Introduction to Advanced Degree Programs in Mathematics at Virginia Tech
Introduction and Overview
Graduate study in mathematics is attractive for many reasons. Centuries of careful scholarship have produced a rich tradition and intellectual depth that is unsurpassed among classical subjects of advanced study. Today, careful modes of advanced mathematical reasoning are vital to many breathtaking technological advances. Indeed, the interplay with emerging sciences and technologies has always been vital to the development of mathematical ideas. That interplay is even more intense today, with computing technology providing a level of interaction that was not possible in the past. Thus the modern world of mathematics consists both of established bodies of knowledge, distilled from the studies of scholars past and present, and of emerging areas of study and application. These advancing areas are often characterized more by unanswered questions and intellectual challenges than by established answers. The world of mathematics is not static, but is part of the same explosion of knowledge and technology that fills the headlines.
The Department of Mathematics at Virginia Tech offers a range of degree programs intended to educate qualified students in the burgeoning world of advanced mathematics, and to prepare them for careers in its use and development. New mathematically related career paths continue to emerge as our technology-based economy evolves, while some traditional ones seem to be declining. Those pursuing careers in teaching will need to prepare their own students to meet the challenges of this changing world. We believe that the best way to prepare for the future is to develop a foundation from which graduates can build careers in many directions, gain some experience with how a specialty is built on those foundations, and learn to use the most up-to-date tools. In what follows you will find an overview of our programs, resources, and faculty. In addition to understanding the opportunities, we also want you to have realistic expectations of the rigors of full-time graduate study and what preparation you should have. If, after reviewing the information below, you have questions or would like to apply for admission, do not hesitate to contact us using the addresses, phone numbers, and electronic addresses which are provided.
Graduate Student Environment and Opportunities
The Mathematics and Computer Science Departments are located in McBryde Hall which is the mathematical sciences building pictured below. There are roughly 70 graduate students actively enrolled in the mathematics program. Over 1/3 of these are women. About 2/3 are in the Master's (M.S.) program. All graduate students have an office/study space for their own use, and are welcome in the Department's Common Room for informal discussion and conversation. These office/study spaces all include convenient access to the computing system. One suite of graduate student office spaces is being equipped with individual workstations, linked by a special high-speed network, for students whose projects make such resources appropriate.
All students start in the Master's program, unless they have already completed a Master's Degree elsewhere. (Those wanting to continue beyond an M.S. will request admission to the Ph.D. program during their second year.) As a new student you will be given an initial advisor appropriate for the mathematical interests you have expressed. However you are free to make your own arrangements with any other faculty member to be your advisor if you wish. We encourage students to consider taking advantage of one of the several national programs providing an internship with a lab or private sector employer. (We can help you identify some possibilities.) Not only does this help clarify career goals and possibilities, it can also be a valuable experience to future employers.
Most of our students become involved in some sort of study or research project in addition to their course work. This can range from independent study on a topic of special interest, under the direction of a faculty member, to supporting work on a faculty research project, to the doctoral dissertation of Ph.D. students. Most M.S. students complete their studies with a presentation based on a reading/study/research project that they have carried out under faculty supervision. These opportunities to study with and learn from both student and faculty colleagues is one unique privilege of graduate work in a major research department.
Each year an average of 15 or so students complete their M.S. degrees. Some of the employers of our past Master's Degree graduates are: Cambridge Technology Partners, Bluffton College, Wingate College, U.S. Army Combined Arms Support Command, Mississippi State University, Johns Hopkins University Applied Physics Laboratory, Naval Air Warfare Center, Allied Signal Communications Systems, GEICO and MIT's Lincoln Laboratories. Of course many go on to further studies.
Some of the most popular areas for Ph.D. dissertation work are, PDEs (Applied), Mathematical Physics, Numerical Computation, Control and Systems Theory, Algebra and Number Theory. An average of about 5 students complete their Ph.D. each year. Some employers of recent graduates include: University of Texas, Washington University (St. Louis), University of California - Berkeley, Georgia Tech, Memphis State University, Iowa State University, University of Richmond, University of California - San Diego; Siemens Energy and Automation Inc., Advanced Technologies, Crown Life Insurance, Hollins College, University of Arkansas, University of St. Thomas, Mississippi State University.
Most graduate students in the Mathematics Department are supported by Graduate Teaching Assistantships (GTAs). GTAs receive a stipend (paycheck) for the 9 month academic year, as well as a waiver of tuition. (GTAs pay only the $682.50 comprehensive fee each semester.) The stipend level is variable, depending on your progress in the program. Informatation on stipends can be found at http://graduateschool.vt.edu/financial/assistantships/index.html#graduate. As a GTA you are an employee of the Mathematics Department with assigned duties related to the teaching of our undergraduate courses. (Additional description of GTA duties is provided below.) Students are typically supported for 2 years in the M.S. program. After admission to the Ph.D. program 4 additional years of GTA support can be anticipated for students making satisfactory progress. There are usually a number of GTA positions available for the summer months, awarded on a competitive basis. Faculty with research grants can often provide summer support for students assisting them on their projects. In each of the last 3 years over 25 graduate students received summer support from faculty research grants.
Some graduate students may find academic year support (in place of a GTA) through their advisor's research grant, or through fellowships or grants. Some of the awards that our graduate students have held are: NSF Graduate Fellowship, DOD Assert grant, Albert Einstein Congressional Fellowship, Willma Lowry Teacher of the year in Mathematics, Presidential Award for Excellence in Science and Mathematics Teaching. The Department maintains a file listing such opportunities.
Preparing for Graduate Study
Naturally the best preparation for graduate study in mathematics is the strongest undergraduate experience you can obtain. The minimal requirements for an undergraduate degree are usually not adequate preparation for graduate school - you should try to take additional advanced classes. In particular you should make every effort to take senior level Modern Algebra and Real Analysis courses. You should try to take an upper-level course that gives you significant experience with numerical computation as well. These preparations will make you a stronger applicant, and you will not need to spend part of your graduate program filling them in if you have taken care of them while an undergraduate.
The combination of challenging academic work and GTA duties can be quite demanding. Thus beyond academic back ground you will need a greater level of commitment to your studies than you may have as an undergraduate. Your effort however will be well-spent. The training and education you obtain as a graduate student will probably have a profound effect on your future. As a student at a major research university like Virginia Tech, you will have more opportunities and resources for developing yourself than you will ever have again. You need to be committed to taking full advantage of them. If you have a family, you and they need to understand and support the commitment of your time while a graduate student. On the encouraging side, you will be part of a community of graduate students that share the same kind of commitment to their studies. Blacksburg is full of hardworking faculty and students, so you will find plenty of understanding for the demands of a graduate student's life among any community or social involvement you form outside the Mathematics Department.
The Admissions Process
To be admitted to the graduate program a formal application must be submitted to the Graduate School. Online applications are required and can be completed following the link below:
Note that a complete application requires 3 letters of recommendation, GRE scores (only the General Test is required), complete official transcripts of all college/university study, a non-refundable $65 application fee, and TOEFL scores for international students. Your application materials will be collected into a packet or dossier. A committee of experienced mathematics faculty will review the application packets, looking for those that have the greatest potential for success, and whose expressed goals are consistent with our programs. All the materials in your application will be considered. Strong academic performance in the past certainly strengthens an application. However other considerations (evidence of strong focus and motivation or relevant work experience) can compensate for a somewhat weaker academic record. Your overall grade point average may not be as important as how thorough your prior mathematics and related subject preparation is. Thus, GRE subject test scores, recommendation letters that specifically address competence in advanced mathematical courses and/or contributions during research experiences, or documented use of mathematical reasoning in work experiences can aid the graduate admissions committee in making their decisions. There is no simple formula that identifies the applications that will be accepted. We rely on the judgment and experience of the reviewing faculty. If we accept your application you will receive a letter from the department describing any offer of financial support that we can make. Once a decision is reached on an application, the Graduate School will notify the applicant.
Most new students begin in the Fall Semester. A small number of students are sometimes admitted to begin in the Spring Semester, but finding additional support at that time of the year can be more difficult. A few applicants are admitted without an offer of financial support, for a variety of reasons. However, in most cases accepted applicants will receive an offer of support.
Students who have not previously completed a graduate degree program are admitted into the Master's Degree program, not directly into the Doctoral Program. Those who desire to continue into the Doctoral Program may apply for that in their second year. The Master's program is structured so that normal progress toward doctoral studies will satisfy the requirements of a (nonthesis) Master's degree by the end of the second year. Students who have previously completed a graduate degree may be admitted directly into the doctoral program. A reasonable amount of graduate work completed at other institutions can be transferred toward your Ph.D. here, provided it is approved by your advisory committee (and was completed with grades of "B" or better).
