A Guide for Graduate Students in Mathematics: Introduction

Applications

Applications are accepted for both fall and spring semesters. The submission deadline for applications for Fall Semester that include a request for financial aid is December 1. The deadline for supplemental materials (including letters of recommendation and transcripts) is January 2. Only applications which are complete will be reviewed. Applications for admission only can be considered up to April 15. The deadline for all applications for spring semester is the preceding October 15. Department of Mathematics reserves the right to close the acceptance of applications at any time. Financial aid is rarely available for students entering in a spring semester. The webpage at http://www.math.uiuc.edu/GraduateProgram/apply_admission_info.html has links to admission requirements and application forms, as well as other useful information.

Orientation Program

An orientation program is offered at the beginning of the Fall semester to introduce incoming graduate students to the Department of Mathematics and its faculty, the operation of the Graduate Program, and the Department computer laboratories and facilities. During this time, all new teaching assistants participate in a three-day training session in which teaching assignments are discussed, practice teaching is done, and practice exams are constructed.

The Advisory System

During the orientation program, each new graduate student will meet with a team consisting of a faculty member from analysis and a faculty member from algebra to determine exactly where he or she belongs in the sequences of graduate courses. Students should be prepared to provide detailed information about their mathematical backgrounds. They should have read the relevant portions of this Guide and should have prepared a tentative plan for a program of studies that they can discuss with the advisors.

Each incoming graduate student is assigned a faculty advisor whose research interests are similar to those indicated on the student's application for admission. New research assistants are advised by the faculty members who offered them employment. Until such time that the student finds a thesis advisor, the faculty advisor helps the student in planning a program of studies. Faculty advisors may be required to submit a report to the Director of Graduate Studies on the progress of their advisees.

This Guide contains a variety of materials which are useful in planning a program of studies. These materials include the background knowledge expected for various degrees, general advice about the fields of specialization and lists of courses considered essential and desirable in each field, and information about advanced courses offered by the Department.

The Master's degree allows an optional thesis and the Ph.D. degree requires one. The Ph.D. thesis records the results of a successful research program conducted under the direction of a thesis advisor. Since the thesis advisor usually requires the student to take specific courses in order to obtain the necessary background for conducting research, it is imperative that students seek out faculty members with research interests similar to theirs and explore potential thesis topics as early as possible in the course of their studies. The Department of Mathematics Graduate faculty and their research interests can be found at http://www.math.uiuc.edu/ResearchAreas/.

Initial Registration

New students may register for courses using UI-Integrate Self-Service as indicated by their Notice of Admission. Occasionally, students find that some of the courses in which they are enrolled are not suited to their interests or previous preparation. With the advisor's consent, the student may drop such courses and add others after the semester has begun. Such changes are handled through on-line registration.

Background Requirements

Certain basic principles must be thoroughly understood by every graduate student in order to make reasonable progress in the Mathematics Graduate Program. A basic understanding of undergraduate mathematics for mathematics, engineering, and science students should include calculus through differential equations and linear algebra. Prospective mathematics graduate students should have mastered these computations and be comfortable with the concepts behind them. However, more is expected of a student who intends to embark on a program of graduate study in mathematics. Accordingly, it is expected that students entering the program will have adequate background at the undergraduate level of the following material: students should be familiar with all of the topics 1-4, as well as at least two of the topics 5-9. Some Master's degree programs also require knowledge of topics 10 or 11.

