The Department of Mathematical Sciences offers a Doctor of Philosophy degree in Mathematical Sciences with four different core-course choices. The requirements for the degree is described in the latest Graduate catalog. This document is to give you some additional information on our doctoral program in mathematical sciences.
Students who enter the doctoral program with a master's degree, or with a very strong undergraduate background, may be able to complete the doctoral program in 4 years of full-time study. This cannot be guaranteed, since it depends crucially on the student's progress in research and in writing a dissertation. Extended time schedules may be designed for part-time students.
Doctoral students focus their work in an area of mathematics ("pure" or "applied"), mathematics education or statistics and probability. While all doctoral students take the same five core courses, the choice of focus will determine most of the coursework (as shown in the Graduate Catalog under "Group A, B, C and D"). The degree requirements involve a significant amount of mathematics, regardless of the student's focus.
All doctoral students must pass three "qualifying exams" as stated in the catalog. Those who focus on an area mathematics must take two exams from among Algebra, Real and Complex Analysis, Functional Analysis and Topology, and a third chosen from any topic. Those who focus on mathematics education must take one exam chosen from among Algebra, Real and Complex Analysis, Functional Analysis and Topology, and two exams on mathematics education. Those who focus on statistics and probability must take the three exams on those areas. See Comprehensive and Qualifying Exams for more detailed information on these exams.
While a high degree of focus and depth is necessary for the preparation of a doctoral dissertation, our doctoral students receive significant breadth of exposure as well. For example, all doctoral students develop a significant application of the computer to their area of research. Doctoral students also complete an Applications Involvement Component (AIC), which is described below. Graduates from our doctoral program obtain a perspective of the mathematical sciences as an integrated whole. The required combination of course work, experience, and research enables a graduate of the program to pursue a careerin either academic or non-academic settings.
The design of our doctoral program recognizes the need for new Ph.D. recipients to be exposed to mathematics in non-academic settings by requiring that all doctoral students complete the Applications Involvement Component (AIC) of our Ph.D. degree.
Typically a student's AIC has three parts. In the first, doctoral students attend the AIC colloquia, where speakers external to our department present accounts of how mathematics is used outside of mathematics departments. The external speakers come from industry, government, and education, and are chosen to present a diverse collection of case studies and viewpoints.
The second part of the AIC experience requires each student to undertake an internship in industry, government, or education. Internships usually take place during one of the summers and are arranged by the department's AIC director in consultation with the student. Some of the organizations with which interns have been placed are listed below.
The third part of the AIC requires each student to give a talk about her/his experience and research results obtained in the internship.
An advantage of our program is that it offers a variety of transitional
courses to bridge the gap between undergraduate work and graduate-level
courses. Working closely with graduate advisers, this allows you to plan
the early part of your graduate career in a way that is appropriate for your
ability and background. Here is a brief description of some of the transitional
courses we offer in various areas. Also refer to the course descriptions
given in the Graduate Catalog.
Algebra: If you have already had a one-year sequence
of courses in abstract algebra (proof-oriented courses on groups, rings,
and fields), then you are probably ready for the initial graduate algebra
sequence, MATH 620 and 621. If not, then you may want to start with MATH
520 and 521, transitional courses which discuss groups, rings, and fields,
and allow you to develop your skills at writing correct proofs. There is
another transitional course, MATH 523, a second course in linear algebra.
Perhaps your first linear algebra course was taken as a sophomore and focused
on matrix theory. The second course is more theoretical, and provides important
background for subsequent graduate courses in virtually all areas of
Analysis: If you have already had a one-year sequence
of courses in advanced calculus (including differentiation and integration
of functions of several variables), with lots of attention paid to writing
your own proofs, then you are probably ready for the initial graduate analysis
courses, MATH 630 and MATH 632. If not, then you may want to start with MATH
530 and 531, transitional courses in advanced calculus which will give you
lots of practice at writing proofs, as well as exposure to the important
techniques of the area. MATH 532 (Advanced Calculus III) also fits in this
category; it is intended as preparation for MATH 642 (Partial Differential
Differential Equations: You have probably had one
course in differential equations as an undergraduate, perhaps with a primary
focus on techniques. If not, you should consider taking the NIU course MATH
336. This is a sub-transitional course (an undergraduate course) and would
not carry graduate credit. The initial graduate courses are MATH 636 (ordinary
differential equations) and MATH 642 (partial differential equations). Before
taking MATH 642, it is recommended that you have appropriate background in
line integrals, surface integrals, and Fourier series. This is provided by
another transitional course, MATH 532 (Advanced Calculus III).
Mathematics Education: Graduate students who are also seeking certification to teach at the middle and secondary school level, in addition to a graduate degree, can be accommodated through courses in methods of teaching (MATH 610 and MATH 612) and the student teaching experience.
Numerical Analysis: If you have never written and debugged your own programs in a high-level programming language such as FORTRAN or C, you may wish to take a programming class such as Computer Science 230, which is sub-transitional and does not carry graduate credit, before taking your first numerical analysis class. However, graduate mathematics students who have some programming experience will receive little benefit by taking this course.
Our introductory numerical analysis courses are MATH 534 (numerical linear algebra), MATH 535 (a survey of approximation techniques, numerical integration, and numerical solution of differential equations), and MATH 662 (numerical analysis). These courses involve programming in FORTRAN (or `C'), and they provide an introduction to theoretical issues in numerical analysis.
MATH 662 is an introductory numerical analysis class, and is required for all doctoral students. This course covers many of the same topics as MATH 534 and MATH 535, but from a more advanced mathematical perspective. Doctoral students who desire a transition to MATH 662 can take MATH 534 or MATH 535, but not both of these. (Students may receive credit for only two of MATH 534, MATH 535, and MATH 662). After completing MATH 662, or MATH 534 and MATH 535, students are ready to take subsequent graduate courses in numerical analysis: MATH 664 (numerical linear algebra), and MATH 666 (numerical differential equations).
In the related area of optimization theory, the course MATH 544 (linear
programming) is sub-transitional, and does not carry graduate credit. The
first graduate course in this area is MATH 668 (nonlinear programming).
Topology: The transitional course is MATH 550, which
should be taken after a theoretical course in advanced calculus (such as
MATH 530). MATH 550 is almost entirely a course in point-set topology. The
initial graduate course is MATH 650, which discusses algebraic as well as
point-set topology. MATH 521 is also a prerequisite for MATH 650. Because
of the algebraic nature of MATH 650, students may benefit by also taking
MATH 620 before taking MATH 650.
The transitional courses described above can be very helpful in facilitating a smooth entry into our program. Keep in mind, however, that a decision to begin at the transitional level will probably delay the completion of your program. If your background has prepared you for the basic graduate courses (this is something your adviser can help you to measure), then you should go ahead and take them.
Examples of Programs of Study. Sample Ph.D. Study Plans provides some examples of programs of study for the Ph.D. in mathematical sciences (90 hours). Please note that many other combinations of courses are possible. Your adviser will have up-to-date information on the semesters when particular courses are normally offered.
Students interested in doing graduate work in statistics should contact the Division of Statistics for information about their program. The Division of Statistics also has its own budget for graduate assistantships; contact the Director of the Division for details.