a sophisticated level: differential equations, flows and maps on Euclidean space and other manifolds. Emphasis will be placed on the fundamental theoretical concepts and the interaction between the geometry and topology of manifolds and global flows. Discrete dynamics includes Bernoulli shifts, elementary Anosov diffeomorphisms and surfaces of sections of flows. Bifurcation phenomena in both continuous and discrete dynamics will be studied. |
| Math 553 | Partial Differential Equations | Basic introduction to the study of partial differential equations; topics include: the Cauchy problem, power-series methods, characteristics, classification, canonical forms, well-posed problems, Riemann's method for hyperbolic equations, the Goursat problem, the wave equation, Sturm-Liouville problems and separation of variables, Fourier series, the heat equation, integral transforms, Laplace's equation, harmonic functions, potential theory, the Dirichlet and Neumann problems, and Green's functions. |
| Math 561 | Probability | Mathematical foundations of probability and stochastic processes; probability measures, random variables, distribution functions, convergence theory, the Central Limit Theorem, conditional expectation, and martingale theory. |
| Math 531 | Analytic Number Theory | Problems in number theory treated by methods of analysis; arithmetic functions, Dirichlet series, Riemann zeta function, L-functions, Dirichlet's theorem on primes in progressions, the prime number theorem. |
| Math 580 | Combinatorics | Fundamental results on core topics of combinatorial mathematics: classical enumeration, basic graph theory, extremal problems on finite sets, probabilistic methods, design theory, discrete optimization. |
Examinations are held three times per year at or near the beginning of each semester, but the examinations for a particular course are only offered just before and after the course has been taught. Although students are allowed to repeat taking exams without penalty, they should complete the Comprehensive Requirements within 2 academic years. A reasonable pace is to satisfy one or two topics per session. The examinations are based on syllabi covering the core material in each course, i.e. subject material that is expected to be covered by all versions of the course. Note that the syllabus for a particular core requirement might differ slightly from the subject material actually taught in any particular section of the course. The syllabi for these courses appear in the appendix of this guide. Each of the core examinations will be prepared and graded by a two or three person committee appointed by the Chair. (In exceptional circumstances a single faculty member may prepare and grade an exam.) Each examination committee will report for each student one of the following grades: pass or fail. In cases where a Ph.D. research program contains an interdisciplinary component, students may be allowed to take one Ph.D. qualifying/comprehensive exam (at a graduate level) from a department other than the Department of Mathematics. This exam would be one of the regularly scheduled Ph.D. qualifying exams offered by the department. This substitution would be subject to the approval of the Graduate Affairs Committee based on the mathematical content of the exam.
All exceptions to the above schedule of examinations and requirements must be approved by the Director of Graduate Studies in consultation with the Graduate Affairs Committee.
The Foreign Language Requirement
Doctoral candidates must demonstrate proficiency in one language chosen from French, German, and Russian. Proficiency can be demonstrated by passing the 501 course in that language with a grade of A or B or by taking the language examination from faculty members within the Department of Mathematics. Professor Maarten Bergvelt is in charge of these exams. For further details on these special options see him or the Director of Graduate Studies.
The department urges students to complete the foreign language requirement before taking the Preliminary Examination. After a student passes the Preliminary Examination it is time to focus on research, which is difficult to do while enrolled in a language course.
No candidate will be allowed to schedule their Final Examination until the language requirement has been satisfied.
The Preliminary Examination
The Preliminary Examination is taken after the Comprehensive Requirements have been completed and after the student has found a potential thesis adviser. The purpose of the exam is (a) to verify that the candidate has chosen a suitable topic for thesis research, (b) to evaluate the candidate's depth of knowledge in a chosen area of specialization and ability to begin or continue research in this chosen area, and (c) to formally create the adviser/student relationship for the thesis.
The exam will be administered by a committee appointed by the dean of the Graduate College upon recommendation of the Director of Graduate Studies. The candidate in consultation with the Director of Graduate Studies must set this committee up at least three weeks before the scheduled time of the exam. It must include at least four voting members, three of whom must be members of the Graduate Faculty and two of whom must be tenured. The potential thesis advisor is a member of this committee.
The exam will be oral and not longer than two hours. It will consist of a short presentation by the candidate describing her/his accomplished and proposed research, followed by a question and answer period. The expected length of the candidate's presentation, and a precise list of topics on which the student is to be examined, must be agreed upon by the student and the committee before the exam. The topics should cover advanced material in the research area and preliminary research results. They should not seriously overlap basic material covered in the Comprehensive Requirements.
The Preliminary Examination Committee shall report a grade of pass or fail or decision deferred. The decision of the committee must be unanimous. On the first attempt a grade of pass with distinction may be reported. Failure can be final, or the committee may grant the student another opportunity to take the examination after completing additional course work, independent study, or research. Finally, the committee may defer its decision for up to six months. If a second exam is allowed, the student may petition the Director of Graduate Studies to have a written exam on the second attempt.