If you are an international student you will be required to submit TOEFL scores as part of your application. The University takes on a certain legal responsibility in authorizing the visa for an international student. It is also important that our GTAs are as effective as possible in their contributions to undergraduate instruction. For reasons such as these we require the strongest credentials and faculty endorsement for admission of an international applicant. Once admitted, the Graduate School will administer additional tests for English language skills. You will be required to take special English courses to strengthen your language skills, if necessary. International students are also required to provide evidence of health insurance coverage, or else participate in the University's optional group policy for graduate students.
For More Information
If you have questions that are not answered in this website, do not hesitate to phone, write or send a message to
Department of Mathematics
460 McBryde Hall Mail Code 0123
Blacksburg, VA 24061-0123
The Virginia Tech home page is a source of information about the University. General Graduate School information is available from the Graduate School's home page. More information about the Mathematics Department specifically is available here at our departmental web site.
You are welcome to come visit us and see the campus, department, and facilities first hand. If you call ahead to let us know you are coming we can be better prepared to show you around. Each Spring we invite a number of our prospective applicants to campus for our annual Visitor's Day program. For that occasion we plan numerous special activities to introduce our guests to the Department, provide a chance to meet the faculty and current graduate students, and see the campus community.
Summary of Programs
We offer programs leading to the Master of Science (M.S.) degree, and a doctoral program leading to the Doctor of Philosophy (Ph.D.) degree. The descriptions below of degree requirements for these programs are only summaries, in tended to communicate the nature of the programs to potential students. They are not complete in every detail. For complete descriptions see the Mathematics Department brochure, Graduate Program in Mathematics: Policies and Degree Requirements. The requirements are described below in terms of numbers of courses. Other university and department publications use "credit hours". The typical graduate course is 3 credit hours. Thus the 10 course requirement for the M.S. degree is stated as 30 credit hours in other publications.
The Master of Science degree represents two years of course work and an optional thesis. This degree is intended to provide a basic foundation of graduate level work from which the student can pursue multiple options for the future. Many M.S. degree holders move on to a technically oriented job with an employer in the private sector or with a government agency. For others it is the gateway into the more intense and in-depth study of the doctoral program. There are 2 M.S. degree options: the nonthesis option and the thesis option. As alternatives to the standard requirements for these options, both are available under a special interdisciplinary plan.
The nonthesis option requires that at least 5 of the 10 courses be graduate level (5000 or higher) mathematics courses, and that an additional 2 be graduate level courses in either mathematics or a related discipline (e.g. computer science, statistics, engineering, economics, finance). The degree requires a minimal basic competency in Modern Algebra (the equivalent of 4124), Real Analysis (the equivalent of 4225,6) and computational experience. These can be satisfied by work completed prior to starting graduate studies, or by including appropriate courses in graduate work. The 10 courses must include one of the following major full-year course sequences:
- Abstract Algebra (5215,6)
- Calculus of Variations (5545,6)
- Complex Analysis (5235,6)
- Functional Analysis (6255,6)
- Mathematical Methods for Modeling and Simulation of Biological Systems (5515-6)
- Numerical Analysis (5465,6)
- Numerical Methods for PDEs (5474,5484)
- Ordinary Differential Equations (5245,6)
- Real Analysis (5225,6)
To complete the degree requirements the student must satisfy some form of final examination requirement. Our current practice is to have the student prepare and give a lecture with written summary on the topic of a short independent study project carried out under the supervision of a faculty member. These Master's Presentations have proven valuable to students in job interviews and in developing professional communication skills, not to mention the interesting mathematical topics they have explored. A standing alternative to the Master's Presentation requirement is the passage of two Ph.D. preliminary examinations. This allows a student to complete the M.S. requirements "automatically" in the first two years of normal progress toward the Ph.D.
