# Graduate Programs

The Mathematics Department is located in McBryde Hall, pictured above (photo credit: Mia Shu). There are roughly 70 graduate students actively engaged in the mathematics graduate programs. 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.

As a new student you will be given an initial advisor appropriate for the research interests you have expressed. However, those interests will envolve, and you are free to make your own arrangements with any other faculty member to be your advisor.

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. Many go on to become Ph.D. students. With good progress, all M.S. students are invited to do so in our department.

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.

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, intended to communicate the nature of the programs to potential students. For complete descriptions see the Policies document linked below.

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 that 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 ensure this, the student must pass Ph.D. preliminary examinations in two of the following four areas: Algebra, Analysis, Partial Differential Equations, and Computational Mathematics. 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.

The normal course load for a first or second year student is 3 courses per semester, with time reserved for research and/or teaching.

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).

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. As a GTA you are an employee of the Mathematics Department with assigned duties related to teaching undergraduate courses. 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.

Some graduate students may find academic year support 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.

Our department welcomes students from across the world, and our graduate programs currently include students from nearly every continent. The accommodate international students and meet university guidelines, a few additional requirements apply. 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. 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.

To be admitted to the graduate program a formal application must be submitted to the Graduate School:

https://graduateschool.vt.edu/admissions/how-to-apply.html

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).

Please contact the Graduate Program Coordinator with further questions about our graduate programs: