# Mathematics

[ Graduate Program] [ Professors] [ Courses]

OFFICE: 7018 Applied Physics and Mathematics Building, Muir College

http://math.ucsd.edu

## The Undergraduate Program

The mathematics department offers a wide range of courses in pure and applied mathematics for its majors and for students in other disciplines. The department offers six majors leading to the B.A. degree: mathematics, applied mathematics, mathematics–applied science, mathematics–computer science, joint major in mathematics and economics, and mathematics–secondary education, and one leading to the B.S. Degree: mathematics–scientific computation. In addition, students can minor in mathematics. The department also has an Honors Program for exceptional students in any of the seven majors. See the sections on major programs and the other areas mentioned above as well as the course descriptions at the end of this section for more specific information about program requirements and the courses that are offered by the department. You may visit our Web site, http://math.ucsd.edu for more information including course Web pages, career advising, and research interests of our faculty.

### First-Year Courses

Entering students must take the Mathematics Placement Exam (MPE) prior to orientation unless they have an appropriate score on an AP calculus exam, a score of 650 or higher on the SAT II Math Level 2C Exam, an appropriate score on the International Baccalaureate Higher Level Mathematics Exam, credit by means of a foreign exam (e.g. GCE), or transferable college credit in calculus. The purpose of the MPE is to recommend placement for entering students in Math. 3C, 4C, 10A, or 20A. For more information about the MPE (test dates, test description, sample exams, online practice tests), see the Mathematics Testing and Placement Web site, http://mathtesting.ucsd.edu.

Prerequisites for Math. 4C, 10A-B-C and 20A-B-C-D-E-F are enforced through TritonLink. Students need to ensure that test scores and transferable college credit are submitted to the Registrar prior to enrollment through WebReg.

Math. 3C is the department’s preparatory course for the Math. 10 sequence, providing a review of algebraic skills, facility in graphing, and working with exponential and logarithmic functions.

Math. 4C is the department’s preparatory course for the Math. 20 sequence, providing a brief review of college algebra followed by an introduction to trigonometry and a more advanced treatment of graphing and functions.

Math. 10A-B-C is one of two calculus sequences. The students in this sequence have completed a minimum of two years of high school mathematics. This sequence is intended for majors in liberal arts and the social and life sciences. It fulfills the mathematics requirements of Revelle College and the option of the general-education requirements of Muir College. Completion of two quarters fulfills the requirement of Marshall College and the option of Warren College and Eleanor Roosevelt College.

The other first-year calculus sequence, Math. 20A-B-C, is taken mainly by
students who have completed four years of high school mathematics or have taken
a college level precalculus course such as Math. 4C. This sequence fulfills all
college level requirements met by Math. 10A-B-C and is required of many majors,
including chemistry and biochemistry, bioengineering, cognitive science, economics,
mathematics, molecular biology, psychology, MAE, CSE, ECE, and physics. Students
with adequate backgrounds in mathematics are strongly encouraged to take Math.
20 since it provides the foundation for Math. 20D-E-F which is required for some
science and engineering majors. **Note:** As of summer 2003, Math. 21C and
21D have been renumbered to Math. 20C and 20D.

Certain transfers between the Math. 10 and Math. 20 sequences are possible, but such transfers should be carefully discussed with an advisor. Able students who begin the Math. 10 sequence and who wish to transfer to the Math. 20 sequence, may follow one of three paths:

- Follow Math. 10A with Math. 20A, with two units of credit given for Math. 20A. This option is not available if the student has credit for Math. 10B or Math. 10C.
- Follow Math. 10B with Math. 20B, receiving two units of credit for Math. 20B.
- Follow Math. 10C with Math. 20B, receiving two units of credit for Math. 20B and two units of credit for Math. 20C.

Credit will not be given for courses taken simultaneously from the Math. 10 and the Math. 20 sequence.

### Major Programs

The department offers six different majors leading to the Bachelor of Arts
degree: (1) mathematics, (2) applied mathematics, (3) mathematics–applied
science, (4) mathematics–computer science, (5) joint major in mathematics
and economics, (6) mathematics–secondary education; and two leading to a
B.S. Degree: (1) mathematics–scientific computation, (2) mathematics–probability
and statistics. The specific emphases and course requirements for these majors
are described in the following sections. All majors must obtain a minimum 2.0
grade-point average in the upper-division courses used to satisfy the major requirements.
Further, the student must receive a grade of C– or better in any course
to be counted toward fulfillment of the major requirements. Any mathematics course
numbered 100–194 may be used
as an upper-division elective. (**Note:** 195, 196, 197, 198, 199, and 199H
cannot be used towards any mathematics major.) All courses used to fulfill the
major must be taken for a letter grade. No more than three upper-division courses
taken externally from UCSD can be counted towards any major. Special exceptions
may be considered via petition.

It is strongly recommended that all mathematics majors review their programs at least annually with a departmental advisor, and that they consult with the Advising Office in AP&M 6016 before making any changes to their programs. Current course offering information for the entire academic year is maintained on the department’s Web page at http://www.math.ucsd.edu. Special announcements are also e-mailed to all majors.

Students who plan to go on to graduate school in mathematics should be advised that only the best and most motivated students are admitted. Many graduate schools expect that students will have completed a full sequence of abstract algebra (Math. 100A-B-C) as well as a full sequence of analysis (Math. 140A-B-C). The advanced Graduate Record Exam (GRE) often has questions that pertain to material covered in the last quarter of analysis or algebra. In addition, it is advisable that students consider Summer Research Experiences for Undergraduates. This is a program funded by the National Science Foundation to introduce students to math research while they are still undergraduates. In their senior year or earlier, students should consider taking some graduate courses so that they are exposed to material taught at a higher level. In their junior year, students should begin to think of obtaining letters of recommendation from professors who are familiar with their abilities.

Math. 20D, E, and F do not need to be taken in order. Math majors are strongly advised to take 20F as early as possible after successfully completing 20C.

### Education Abroad

Students may be able to participate in the UC Education Abroad Program (EAP) and UCSD’s Opportunities Abroad Program (OAP) while still making progress towards the major. Students interested in this option should contact the Programs Abroad Office in the International Center and discuss their plans with the mathematics advising officer before going abroad.

The department must approve courses taken abroad. Information on EAP/OAP can be found in the Education Abroad Program section of the UCSD General Catalog and the Web site http://pao.ucsd.edu.

### Major in Mathematics

The upper-division curriculum provides programs for mathematics majors as well as courses for students who will use mathematics as a tool in the biological, physical and behavioral sciences, and the humanities.

#### Required Courses:

##### Lower-Division

- Calculus: Math. 20A-B-C-D-E-F

##### Upper-Division

- Mathematical Reasoning: Math. 109
- One of the following sequences:
- Foundations of Analysis: Math. 140A-B
- Advanced Calculus: Math. 142A-B and Math. 120A (Math. 140A-B-C recommended for graduate school)
- One of the following sequences:
- Modern Algebra: Math. 100A-B
- Modern Applied Algebra: Math. 103A-B and Math. 102 (Math. 100A-B-C recommended for graduate school)
- Upper-division electives to complete thirteen four-unit courses, chosen from any mathematics course numbered between 100 and 194 (including those taken from the requirements listed above.)

As with all departmental requirements, more advanced courses on the same material may be substituted with written approval from the departmental advisor.

To be prepared for a strong major curriculum, students should complete the last three quarters of the 20 sequence and Math. 109 before the end of their sophomore year. Either Math. 140A-B or 100A-B should be taken during the junior year.

### Major in Applied Mathematics

A major in applied mathematics is also offered. The program is intended for students planning to work on the interface between mathematics and other fields.

#### Required Courses:

##### Lower-Division

- Calculus: Math. 20A-B-C-D-E-F
- Programming (one of the following)
- CSE 8AL-B (Intro to Computer Sci: Java)
- CSE 11 (Intro to Computer Sci: Java, Accelerated Pace)
- MAE 9 (C/C++ Programming)
- MAE 10 (FORTRAN for Engineers)

##### Upper-Division

- Mathematical Reasoning: Math. 109
- Linear Algebra: Math. 102 or Math. 170A
- Advanced Calculus: Math. 140A-B or Math. 142A-B
- One of the following sequences:
- Math. 180A-B-C-181A
- Math. 180A-181A and any two from Math. 181B-C-D-E
- (Math. 183 or Math. 180A-181A) and any three from Math. 170A-B-C-172-173\
- One additional sequence which may be chosen from the list (#6) above or the following list: Math. 110-120A-130A, 120A-B, 130A-132A, 155A-B, 171A-B, 193A-B.
- Upper-division electives to complete at least thirteen four-unit courses, chosen from any mathematics course numbered between 100 and 194 (including those taken from the requirements listed above) except:
- Up to twelve units may be taken from outside the department in an approved applied mathematical area. A petition specifying the courses to be used must be approved by an applied mathematics advisor. No such units may also be used for a minor or program of concentration.
- MAE 107, Econ. 120A-B-C, cannot be counted toward the thirteen required courses.

To be prepared for a strong major curriculum, students should complete the last three quarters of the 20 sequence and Math. 109 before the end of their sophomore year.

### Major in Mathematics–Scientific Computation

This major is designed for students with a substantial interest in scientific computation. The program is a specialized applied mathematics program with a concentration in computer solutions of scientific problems.

**Required Courses: **

**Lower-Division **

- Calculus: Math. 20A-B-C-D-E-F
- Computer Programming: MAE 9 or MAE 10 and CSE 8AL-B or CSE 11
- Basic Computation: Math. 15A (or CSE 20) and Math. 15B (or CSE 21) and CSE 12

**Upper-Division **

- Mathematical Reasoning: Math. 109
- Linear Algebra: Math. 102
- Probability and Statistics: Math. 183 or 180A-181A (Note: No credit for Math. 183 if Math. 180A or 181A taken prior or concurrently.)
- Analysis: Math. 140A-B or 142A-B (Note: Students planning to go to grad school should take 140A-B)
- Numerical Analysis: Math. 170A-B-C or Math. 170A-B/Math. 172
- Optimization: Math. 171A-B
- Scientific Computing: Math. 173
- Additional elective upper-division courses to total 15 chosen from the following: Math. 107A-B, 110, 120A-B, 130A-B, 131, 132A-B, 152, 155A-B, 170C, or 172.

At least 15 upper-division mathematics courses are required for the major, except:

- Up to 3 upper-division courses may be taken outside the department in an approved scientific computation area in the sciences or engineering. A petition specifying the courses to be used must be approved by a mathematics-scientific computation advisor. No such units may also be used by a minor or program of concentration.
- MAE 107, Econ. 120A-B-C, Math. 195, 196, 197, 199, and 199H cannot be counted toward the 13 four-unit upper-division courses.

### Major in Mathematics–Probability and Statistics

#### Effective Winter 2007

This major is designed for students with a substantial interest in probability theory and statistics. It is useful preparation for many fields of employment as well as graduate school.

**Required Courses: **

**Lower-Division **

- Calculus:
Math. 20A-B-C-D-E-F

- Programming
(one of the following)

a. CSE 8A-8AL-8B (Java)

b. CSE 11 (Java)

c. MAE 9 (C/C++)

d. MAE 10 (Fortran)

**Upper-Division **

- Mathematical Reasoning: Math. 109
- Linear Algebra: Math. 102 or Math. 170A
- Analysis/Advanced Calculus: Math. 140A-B or Math. 142A-B
- Probability: Math. 180A-B-C
- Mathematical Statistics: Math. 181A-B
- One of the following: Math. 181C, 181E, 193A, 193B, 194
- Computational Statistics: Math. 185
- Upper-division electives to complete 15 upper-division courses from the following list: Math. 100A-B, 103A-B, 110, 120A-B, 130A, 131, 132, 152, 155A-B, 161, 170A-B-C, 171A-B, 172, 173, 176, 181C, 181E, 184A, 187, 188, 193A-B, 194.

At least 15 four-unit upper-division mathematics courses are required, except:

- Two upper-division electives may be outside the department in an approved applied mathematical area. A petition approved by a math advisor is required. No such units may also be used for a minor or program of concentration.
- MAE 107, Econ. 120A-B-C, Math. 195-199 cannot be counted toward the upper-division requirements.

To be prepared for a strong major curriculum, students should complete the last three quarters of the 20 sequence and Math. 109 before the end of their sophomore year.

### Major in Mathematics–Applied Science

This major is designed for students with a substantial interest in mathematics and its applications to a particular field such as physics, biology, chemistry, biochemistry, cognitive science, computer science, economics, management science, or engineering.

#### Required Courses:

##### Lower-Division

- Calculus: Math. 20A-B-C-D-E-F
- Programming (one of the following is recommended):
- CSE 8AL-B (Intro to Computer Sci: Java)
- CSE 11 (Intro to Computer Sci: Java, Accelerated Pace)
- MAE 9 (C/C++ Programming)
- MAE 10 (FORTRAN for Engineers)

##### Upper-Division Mathematics Requirements:

- Mathematical Reasoning: Math. 109
- Linear Algebra: Math. 102 or Math. 170A
- Any two-quarter, upper-division math sequence
- Upper-division electives to complete at least seven four-unit courses, chosen from any mathematics course numbered between 100 and 194 (including those taken from the requirements listed above.)

##### Upper-Division Applied Science Requirements:

- Seven upper-division courses selected from one or two other departments (these cannot be from mathematics). At least three of these seven upper-division courses must require at least Math. 20C as a prerequisite.

Students must submit an individual plan for approval in advance by a mathematics department advisor, and all subsequent changes in the plan must be approved by a mathematics department advisor.

### Major in Mathematics–Computer Science

The program provides for a major in computer science within the Department of Mathematics. Graduates of this program will be mathematically oriented computer scientists who have specialized in the mathematical aspects and foundations of computer science or in the computer applications of mathematics.

As of fall 2000, a mathematics-computer science major is not allowed to also minor in computer science in the Computer Science and Engineering department.

The detailed curriculum is given in the list below:

**Mathematics–Computer Science Pre-Major **

In October 2001, the Academic Senate approved a minimum GPA requirement of 2.5 in the lower-division mathematics courses required for the mathematics–computer science major. The 2.5 minimum GPA in the lower-division math courses reflects minimal preparation for the upper-division courses required for the major.

Therefore, students entering UCSD as first- year students for the fall 2002 quarter and later and students entering as transfer students for the fall 2003 quarter and later will be held to this requirement. Applications from students entering UCSD on or after the effective dates above will be held until all lower-division math courses for the major are completed and the minimum GPA in those courses of 2.5 can be verified. Students meeting the 2.5 minimum GPA requirement will be accepted into the mathematics–computer science major.

**Required Courses:**

##### Lower-Division:

- Calculus: Math. 20A-B-C-D-E-F
- Intro to Computer Science—CSE 8AL-B Introduction to Computer Science: Java, or CSE 11 Introduction to Computer Science: Java (Accelerated)
- Basic Data Structures and Object-oriented Programming: CSE 12
- Computer Organization and Systems Programming: CSE 30 (Note: CSE 30 requires CSE 20 or Math. 15A as a prerequisite.)

##### Upper-Division:

- Mathematical Reasoning: Math. 109
- Modern Applied Algebra: Math. 103A-B or Modern Algebra: Math. 100A-B
- Theory of Computability: Math. 166 (or CSE 105)
- Intro to Probability: Math. 180A or 183
- Mathematical Foundations of Computer Science: Math. 184A
- Computer Implementations of Data Structures: Math. 176 (or CSE 100)
- Design & Analysis of Algorithms: Math. 188 (or CSE 101)
- Eight units from: Math. 170A-B-C, 172, 173, 174 (Note duplication of credit between Math. 174 and 170A-B-C)
- Eight units from: Math. 107A-B, 152, 154, 155A-B, 160A-B, 168A-B, 187, CSE 120-121, 130, 131A-B, 140-140L, 141-141L, 167
- Eight additional units from: any course in list #12 or #13 above or Math. 102, 110, 111A-B, 120A-B, 130A-B, 131, 132A-B, 140A-B, 142A-B, 150A-B, 181A-B-C

In order to graduate by the end of their senior year, students should complete Math. 103A-B by the end of their junior year.

### Joint Major in Mathematics and Economics

Majors in mathematics and the natural sciences often feel the need for a more formal introduction to issues involving business applications of science and mathematics. Extending their studies into economics provides this application and can provide a bridge to successful careers or advanced study. Majors in economics generally recognize the importance of mathematics to their discipline. Undergraduate students who plan to pursue doctoral study in economics or business need the more advanced mathematics training prescribed in this major.

This major is considered to be excellent preparation for Ph.D. study in economics and business administration, as well as for graduate studies for professional management degrees, including the MBA. The major provides a formal framework making it easier to combine study in the two fields.

Course requirements of the Joint Major in Mathematics and Economics consist principally of the required courses of the pure mathematics major and the economics/management science major.