The thesis option requirements differ from the above mainly in that the writing of a Master's Thesis replaces some of the course work. The Master's Thesis is a written report resulting from a significant independent study or research project, conducted under the guidance of a faculty advisor. Two (or 3) of the 10 required courses are replaced by Math 5994, a special course designation to account for the time spent in independent study and writing of the thesis. In addition the thesis option allows a little more flexibility in allowing you to take courses outside the Mathematics Department that may be relevant to your work. The basic competency and sequence requirements are the same as for the nonthesis option. The final examination is replaced by a presentation of the thesis and response to questions (the "thesis defense"). This option pro vides more opportunity for in-depth work in a specific topic or application than would be possible under the nonthesis option.
The interdisciplinary plan is intended (only) for students having clearly defined interdisciplinary career goals which cannot be adequately served under either the standard thesis or nonthesis options above. The basic competency requirement in Real Analysis (the equivalent of 4225,6) again applies, and the 10 courses must include at least 7 at graduate level, but there are no specific requirements beyond these. Instead the student's program of study is designed by the student and an advisory committee (including at least one faculty member from the related discipline). This allows a program of study to be customized to the student's specific interdisciplinary goals, while insuring a level of quality comparable to the other master's degree options. It is important that the student interested in this option take the initiative to form the advisory committee and plan a program of courses at the very beginning of their graduate studies.
The Doctor of Philosophy (Ph.D.) degree takes the student far beyond the Master's degree in several ways. For one, a broader and more solid basis of advanced mathematical knowledge must be demonstrated. To insure this the student must pass Ph.D. preliminary examinations in a selection of the subject areas covered by the full-year course sequences listed under the nonthesis M.S. option above. With the prelims complete the student begins the preparations for his/her doctoral research project. The transition into the research phase is marked by the comprehensive exam. This exam is administered by the student's advisory committee. Its structure may vary, but the exam generally includes a discussion of the planned research project. There is also a foreign language requirement (1 language) whose design is determined by the advisory committee. The time required to complete the Ph.D. program is not fixed but typically involves 3 to 5 years of study after completion of the M.S.
The centerpiece of the Ph.D. is the dissertation. Under an advisor's guidance, the doctoral candidate engages in a major research project. The dissertation itself is the written document, following professional standards, resulting from this project. Through this dissertation work the doctoral student moves beyond the relatively passive role of receiving knowledge presented in courses to become an active, self-motivated scholar, making a significant contribution to their area of specialty. The work of the dissertation is expected to be of such quality as to merit publication in a scholarly journal, after appropriate revisions. For those continuing in academic research the dissertation topic may initiate a more lengthy research program that forms the beginning of a scholarly career. For those who continue in a nonacademic direction, the dissertation experience is valued because it requires the highest level of creativity and independent thinking.
Typical Programs of Study
The normal course load for a first or second year student is 3 courses per semester. (An additional 3 hours of Math 5994 (Research and Thesis) is added to this to bring the total registration to 12 course hours per term. There are several reasons for this. One is so that the schedule will have some representation of the time spent in fulfillment of GTA duties.) A typical first year course schedule (for a student who has not previously had the equivalent of 4124 Algebra or 4225,6 Real Analysis) might look like:
- 4124 Algebra
- 4225 Real Analysis
- 5xxx (sequence, part 1)
- /GTA duties
- xxxx (single semester course)
- 4226 Real Analysis
- 5xxx (sequence, part 2)
- /GTA duties
There are several "single semester course" opportunities. A Topics in Algebra (5114) course is frequently offered in the Spring for students following the above pattern. Also either Combinatorics (5464) or Graph Theory (5454) is usually available. Some students may prefer to reverse the order, taking Matrix Theory (5224) or the first part of a 5000-level sequence in the Fall and then 4124 in the Spring instead.
Many graduate-level course sequences require either 4124 or 4225,6 as prerequisites. Students entering without either of these may need to delay their primary 5000-level sequence(s) until the second year. First year students in this situation may find Combinatorics (5464) and Graph Theory (5454) or Numerical Analysis (5465,6) attractive since their dependence on the real analysis and algebra prerequisites is not as strong. Another choice is the Topology, Geometry Combination (4324 & 5344).
The following is not a complete listing of courses offered, but includes the regular offerings that would be most relevant to a prospective graduate student. Not listed are special topics courses, for which the subject content varies from one term to the next, at the discretion of the instructor. Some subjects of recent special topics courses include group representations and applications, Galois theory, control and systems theory, H-infinity control theory, algebraic number theory.
