Mathematics

[ undergraduate program | courses | faculty ]

OFFICE: 7018 Applied Physics and Mathematics Building, Muir College
http://math.ucsd.edu

The Graduate Program

The Department of Mathematics offers graduate programs leading to the M.A. (pure or applied mathematics), M.S. (statistics), and Ph.D. degrees.

The application deadline for fall admission is January 7 for Ph.D. candidates, and February 7 for M.A./M.S. candidates. Candidates should have a bachelor’s or master’s degree in mathematics or a related field from an accredited institution of higher education or the equivalent. A minimum scholastic average of B or better is required for course work completed in upper-division or prior graduate study. In addition, the department requires all applicants to submit scores no older than twelve months from both the GRE General Test and Advanced Subject Test in Mathematics. Completed files are judged on the candidate’s mathematical background, qualifications, and goals.

Departmental support is typically in the form of teaching assistantships, research assistantships, and fellowships. These are currently only awarded to students in the Ph.D. program.

General Requirements

All student course programs must be approved by a faculty advisor prior to registering for classes each quarter, as well as any changes throughout the quarter.

Full-time students are required to register for a minimum of twelve (12) units every quarter, eight (8) of which must be graduate-level mathematics courses taken for a letter grade only. The remaining four (4) units can be approved upper-division or graduate-level courses in mathematics-related subjects (Math. 500 may not be used to satisfy any part of this requirement). After advancing to candidacy, Ph.D. candidates may take all course work on a Satisfactory/Unsatisfactory basis. Typically, students should not enroll in Math. 299 until they have satisfactorily passed both qualifying examinations (see “Ph.D. in Mathematics”) or obtained approval of their faculty advisor.

Master of Arts in Pure Mathematics

[Offered only under the Comprehensive Examination Plan.] The degree may be terminal or obtained on the way to the Ph.D. A total of forty-eight units of credit is required. Twenty-four of these units must be graduate-level mathematics courses approved in consultation with a faculty advisor.

In the selection of course work to fulfill the remaining twenty-four units, the following restrictions must be followed:

  1. No more than eight units of upper-division mathematics courses.
  2. No more than twelve units of graduate courses in a related field outside the department (approved by the Department of Mathematics).
  3. No more than four units of Math. 295 (Special Topics) or Math. 500 (Apprentice Teaching).
  4. No units of Math. 299 (Reading and Research) may be used in satisfying the requirements for the master’s degree.

Comprehensive Examinations

Seven written departmental examinations are offered in three areas (refer to “Ph.D. In Mathematics,” Areas 1, 2, and 3, for list of exams). A student must complete two examinations, one from Area 1 and one from Area 2, both with an M.A. pass or better.

Foreign Language Requirement

A reading knowledge of one foreign language (French, German, or Russian) is required. In exceptional cases other languages may be substituted. Testing is administered by faculty in the department who select published mathematical material in one of these languages for a student to translate.

Time Limits

Full-time students are permitted seven quarters in which to complete all degree requirements. While there are no written time limits for part-time students, the department has the right to intervene and set individual deadlines if it becomes necessary.

Master of Arts in Applied Mathematics

[Offered only under the Comprehensive Examination Plan] The degree may be terminal or obtained on the way to the Ph.D. Out of the total forty-eight units of required credit, two applied mathematics sequences comprising twenty-four units must be chosen from the following list (not every course is offered each year):

202A-B-C. (Applied Algebra)

210A-B-C. (Mathematical Methods in Physics and Engineering)

261A-B-C. (Probabilistic Combinatorics and Algorithms)

264A-B-C. (Combinatorics)

270A-B-C. (Numerical Mathematics)

271A-B-C. (Numerical Optimization)

272A-B-C. (Numerical Partial Differential Equations)

273A-B-C. (Advanced Techniques in Computational Math)

In certain cases, a petition may be approved to substitute one of these requirements from the following list of sequences:

220A-B-C. (Complex Analysis)

231A-B-C. (Partial Differential Equations)

240A-B-C. (Real Analysis)

280A-B-C. (Probability Theory)

281A-B-C. (Mathematical Statistics)

282A-B. (Applied Statistics)

In choosing course work to fulfill the remaining twenty-four units, the following restrictions must be followed:

  1. At least eight units must be approved graduate courses in mathematics or other departments [a one-year sequence in a related area outside the department such as computer science, engineering, physics, or economics is strongly recommended];
  2. A maximum of eight units can be approved upper-division courses in mathematics; and
  3. A maximum of eight units can be approved upper-division courses in other departments.
  4. A maximum of four units of Math. 500 (Apprentice Teaching).
  5. NO UNITS of Math. 295 (Special Topics) or Math. 299 (Reading and Research) may be used.

Students are strongly encouraged to consult with a faculty advisor in their first quarter to prepare their course of study.

Comprehensive Examinations

Two written comprehensive examinations must be passed at the master’s level in any of the required applied mathematics sequences listed above. The instructors of each course should be contacted for exam details.

Foreign Language Requirement

There is no foreign language requirement for the M.A. in applied mathematics.

Time Limits

Full-time M.A. students are permitted seven quarters in which to complete all requirements. While there are no written time limits for part-time students, the department has the right to intervene and set individual deadlines if it becomes necessary.

Master of Science in Statistics

[Offered only under the Comprehensive Examination Plan.] The M.S. in statistics is designed to provide recipients with a strong mathematical background and experience in statistical computing with various applications. Out of the forty-eight units of credit needed, required core courses comprise twenty-eight units, including

and any two topics comprising eight units chosen freely from Math. 287A-B-C and 289A-B (see course descriptions for topics).

The following guidelines should be followed when selecting courses to complete the remaining twenty units:

  1. For a theoretical emphasis, Math. 280A-B-C (Probability Theory) is required.
  2. For an applied orientation, Math. 270A-B-C (Numerical Mathematics) is recommended.
  3. A maximum of eight units of approved upper-division applied mathematics courses (see faculty advisor) and Math. 500 (Apprentice Teaching).

Upon the approval of the faculty advisor, the rule above, limiting graduate units from other departments to eight may be relaxed in making up these twenty non-core units.

Comprehensive Examinations

Two written comprehensive examinations must be passed at the master’s level in related course work (approved by a faculty advisor). Instructors of the relevant courses should be consulted for exam dates as they vary on a yearly basis.

Foreign Language Requirement

There is no foreign language requirement for the M.S. in statistics.

Time Limits

Full-time M.S. students are permitted seven quarters in which to complete all requirements. While there are no written time limits for part-time students, the department has the right to intervene and set individual deadlines if it becomes necessary.

Ph.D. in Mathematics

Written Qualifying Examinations

The department offers written qualifying examinations in seven subjects. These are grouped into three areas as follows:

Area #1

Complex Analysis (Math. 220A-B-C) Real Analysis (Math. 240A-B-C)

Area #2

Algebra (Math. 200A-B-C) Applied Algebra (Math. 202A-B-C) Topology (Math. 290A-B-C)

Area #3

Numerical Analysis (Math. 270A-B-C) Statistics (Math. 281A-B-C)

  1. Three qualifying exams must be passed. At least one must be passed at the Ph.D. level, and a second must be passed at either the Ph.D. or Provisional Ph.D. Level The third exam must be passed at least at the master’s level.
  2. Of the three qualifying exams, there must be at least one from each of Areas #1 and #2. Algebra and Applied Algebra do not count as distinct exams in Area #2.
  3. Students must pass a least two exams from distinct areas with a minimum grade of Provisional Ph.D. (For example, a Ph.D. pass in Real Analysis, Provisional Ph.D. Pass in Complex Analysis, M.A. pass in Algebra would NOT satisfy this requirement, but a Ph.D. Pass in Real Analysis, M.A. pass in Complex Analysis, Provisional Ph.D. Pass in Algebra would, as would a Ph.D. Pass in Numerical Analysis, Provisional Ph.D. Pass in Applied Algebra, and M.A. pass in Real Analysis.)
  4. All exams must be passed by the September exam session prior to the beginning of the third year of graduate studies. (Thus, there would be no limit on the number of attempts, encouraging new students to take exams when they arrive, without penalty.)

Department policy stipulates that at least one of the exams must be completed with a Provisional Ph.D. Pass or better by September following the end of the first year. Anyone unable to comply with this schedule will lose their funding as a Ph.D. student. They will be terminated from the doctoral program and transferred to one of our master’s programs.

Any master’s student can submit for consideration a written request to transfer into the Ph.D. Program when the qualifying exam requirements for the Ph.D. Program have been met and a dissertation advisor is found. Approval by the Qualifying Exam and Appeals Committee (QEAC) is not automatic, however.

Exams are typically offered twice a year, one scheduled in the spring quarter and again in early September (prior to the start of fall quarter). Copies of past exams are made available for purchase in the Graduate Office.

In choosing a program with an eye to future employment, students should seek the assistance of a faculty advisor and take a broad selection of courses including applied mathematics, such as those in Area #3.

Foreign Language Requirement

A reading knowledge of one foreign language (French, German, or Russian) is required prior to advancing to candidacy. In exceptional cases other languages may be substituted. Testing is administered within the department by faculty who select published mathematical material in one of these languages for a student to translate.

Advancement to Candidacy

It is expected that by the end of the third year (nine quarters), students should have a field of research chosen and a faculty member willing to direct and guide them. A student will advance to candidacy after successfully passing the oral qualifying examination, which deals primarily with the area of research proposed but may include the project itself. This examination is conducted by the student’s appointed doctoral committee. Based on their recommendation, a student advances to candidacy and is awarded the C.Phil. degree.

Dissertation and Final Defense

Submission of a written dissertation and a final examination in which the thesis is publicly defended are the last steps before the Ph.D. degree is awarded. When the dissertation is substantially completed, copies must be provided to all committee members at least four weeks in advance of the proposed defense date. Two weeks before the scheduled final defense, a copy of the dissertation must be made available in the department for public inspection.

Time Limits

The normative time for the Ph.D. In mathematics is five years. Students must have a dissertation advisor by the end of nine quarters. Students must be advanced to candidacy by the end of eleven quarters. Total university support cannot exceed six years. Total registered time at UCSD cannot exceed seven years.

A student making normal progress must meet the time limits described below. Ph.D. students who fail to meet these time limits may lose their TA funding.

  1. Pass Qualifying Exams requirement by the fall quarter of the beginning of the third year.
  2. Find Thesis advisor by the end of nine quarters.
  3. Advance to Candidacy by the end of eleven quarters.
  4. Final Defense by the end of the fifth year.

Ph.D. in Mathematics with Specialization in Computational Science

The Ph.D. in mathematics, with a Specialization in Computational Science is designed to allow a student to obtain standard basic training in his or her chosen field of science, mathematics, or engineering with training in computational science integrated into those graduate studies. The Specialization in Computational Science recognizes the nation’s growing and continuing need for broadly trained advanced computational scientists in academic, industry, and government laboratories. Its graduates will be well-positioned to compete effectively for the best jobs in these areas.

Computational science refers to the use of computer simulation and visualization for basic scientific research, product development, and forecasting. It is an interdisciplinary field that combines mathematics (mathematical modeling, numerical analysis) and computer science (architecture, programming, networks, graphics) with one of the scientific or engineering disciplines.

The specialization draws upon the expertise of faculty from bioengineering, biological sciences, chemistry and biochemistry, computers and engineering, electrical and computer engineering, mathematics, mechanical and aerospace engineering, physics, Scripps Institution of Oceanography, structural engineering as well as research staff from the San Diego Supercomputer Center.

Admission

Prospective students must apply to the Ph.D. program of a participating home department, be admitted to that department and then be admitted to the specialization. The five participating academic departments that have a Specialization in Computational Science are chemistry and biochemistry, computer science and engineering, mathematics, mechanical and aerospace engineering, and physics. Requirements consist of those of the admitting home department (one of the five participating departments) as well as the proficiency, qualifying. and elective course requirements as outlined below. Requirements and policies relating to the home department can be found in the UC San Diego General Catalog under that department’s name. In the case of the mathematics department, the admission requirements for the mathematics doctoral program are those outlined above.