**Required Courses:**

##### Lower-Division

- Calculus: Math. 20A-B-C-D-F
- Intro to Economics: Econ. 1 and 3

##### Upper-Division

Fifteen upper-division courses in mathematics and economics, with a minimum of seven courses in each department, chosen from the courses listed below (prerequisites are strictly enforced):

- Mathematical Reasoning: Math. 109
- One of the following: Applied Linear Algebra: Math. 102 Numerical Linear Algebra: Math. 170A Modern Algebra: Math. 100A-B
- One of the following: Foundations of Analysis: Math. 140A Advanced Calculus: Math. 142A
- One of the following: Ordinary Differential Equations: Math. 130A, Foundations of Analysis: Math. 140B Advanced Calculus: Math. 142B
- Microeconomics: Econ. 100A-B-C
- Econometrics: Econ. 120A-B-C or Math. 180A and Econ. 120B-C or Probability: Math. 180A, 181A and Econ. 120C
- One of the following:

- Macroeconomics: Econ. 110A-B
- Mathematical Programming: Numerical Optimization: Math. 171A-B

*or*two courses from the following:- Decisions Under Uncertainty: Econ. 171
- Introduction to Operations Research: Econ. 172A-B, (
**Note:**172A is a prerequisite for 172B-C)

Other courses which are strongly recommended are: Math. 130B, 131, 181B, 193A-B and 194 and Econ. 109, 113, 175, and 178.

### Major in Mathematics–Secondary Education

This major offers excellent preparation for teaching mathematics in secondary schools. Students interested in earning a California teaching credential from UCSD should contact the Teacher Education Program (TEP) for information regarding prerequisites and requirements. It is recommended you contact TEP as early as possible.

##### Lower-Division Requirements

- Calculus 20A-B-C-D-E-F

Recommended:

- One of the following: Introduction to Computer Science: Java: CSE 8AL-B, FORTRAN: MAE 10 C/C++ Programming: MAE 9

##### Upper-Division Requirements:

- Mathematical Reasoning: Math. 109
- Number Theory: Math 104A
- History of Mathematics: Math. 163
- Practicum in Learning: EDS 129A-B-C (can use EDS 136 and EDS 138 intead of EDS 129A)
- One of the following: Computer Algebra: Math. 107A, Computer Graphics: Math. 155A, Mathematical Computing: Math. 161, Numerical Linear Algebra: Math. 170A, Intro. to Cryptography: Math. 187
- One of the following: Intro. to Probability: Math. 180A, Statistical Methods: Math. 183
- One of the following: Differential Geometry: Math. 150A, Topics in Geometry: Math 151, Geometry for Secondary Teachers: Math. 153, Intro. to Topology: Math. 190
- One of the Following: Modern Algebra: Math. 100A, Applied Linear Algebra: Math. 102, Modern Applied Algebra: Math. 103A
- One of the following: Foundations of Analysis: Math. 140A, Advanced Calculus: Math. 142A
- Upper-division courses must total thirteen courses. Upper-division courses
must include at least one two-quarter sequence from the following list:
100A-B; 103A-B, 103A-102; 104A-B; 110-120A; 110-130A; 110-132A; 110-131; 120A-B; 130A-132A; 132A-B; 140A-B;142A-B; 150A-B; 155A-B; 160A-B; 170A-B; 170A-172; 170A-173; 170A-171A; 171A-B; 180A-B; 180A-181A; 193A-B.

### Minor in Mathematics

The minor in mathematics consists of seven or more courses. At least four of these courses must be upper-division courses taken from the UCSD Department of Mathematics. Acceptable lower-division courses are Math. 20D, 20E, and 20F.

Math. 195, 196, 197, 198, 199, and 199H are not acceptable courses for the mathematics minor. A grade of C– or better (or P if the Pass/No Pass option is used) is required for all courses used to satisfy the requirements for a minor. There is no restriction on the number of classes taken with the P/NP option. Upper-division courses cannot overlap between major and minor programs.

### Minor in Mathematic Education

The education studies mathematics education minor is intended for students
interested in understanding how people learn mathematics, including: students
considering K-12 teaching as a career; students interested in teaching at the
college level; and students who are interested in becoming better, more reflective
learners. **All majors are welcome**, but the Calculus 10 or 20 sequence is
a prerequisite for two of the upper-division courses required for the minor.
For more information contact the Department of Education Studies:

### Mathematics Honors Program

The Department of Mathematics offers an honors program for those students who have demonstrated excellence in the major. Successful completion of the honors program entitles the student to graduate with departmental honors (see Department Honors in the Academic Regulations section). Application to the program should be made the spring quarter before the student is at senior standing.

**Requirements for admission to the program are: **

- Junior standing
- An overall GPA of 3.0 or higher
- A GPA in the major of 3.5 or higher
- Completion of Math. 109 (Mathematical Reasoning) and at least one of Math. 100A, 103A, 140A, or 142A. (Completion of additional major courses is strongly recommended.)