Algebra and Combinatorics Courses:
- 4124 Introduction to Abstract Algebra
- 5125,6 Abstract Algebra
- 5135,6 Number Theory
- 5454 Graph Theory
- 5464 Combinatorics
- 5225 Matrix Theory
- 4324 Elementary Topology
- 5344 Specialized Topics in Geometry and Topology
- 6324 Topics in Topology and Geometry
- 4225,6 Elementary Real Analysis
- 5225,6 Real Analysis
- 5235,6 Complex Analysis
- 5245,6 Ordinary Differential Equations
- 6255,6 Functional Analysis
Numerical Analysis Courses:
- 5465,6 Numerical Analysis
- 5444 Numerical Methods for Ordinary Differential Equations (Summer only)
- 5474 Finite Difference Methods for PDEs
- 5484 Finite Element Methods for PDEs
- 5485,6 Numerical Analysis and Software
Applied Analysis Courses:
- 5244 Systems and Stability of Differential Equations (Summer only)
- 5415,6 Applied PDEs
- 5545,6 Calculus of Variations and Optimal Control Theory
The University and the Department provide orientation for GTAs new to these responsibilities. A beginning GTA may be assigned to assist in the Department's walk-in tutoring lab or in one of the computer labs provided for undergraduates in computer-intensive courses. New GTAs sometimes assist a faculty member with grading papers for courses they are teaching. Of course such assignments will be made based on an assessment of what you would be best at. New GTAs also begin the Department's teaching certification and mentoring program. This program is designed to insure that GTAs are prepared and qualified to teach their own course. (This certification process must be completed by the end of the second year to be eligible for continued GTA support.) GTAs with teaching responsibilities are provided with faculty mentors and have an additional resource in the senior GTAs. Senior GTAs, selected from among the most experienced GTAs, have special duties including consultation with less experienced GTAs.
Each year the Graduate Program committee reviews the progress of all graduate students. Continuation of GTA support is dependent on making satisfactory progress.
In the M.S. program you can expect 2 years of GTA support (provided your progress is satisfactory). After admission to the Ph.D. program 4 additional years of GTA support can be anticipated for students making satisfactory progress.
Benefits and the Blacksburg Community
Blacksburg is located on a plateau between the Allegheny and Blue Ridge mountains, only a few miles from the New River (the second oldest river valley in the world). In addition to its scenic beauty and outdoor recreational opportunities, this location is close to numerous centers of traditional Appalachian culture and early American history. Blacksburg itself is nearly 200 years old. The Smithfield Plantation house (maintained as a historic site on the Virginia Tech campus) was built in 1772 when this part of Virginia was still the western frontier. Today's Blacksburg still reflects the heritage of its past, with numerous historic buildings and sites. The town offers a comfortable blend of quiet rural charm, modern technology and advanced learning. A regular schedule of cultural events and activities is available throughout the year, including concerts of all kinds from classical to bluegrass, indoor and outdoor activities, not to mention sporting events. For more about modern-day Blacksburg, you may enjoy exploring the web site for the Blacksburg Electronic Village.
There is a limited amount of on-campus housing designated for graduate students, but most live off-campus. The cost of living is about 5% below the national average, comparing favorably with other major university towns in the region. (Richmond and Raleigh are 4% and 10% above the national average, respectively.) Students may participate in optional group health insurance coverage, with a choice of several levels of coverage. For more information and rates contact Trigon Blue Cross Blue Shield of Virginia, (540) 953-3405.
Department and University Resources
The Mathematics Department maintains its own computing system, dedicated to the research and instructional needs of the Department. At its heart is a multiprocessor UNIX system supporting a variety of workstations and personal computers. Interconnected to this is a campus-wide network joining numerous Macintosh machines, which are the platform for the University's advances in technology-based instruction. Of course all of this is joined to the larger campus network which includes the large scale systems managed by the Computing Center. The Department maintains site licenses for numerous software tools that are vital to its mission (e.g. Mathematica, MATLAB, Microsoft Office, TeX, ... ). All graduate students have accounts on the UNIX system, which they can access through numerous machines located in offices and the graduate student computer lab(s). This provides (high speed) internet access, e-mail (which is the primary means of departmental news and communication), use of advanced software, laser printing and scanning services. The Department has a professional staff dedicated to maintaining and upgrading its computing system, and to providing training in its use.