Specialization in Computational Science Policies

  1. The specialization requires that students complete all home department requirements for the Ph.D. along with satisfying the CSME proficiency, qualifying, and elective requirements. In the case of the mathematics department, the requirements and timelines for the normal mathematics Ph.D. program are as described above.
  2. CSME Proficiency (see below) must be satisfied by the end of the first year.
  3. The CSME qualifying exams must be passed by the end of the second year or, on petition, by end of the third year.
  4. The CSME qualifying exams can be attempted repeatedly but no more than once per quarter per subject.
  5. The regular qualifying exams in the home department and the CSME qualifying exams must all be passed before the student is permitted to take the candidacy (Senate) exam.
  6. Two CSME electives outside the home department must be taken.
  7. The two CSME electives can be taken at any time before defending the thesis.
  8. One of the CSME electives may be taken Pass/Fail; the other must be taken for a letter grade.
  9. No CSME field examination requirements will be imposed beyond those of the admitting home department.
  10. No CSME foreign language requirements will be imposed beyond those of the admitting home department.

Proficiency Requirements: All Ph.D. students participating in the CSME doctoral program must demonstrate advanced undergraduate level proficiency in numerical analysis and in computer algorithms and data structures. Proficiency may be demonstrated by taking UCSD’s courses in both subjects while enrolled in the graduate program (four units per course):

  1. Numerical Methods (Math. 174/274 or MAE 290A)
  2. Data Structures and Algorithms (CSE 100/101)

Alternatively, proficiency in the material contained in these courses may be satisfied by having previously taken these or equivalent courses at other institutions, or through other evidence of sufficient knowledge of this material. Demonstrating proficiency without taking these courses at UCSD is subject to approval by the CSME Executive Committee on an individual basis.

Qualifying Requirements: In addition to the home department qualifying exam requirements, Ph.D. students participating in the CSME doctoral program must pass the final exams in all three qualifying exam courses listed below. The three qualifying exam courses have been selected to provide a general broad set of tools in computational science. It is expected that most students will register for and take these courses (four units per course), but the CSME Qualifying Exam Committee may allow an exceptionally well-prepared student to take the final exams without taking the courses. Students must pass the qualifying examinations by the end of the second year or, on petition, by the end of the third year.

  1. Math. 275 or MAE 290B (Numerical PDEs)
  2. Phys. 244 or CSE 260 (Parallel Computing)
  3. Course to be selected from List A
LIST A: CSME Qualifying Exam Courses

Courses taken to satisfy the qualifying requirements cannot count toward the elective requirements.

  1. Phys. 243 (Stochastic Methods)
  2. Math. 270A, B, or C (Numerical Analysis; not permitted for math. students, who typically take Math. 270A-B-C as a normal mathematics qualifying course.)
  3. Math. 272A, B, or C (Advanced Numerical PDEs)
  4. MAE 223 (Computational Fluid Dynamics)
  5. MAE 232A or B (Computational Solid Mechanics)
  6. MAE 280A or B (Linear Systems Theory)
  7. (To be determined by Executive Committee)

Elective Requirements: To encourage Ph.D. students specializing in computational science to both broaden themselves in an area of science or engineering as well as to obtain more specialized training in specific areas of computational science, students will be required to take and pass two elective courses from the following approved List B (four units per course), both of which must be outside of their home department, the first of which must be taken for a letter grade, and the second of which may be taken pass/fail. The CSME Executive Committee may approve the use of courses not appearing on the following list on a case-by-case basis. Courses taken to satisfy the elective requirements cannot count toward the qualifying requirements.

LIST B: Elective Graduate Courses in Mathematics, Science, and Engineering
  1. Math. 270A-B-C (Numerical Analysis; not permitted for math. students, who typically take Math. 270A-B-C as a normal mathematics qualifying course.)
  2. Math. 271A-B-C (Optimization)
  3. Math. 272A-B-C (Advanced Numerical PDEs)
  4. Math. 273A-B-C (Computational Mathematics Project)
  5. Phys. 141/241 (Computational Physics I)
  6. Phys. 142/242 (Computational Physics II)
  7. Phys. 221 AB (Nonlinear dynamics)
  8. Chem. 215 (Modeling Biological Macromolecules)
  9. BGGN 260 (Neurodynamics)
  10. (To be determined by Executive Committee)

Thesis/Dissertation: Students participating in the Ph.D in mathematics with a Specialization in Computational Science must complete a dissertation that meets all requirements for the regular Ph.D. in the home department. In addition, it is expected that the Ph.D. dissertation will be interdisciplinary in nature and involve some aspect of computational science.

Final Examination: Students participating in the Ph.D. in mathematics with a Specialization in Computational Science must meet the regular final examination requirements of the home department.