1. Real Analysis. Completeness properties of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration.
2. Abstract Algebra. Modular arithmetic, permutations, group theory through the isomorphism theorems, ring theory through the notions of prime and maximal ideals; additional topics such as unique factorization domains and classifications of groups of small order.
3. Complex Analysis. Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra, and the argument principle.
4. Abstract Linear Algebra. Linear equations, vector spaces, linear transformations, matrices, determinants, invariant subspaces, direct sum decompositions, canonical forms, inner product spaces and bilinear forms.
5. Geometry and Topology. The shape and curvature of curves and surfaces, vector fields, differential forms on euclidean spaces; moving frames. Informal set theory, cardinal and ordinal numbers and the axiom of choice; topology of metric spaces and introduction to general topological spaces.
6. Number Theory. Divisibility, primes, congruences, quadratic reciprocity and standard mathematical functions.
7. Differential Equations. Existence and uniqueness of solutions and the general theory of systems of linear differential equations.
8. Logic. Experience with informal mathematical reasoning in an abstract setting as well as with formal or symbolic expression of mathematical statements.
9. Combinatorics. Permutations and combinations, generating functions, recurrence relations, inclusion and exclusion, Polya's theory of counting, and block designs.
10. Statistics and Probability. Calculus of probability, combinatorial analysis, random variables, expectation, distribution functions, moment-generating functions and central limit theorem.
11. Computer Science. Concepts, principles and use of computing data structures; pointers, lists, stacks, trees, hashing, graphs, and sorting.

The goal of the Ph.D. degree program is to bring students to the level where they can join in mathematical research and development either in industry or in academia. In industry, mathematicians frequently work in teams with scientists and engineers. They are responsible for the mathematical details of models of situations of interest in the industry and for drawing quantitative conclusions from the models. In the academic world, mathematicians divide their time between teaching and research, in different proportions depending on the institution where they are employed. There is general agreement that teaching mathematics at the college level requires a personal understanding of what it means to carry out mathematical research. Such an understanding can only be acquired by doing mathematical research. In particular, academic mathematicians will be responsible for the next generation of undergraduates who go on to graduate work in mathematics. They will also carry the heavy responsibility of training the next generation of mathematics teachers at the K-12 level, an important task that has been identified by the government as a national goal.

A student whose background is deficient in one or more of the basic areas, or whose knowledge in the additional areas is inadequate may need to acquire course work that cannot be counted for graduate credit. In some instances, such a student may be required to take a proficiency examination in lieu of additional coursework. A grade-point average of 3.0 or more for Master's degree candidates and 3.25 for Ph.D. degree candidates must be achieved for the totality of courses taken.

General Requirements Applying to All Students

Registration

There are no general registration requirements for Master's degree students, other than those specific to each program. However, candidates for the Master's degree with thesis option must be registered in Math 599 (for four hours) while thesis work is carried out. Students in related areas at UIUC who have accumulated sufficient credits in mathematics for a Master's degree must be registered as students in the Department of Mathematics for at least one semester before they receive their degree.

Ph.D. candidates normally register for Math 599 while working on their thesis. The Graduate College requires that they be registered in Math 599 (for zero or more hours) at the time of the Final Examination. A candidate who registers for a term (semester or summer session) is considered to have satisfied this requirement if the Final Examination is taken after the end of that term, but before the first day of classes for the following term.

Satisfactory Academic Progress

All graduate students must maintain a cumulative grade-point average (GPA) of a least 3.0 out of 4.0 for Master's degree candidates and 3.25 for Ph.D. degree candidates. (See Section 2.3 below.) The cumulative GPA is computed on all courses taken for unit credit except thesis units in which DF is recorded until the thesis is completed. A cumulative GPA of at least 3.0 is also required in undergraduate courses taken for hours credit.

A student who fails to maintain a cumulative GPA of at least 3.0 for Master's degree candidates and 3.25 for Ph.D. degree candidates is placed on probation. If the student's cumulative GPA is still less than 3.0 (respectively 3.25) after one semester on probation, then the student may not be allowed to continue in the Mathematics Graduate Program, and further registration may be prohibited.

All degree candidates except doctoral candidates who have passed the Preliminary Examination must earn at least eight hours of credit (other than credit in Math 499) each semester, and at least twenty hours of credit (other than credit in Math 499) during each calendar year. Students who fail to earn eight hours in a semester or twenty hours in a calendar year may receive a letter warning them of the deficiency. If the progress during the next academic term is not satisfactory, the student may not be allowed to continue in the Mathematics Graduate Program, and may be prohibited from further registration.