The Dissertation Committee
When the Preliminary Examination has been passed, or shortly thereafter, a Dissertation Committee will be appointed for each student by the Dean of the Graduate College upon recommendation of the Director of Graduate Studies. The Dissertation Committee shall consist of a minimum of four voting members, three of whom must be members of the Graduate Faculty of the UIUC and two of whom must be tenured. The thesis advisor is a member of this committee and the chair of the committee must be a member of the Graduate Faculty other than the thesis adviser. A contingent chair should be designated to serve as the chair of the Dissertation Committee should the original chair leave the university. Normally, this committee consists of the examiners for the Preliminary Examination, but it can also have other faculty members. It, in turn, will form the nucleus of the committee administering the Final Examination.
The purposes of the committee are:
a. to provide advice to the student on her/his doctoral research and
preparation of the dissertation.
b. to ensure that the quality of the doctoral dissertation meets a
high academic standard.
c. to judge the performance of the student in the final oral examination,
which is based on the doctoral dissertation.
In order for the committee to properly fulfill its advice and oversight functions, students should meet with their committees at least once a year. The first such progress review should take place no later than one year from the date of the preliminary exam. The format of the meetings may vary. Students may convene a special meeting of the entire committee, or may meet individually with each committee member, or the student may invite the committee to attend a seminar in which he/she presents his/her latest results. The Director of Graduate Studies can offer assistance if a student is having any difficulties meeting with their committee.
The Doctoral Thesis
The goal of doctoral study is the development of a student into a scholar who can conduct independent research. Students gain the necessary basic knowledge by taking courses. However, many students find that there is a hurdle to be overcome in making the transition from studying and learning mathematics to doing mathematics. Problems assigned as homework in advanced courses are usually ones for which the answer is known. Problems that are suitable for thesis topics are ones for which the answer is not known and for which the appropriate methods of attack may not be clear. "Excellent students" may discover at this point that mathematical research is not their true calling, whereas "average students" may find that they excel in working on a single topic in extraordinary depth. Learning to be a scholar conducting independent research is facilitated by participating in research seminars where this process can be observed in action, but the main responsibility lies with the thesis advisor who guides the student in conducting a research program on a topic selected in consultation between the student and the advisor.
Students are responsible for finding their own thesis advisers. Usually a student will approach a faculty member whose work and interests are known to the student through attendance at courses and seminars given by the faculty member and arrange to take Math 597 with the faculty member. While doing reading courses, the student and faculty member will determine whether they can work together on a research program leading to a thesis.
Finding an adviser and/or topic is an important and sometimes difficult process for students. One role of the Director of Graduate Studies is to assist and support a student at this critical stage when needed.
An original thesis must be written in an approved area, normally chosen from one of the research areas represented in the Department, and must be read and approved by the Dissertation Committee. While conducting research on the chosen topic and also while writing their thesis, a student should consult frequently with their thesis adviser and members of the Dissertation Committee. Learning how to write technical papers (including a thesis) is an important part of the research training of a student. Thus, during the course of the research, the thesis advisor may require the student to write one or more papers to report on the research work. Since one measure of success in a research program is the publication of the results in a reputable technical journal with rigorous review procedures, the Department expects that the results in a Ph.D. thesis will be published in one or more journal articles.
Subject to advance approval of the Chair, doctoral students may do their thesis research under the direction of members of the graduate faculty in departments other than Mathematics. To obtain such approval, students should consult with the Director of Graduate Studies to arrange a meeting between the Chair, the thesis supervisor, the student, and any other members of the Department of Mathematics as the Chair may indicate. The purpose of this meeting will be to discuss the nature and direction of the intended thesis research.
Preparation of the Thesis for Submission
The general format of theses is specified by the Graduate College in a manual titled Instructions for the Preparation of Theses. This manual gives complete details on all the materials to be submitted to the Graduate College. Before preparing the thesis, the student should obtain the latest version of the manual from the Graduate college, and read it carefully. All the requirements stated in the manual must be met in full.
In mathematics, a majority of scientific documents are prepared in TeX, so it is well worth the time it takes to learn this word processor. TeX macros are available through the Department computer labs containing all of the formatting instructions required for theses. Other word processors, especially those with style sheet facilities such as Word, Word Perfect, Mathematica, etc., are also suitable but, up to now, nobody has programmed the style instructions for theses.
The Final Examination
The final examination is oral, and covers the material in the dissertation. It should last not more than two hours and is administered by a committee consisting of the Dissertation Committee plus, if appropriate, one additional member who:
- need not be a voting member,
- may be an external reader, or a University of Illinois at Urbana-Champaign faculty member who is off-campus,
- may be someone who can make a significant contribution to the research and assessment of the dissertation.
This committee must be set up at least four weeks before the scheduled time of the exam by the candidate in consultation with the Director of Graduate Studies.
The chair of the Dissertation Committee is responsible for convening the committee, conducting the examination and submitting the Certificate of Result to the Department of Mathematics and to the Graduate College.
All voting members of the Dissertation Committee must be present at the final exam or participate in the exam via appropriate electronic communication technology. A unanimous vote is required to determine the result of the final examination, which is recorded on the Certificate of Result and signed by the voting members of the Dissertation Committee. Non-voting members need not be present at the final examination. The Online Version of the Certificate of Committee Approval Form (formerly called the red-bordered form) that becomes a part of the thesis document may be signed by both voting and non-voting committee members.
Ph.D. degrees are awarded in May, August and December. Please contact the Graduate Office at least three months prior to your planned graduation date. You must also apply for the degree online using Enterprise near the beginning of the semester in which you intend to graduate.