**Completion of the honors program requires the following: **

- At least one quarter of the student colloquium, Math. 196 (Note: Math. 196 is only offered in the fall quarter.)
- The minimum 3.5 GPA in the major must be maintained
- An Honors Thesis. The research and writing of the thesis will be conducted over at least two quarters of the junior/senior years under the supervision of a faculty advisor. This research will be credited as eight to twelve units of Math. 199H. The completed thesis must be approved by the department’s Honors Committee, and presented orally at the Undergraduate Research Conference or another appropriate occasion.

The department’s Honors Committee will determine the level of honors to be awarded, based on the student’s GPA in the major and the quality of the honors work. Applications for the mathematics department’s Honors Program can be obtained at the mathematics department Undergraduate Affairs Office (AP&M 7018) or the Mathematics Advising Office (AP&M 6016). Completed applications can be returned to the Mathematics Advising Office.

### Duplication of Credit

In the circumstances listed below, a student will not receive full credit for a Department of Mathematics course. The notation “Math. 20A [2 if Math. 10A previously/0 if Math. 10A concurrently/0 if Math. 10B or 10C]” means that a student already having credit for Math. 10A will receive only two units of credit for Math. 20A, but will receive no units if he or she has credit for Math. 10B or 10C, and no credit will be awarded for Math. 20A if Math. 10A is being taken concurrently.

- Math. 4C [2 if 3C previously/0 if Math. 10A or Math. 20A previously or concurrently]
- Math. 10A [0 if Math. 20A previously or concurrently]
- Math. 10B [0 if Math. 20B previously or concurrently]
- Math. 10C [0 if Math. 20C previously or concurrently]
- Math. 15A [0 if CSE20 previously or concurrently]
- Math. 15B [0 if CSE21 previously or concurrently]
- Math. 20A [2 if Math. 10A previously/0 if Math. 10A concurrently/0 if Math. 10B or 10C]
- Math. 20B [2 if Math. 10B or 10C previously/0 if Math. 10B concurrently]
- Math. 20C [2 if Math. 10C previously/0 if Math. 10C concurrently]
- Math. 20D [0 if Math. 2DA previously]
- Math. 20E [0 if Math. 2F previously]
- Math. 20F [0 if Math. 2EA previously]
- Both Math. 100 and Math. 103 cannot be taken for credit
- Math. 140A-B and Math. 142A-B cannot both be taken for credit
- Math. 155A [0 if CSE 167]
- Math. 166 [0 if CSE105]
- Math. 174 [0 if 170A or B or C previously]
- Math. 176 [0 if CSE 100 previously or concurrently]
- Math. 180A [2 if Econ. 120A previously/ 0 if Econ. 120A concurrently]
- Math. 181A [2 if Econ. 120B/0 if Econ. 120B concurrently]
- Math. 183 [0 if Econ. 120A or ECE 109 or Math. 180A or Math. 181A or Math. 186 has been taken previously or concurrently. Full credit for Math. 183 will be given if taken previously to Math. 180A or Math. 181A.]

### Advisors

advisors change yearly. Contact the undergraduate office at (858) 534-3590 for current information.

## 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 15. 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:

- No more than eight units of upper-division mathematics courses.
- No more than twelve units of graduate courses in a related field outside the department (approved by the Department of Mathematics).
- No more than four units of Math. 295 (Special Topics) or Math. 500 (Apprentice Teaching).
- 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.

### Ph.D. in Mathematics with Specialization in Computational Science

As of fall 2007, the UC San Diego campus is offering a new comprehensive Ph.D. specialization in computational science that will be available to doctoral candidates in participating science, mathematics, and engineering departments at UC San Diego.

This Ph.D. specialization is designed to allow a student to obtain standard basic training in their chosen field of science, mathematics, or engineering with a specialization in computational science integrated into their graduate studies. Prospective students must apply and be admitted into the Ph.D. program in mathematics described in the previous section. (See Mathematics Department for more information.)

### 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. (Combinatorial 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. (Scientific Computation)

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:

- 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];
- A maximum of eight units can be approved upper-division courses in mathematics; and
- A maximum of eight units can be approved upper-division courses in other departments.
- A maximum of four units of Math. 500 (Apprentice Teaching).
- 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:

- Math. 281A-B-C. (Mathematical Statistics)
- Math. 282A-B. Applied Statistics)

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

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

- For a theoretical emphasis, Math. 280A-B-C (Probability Theory) is required.
- For an applied orientation, Math. 270A-B-C (Numerical Mathematics) is recommended.
- 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)

- 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.
- 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.
- 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.)
- 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 a 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 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 late 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 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.