Program Description

Mathematics is both a fundamental discipline and an essential tool for students of biology, chemistry, computer engineering, computer science, Earth sciences, economics, electrical engineering, information systems management, physics, and psychology. Researchers in all these areas are constantly developing new and fascinating ways of applying mathematics. A strong mathematics background is prerequisite to advanced study in the physical and biological sciences and is often helpful in studying the social sciences.

The UC Santa Cruz mathematics program offers a wide variety of undergraduate mathematics courses:

• Courses 2 and 3 do not require thorough preparation in mathematics at the high school level. However, students interested in studying mathematics are strongly encouraged to take algebra, geometry, and trigonometry before entering the university. Prospective freshmen are also encouraged to take the mathematics placement exam during their senior year of high school at a UCSC-scheduled exam. If they place into course 2 or 3, they should take those courses at UC Santa Cruz during the summer, so they can begin the calculus series when they enter in the fall.
  
  
• Lower-division courses with numbers in the range 11A-B through 30 (calculus, linear algebra, multivariable calculus, differential equations, and problem solving) prepare students for further study in mathematics, the physical and biological sciences, or quantitative areas of the social sciences. Science majors take a combination of these courses as part of their undergraduate studies.
  
  
• Upper-division courses, with numbers in the range 100–199, are intended for majors in mathematics and closely related disciplines. Some of these courses provide students with a solid foundation in key areas of mathematics such as algebra, analysis, geometry, and number theory, whereas others introduce students to more specialized areas of mathematics. Calculus, linear algebra, multivariable calculus, and proof and problem solving are prerequisite to most of these advanced courses.

Within the major, there are three concentrations leading to the B.A. degree: pure mathematics, computational mathematics, and mathematics education. These programs are designed to give students a strong background for graduate study, for work in industry or government, or for teaching. Each concentration requires nine courses, one of which must be a senior thesis or senior seminar. Please read the pure mathematics, computational mathematics, and mathematics education descriptions below for specific information about course requirements. A minor in mathematics is also offered.

The mathematics program also provides an excellent liberal arts background from which to pursue a variety of career opportunities. UCSC graduates with degrees in mathematics hold teaching posts at all levels, as well as positions in law; civil service; insurance and actuarial science; and other professions where skills in logic, numerical analysis, and computing are required.

Requirements

Students who plan to take a precalculus course or the first quarter of calculus must enroll in the course indicated by either their score on the mathematics placement exam (MPE) or the College Entrance Examination Board Advanced Placement (AP) calculus exam or pass the appropriate prerequisite course.

UC Santa Cruz Mathematics Placement Exam

Mathematics placement exam scores are valid for one year. Students whose areas of study require precalculus or calculus courses are strongly advised to take the placement exam and the required courses early in their academic careers. The placement exam is given just prior to the beginning of each quarter, in the sixth and seventh weeks of each quarter, and at prospective student orientations. Bring photo identification for entry into the placement exam.

Students who plan to major in computer engineering, computer science, electrical engineering, information systems management, mathematics, or physics and who receive a score in the range 31–39 on the MPE are strongly encouraged to take courses 3 and 19A-B rather than courses 11A-B.

College Board Advanced Placement Calculus Exams

Students who have received 4 credits for the College Entrance Examination Board Advanced Placement (AP) calculus exam should normally enroll in course 19B, whereas those with 8 credits should normally enroll in course 23A. However, students who are considering a major in the physical and biological sciences and who received a score of 3 on either the calculus AB or BC AP exam are also encouraged to enroll in course 19A or 19B, respectively, in order to improve their knowledge of calculus before continuing their studies. Students wishing to challenge themselves and who received a score of 5 on the AB exam or 4 on the BC exam may choose courses 20A and 20B, Honors Calculus.

Prerequisite Courses

Students who have passed course 2 may enroll in course 3. Students who have passed course 3 may enroll in course 11A or 19A. Students who have passed an articulated precalculus course at a college or university may enroll in course 11A or 19A (they must verify eligibility of the course and course completion with the Mathematics Department staff).

Premajor Requirements

Premajor requirements for all concentrations in the major are courses 20A and 20B, Honors Calculus; or 19A-B, Calculus for Science, Engineering, and Mathematics; 21, Linear Algebra; and 23A-B, Multivariable Calculus. The mathematics education concentration has one additional premajor requirement, Engineering 5, Statistics, which counts as the ninth required course for the major. For non-mathematics majors, courses 11A-B can be substituted for 19A-B, but they are not recommended for students planning to major in computer engineering, computer science, electrical engineering, information systems management, or physics. Although not considered a premajor requirement, course 100 is a prerequisite for most upper-division mathematics courses.

Major Requirements

Pure Mathematics

This concentration is intended for those who desire a comprehensive understanding of mathematics, including those considering going to graduate school in mathematics. Students are required to complete at least nine courses (with laboratories, if appropriate) from among those numbered 100 or higher; course 24 or Engineering 27 (not both) can also be counted as one of the nine courses toward fulfillment of the major requirements. Five of these courses must be 100, Introduction to Proof and Problem Solving; 103, Complex Analysis; 105A, Real Analysis; 111A, Algebra; and either 194, Senior Seminar, or 195, Senior Thesis. The remaining four courses are selected by the student from among courses 24 and 30 and Engineering 27 and mathematics courses numbered 100 or higher.

A typical program for a pure mathematics major might include the following:

The first two years of a typical program for a pure mathematics major who begins mathematics studies with precalculus might include the following:

Computational Mathematics

This concentration is intended to prepare students for technical careers in industry or government while providing a solid mathematical background. Students are required to complete a minimum of seven mathematics courses (with laboratories, if appropriate) as follows: 24, Ordinary Differential Equations; 100, Introduction to Proof and Problem Solving; 103, Complex Analysis, or 105A, Real Analysis; 110, Introduction to Number Theory; 111A, Algebra; 148, Numerical Analysis, or 145, Introductory Chaos Theory, or Engineering 146, Chaotic Dynamical Systems, or Engineering 147, Computational Methods and Applications; and either 194, Senior Seminar, or 195, Senior Thesis. In addition, students must complete two courses in computer engineering or computer science selected from the following:

Computer Engineering 107, 108, 153, 172, 177
Computer Science 101, 109, 112, 130

A typical program for a computational mathematics major might include the following:

in addition to two courses from computer engineering or computer science, as listed above.

Mathematics Education

This concentration is intended to prepare students for teaching kindergarten through high school (K–12). In addition to the premajor requirements, students are required to complete the following eight courses: either 103, Complex Analysis, or 105A, Real Analysis; 110, Introduction to Number Theory; 111A, Algebra; 128A, Classical Geometry; Engineering 131, Introduction to Probability Theory; 181, History of Math; 188, Supervised Teaching Experience; and either 194, Senior Seminar, or 195, Senior Thesis.

UCSC students can pursue a degree in mathematics while preparing to teach at the secondary level. In California, students seeking a single-subject credential (for secondary teaching) in mathematics are required to take the CSET series of exams (formerly The National Teachers Examination). By the time they begin student teaching in their credential program, candidates must have taken the CSET series. Contact the UCSC Education Department for details regarding teaching credentials.

A typical program for a mathematics education major might include the following:

Minor Requirements

The minor is intended for students who are interested in mathematics and want a strong mathematical foundation for studying in areas that rely heavily on analytical skills. Students are required to complete at least eight courses numbered 21 or above, with course 100 required. The courses that may be counted toward fulfillment of the minor requirements are courses 21, 23B, course 24 or Engineering 27 (not both), and those numbered 100 or higher. No senior seminar or thesis is required.

A typical mathematics minor program for a physics major might be:

Courses

Course 2, College Algebra for Calculus, is designed for students who do not meet the requirements for admission to course 3, Precalculus, and who need comprehensive and careful preparation for calculus. Course 2 emphasizes algebra, graphs, and functions. The prerequisite for course 2 is a minimum placement examination score of 12. Students not meeting the minimum requirement should take a college algebra course at a community college.

Course 3, Precalculus, is recommended for students who need some preparation in algebra and trigonometry prior to taking calculus. This course covers functions and their inverse, exponentials, logarithms, and trigonometry.

Statistics. Engineering 5 is a standard course in applied statistics and is intended for majors in the social sciences. Engineering 7 is intended for undergraduate and graduate students in the biological sciences. Students cannot receive credit for both Engineering 5 and 7. An advanced course in probability is also offered in alternate years.

Courses 11A-B, Calculus with Applications, are intended for biology and Earth sciences majors. However, students in these majors who score 40 or more points on the Mathematics Placement Exam are strongly encouraged to take the 19A-B sequence, which is required for all mathematics courses 23 and above. Laboratory sections are mandatory.

Courses 19A-B, Calculus for Science, Engineering, and Mathematics, are intended for chemistry, computer engineering, computer science, electrical engineering, information systems management, mathematics, and physics majors. Laboratory sections are mandatory.

Courses 20A and 20B, Honors Calculus, are intended for students who would enjoy delving particularly deeply into the foundational and theoretical issues of calculus. Laboratory sections are mandatory.

Course 21, Linear Algebra, covers vector spaces, matrices, determinants, systems of linear equations, and eigenvalues. It is intended for students in the physical and biological and social sciences and is prerequisite to course 111A.

Course 22, Introduction to Calculus of Several Variables, is intended for science students whose schedules do not permit a full and comprehensive two quarters of multivariable calculus. Students who intend to pursue further studies in mathematics must take 23A-B and not 22. Laboratory sections are mandatory.

Courses 23A-B, Multivariable Calculus, are intended for mathematics majors and students in computer engineering, computer science, electrical engineering, information systems management, and physics who desire more rigorous mathematical training. Laboratory sections are mandatory.

Course 100, Introduction to Proof and Problem Solving, is an introduction to the methodology of advanced mathematics, emphasizing proof techniques. Basic areas such as set theory and logic are introduced, together with extensive applications within mathematics. A waiver of this course is possible; students must see the faculty undergraduate vice chair.

Graduate-level courses. All graduate courses are open to undergraduates who have taken the recommended prerequisites; students should consult with the course instructor. Advanced undergraduates are strongly advised to take or audit graduate courses that interest them.

Graduate Program

The Mathematics Department offers programs leading to the M.A. and Ph.D. degrees. Prospective students should contact the department for both graduate programs prior to applying.

M.A. Degree Requirements

Students are required to complete courses 200, 201, 202, 203, and 204 and pass an M.A. level examination. In addition, students must do either of the following:

• complete five additional courses in mathematics. No more than two such courses may be independent studies;
  
• write a master’s thesis and complete three additional graduate courses in mathematics. No more than one course may be an independent study or thesis research course.

Ph.D. Degree Requirements

All of the following are required:

• pass the algebra and analysis written examinations;
• satisfy the foreign language requirement;
• pass the qualifying examination;
• complete three quarters as a teaching assistant;
• complete six graduate courses in mathematics other than 200, 201, 202, 203, 204, and 209. No more than three courses may be independent study or thesis research courses;
• write a Ph.D. thesis and present the thesis defense.

Students admitted to the Ph.D. program may receive an M.A. degree en route to the Ph.D.; students admitted to the M.A. program may transfer to the Ph.D. program upon passing the preliminary examinations, at the Ph.D. level.

Further information on the M.A. and Ph.D. programs, as well as on university application procedures, may be obtained from the Division of Graduate Studies.