The Virginia Tech Computing Center operates the largest collection of computing and communication hardware and services in Virginia higher education. In addition the Computing Center provides coordination and leadership for the expanding network of computing systems which are distributed among individual departments, research laboratories and instructional facilities. Communications Network Services maintains and enhances the campus' extensive communications system. This includes a backbone of 1,000 miles of optical fiber linking every office, lab and dorm room to computing, audio and video data.
The University has made a major commitment to provide and encourage the use of state of the art information technology in instruction. In particular all freshman calculus classes are now taught using powerful mathematical software as a fundamental tool. This is significant for mathematics graduate students, since many of them are involved in the teaching of these technology-intensive calculus courses.
The University Library provides an impressive array of resources in support of its research and graduate programs. The collection includes 3.1 million bound volumes, extensive collections of videos, microfilms, and more than 45,000 journal and periodical subscriptions. Beyond this is access to numerous major databases as well as a wealth of internet based resources. The library's electronic holdings and services are growing dramatically, including such services as "electronic reserve" which allows documents placed on reserve in electronic format to be accessed over network connections from anywhere on campus, or elsewhere in the world. The University Library system is a member of the Association of Research Libraries, an organization of the 120 largest research libraries in North America.
About the Mathematics Faculty
A faculty active in current research is essential for graduate education. Even if your own goals do not include research, only a faculty intimately involved with the latest developments and trends is in a position to prepare you for the future with an up-to-date graduate education. The Mathematics Department's large faculty includes over 40 who carry a significant responsibility for research. The funding through research grants obtained by these faculty averages well over $1 million per year. Among our faculty are the editors of 20 scholarly journals, and committee members for several of the major professional associations of mathematicians (AMS, SIAM, MAA). Through their interactions with scholars at other research centers, our faculty invite to campus some of the most prominent contemporary mathematicians. The Mathematics Colloquium provides a weekly lecture on current topics by these visitors and our own faculty. In addition there is usually an assortment of seminars available on particular topics of interest to faculty and graduate students (Current schedules can be viewed by anyone with internet access). Other departments on campus have similar programs. One of the great opportunities that only a major research university can provide is the chance to hear the world's leading scholars, in any of a broad range of disciplines, on a weekly basis.
The research interests of the mathematics faculty are too many to list in their entirety. Brief sketches of many prominent research faculty are provided below. The Department has identified 4 "focus" areas that represent the particularly strong concentrations of research expertise among our faculty. In addition to the Mathematics faculty listed for each of the focus areas below, there are numerous faculty from other departments that are closely associated with these research concentrations.
There are two specially funded centers for specialized and/or interdisciplinary research affiliated with the Mathematics Department. These centers may have special facilities and resources that you may benefit from if you work with the center's faculty.
The Interdisciplinary Center for Applied Mathematics (ICAM) was formed in August 1987 to promote and facilitate interdisciplinary research and education in applied mathematics at Virginia Tech. A major goal of ICAM is the enhancement of the historical links among mathematics, engineering and the sciences. ICAM assists the development of coherent graduate educational programs by providing graduate students with research experience and exposure to world renowned mathematicians, engineers and scientists through its visitor programs. The ICAM facility provides a single location on campus where students from mathematics, engineering and the sciences can spend a significant portion of their time working on a daily basis with students from other disciplines. The Center also promotes multidisciplinary research projects among faculty and students at Virginia Tech and provides high quality computational facilities for interdisciplinary research projects. The present scientific and technological environment offers unprecedented challenges and opportunities for inter disciplinary research. For more information contact Prof. Terry Herdman, Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061-0531.
The Center for Transport Theory and Mathematical Physics (CTTMP) is devoted to research which overlaps the basic areas of mathematics, physics and engineering. Current research areas include transport theory, plasma and gas dynamics, wave propagation, statistical mechanics, quantum mechanics, dynamical systems and chaos theory. The Center provides an active seminar and colloquium program, featuring visiting scholars of international stature. The Center coordinates the Mathematical Physics doctoral program, a joint program of the Mathematics and Physics Departments, which is available to students with undergraduate degrees in either mathematics or physics. Students are enrolled for the Ph.D. in one of the participating departments, and follow a course of study overlapping the two departments. Research is carried out under the direction of one of the participating CTTMP faculty members. For more information contact Prof. George Hagedorn, Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123.