Mathematics
201415 General Catalog
4111 McHenry
(831) 4592969
http://www.math.ucsc.edu
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 cuttingedge ways of applying mathematics to their fields. A strong mathematics background is vital to the advanced study of the physical and biological sciences and plays an integral role in studying the social sciences.
The UCSC 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. Students requiring mathematics courses are encouraged to take the mathematics placement examination (MPE) as early as possible. Students concerned about their ability to place into courses above Mathematics 2 or Mathematics 3 should consider taking these courses before they enter UCSC. Failure to begin the calculus series in the fall could delay progress in some majors.

Lowerdivision courses with numbers in the range 11AB 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.

Upperdivision courses, with numbers in the range 100199, 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 bachelor of arts (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 or ten courses, one of which must be a senior thesis or senior seminar. Please read the pure mathematics, computational mathematics, and mathematics education program 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, government, civil service, insurance, software development, business, banking, actuarial science, forensics, and other professions where skills in logic, numerical analysis, and computing are required. In particular, students of mathematics are trained in the art of problemsolving, a skill absolutely essential to all professions.
Academic Advising
Academic advising is available at the Undergraduate Affairs Office . The undergraduate adviser provides information about requirements, prerequisites, policies and procedures, learning support, scholarships, and special opportunities for undergraduate research. In addition, the adviser assists with the drafting of study plans, as well as certifying degrees and minors. Students are urged to stay informed and involved with their major, as well as to seek advice should problems arise.
The Undergraduate Affairs web site is a critical resource for students. Here you will find a link to the undergraduate program; the materials at that link constitute the undergraduate handbook. Students should visit this first to seek answers to their questions, because it hosts a wealth of information. Each student in the major is encouraged to regularly review the materials posted to stay current about requirements, course curriculum, and departmental policy.
Requirements
Students who plan to take a mathematics course at UCSC must demonstrate sufficient preparation by their score on either the mathematics placement examination (MPE), the College Entrance Examination Board Advanced Placement (AP) calculus examination, the International Baccalaureate Higher Level Mathematics Examination, or by passing the appropriate prerequisite course.
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 a precalculus course at a college or university may enroll in course 11A or 19A, but they must first verify eligibility of the course and course completion with the mathematics adviser (mathadvising@ucsc.edu).
Transfer students interested in a mathematics or science major should take courses equivalent to course 19A, 19B, and 21 before enrolling at UCSC. Courses equivalent to 23A, 23B, and 24 are also recommended before transferring to UCSC.
UCSC Mathematics Placement Examination
The Mathematics Placement Examination (MPE) is a mathematics test designed to assess student readiness for UCSC mathematics classes. The MPE is available during several, short windows each year. Each student may take the exam twice during their academic career. Only the highest MPE score is used for placement. MPE scores are valid for one year after the window closes.
Additionally, if a student receives a D, F, or NP in a course, the placement examination may not be used to place them out of that course. Students whose areas of study require precalculus or calculus courses are strongly advised to take the placement examination and the required courses early in their academic careers. Information on registering and taking the MPE can be found at this web site.
If your MPE score is 
You may enroll in this course 

100 = Not precalculus ready 
MATH 2 
200 = Precalculus ready 
MATH3 
300 = Applied Calculus ready 
MATH11A* 
400 = Calculus ready 
MATH19A 
500 = Honors Calculus ready 
MATH 20A 
* Some majors in science and engineering require Math 19AB. Students planning to concentrate in those majors who receive 300, an Applied Calculus ready score on the MPE, should take courses 3 and 19AB rather than courses 11AB. All students are advised to consult with the Department sponsoring their major before deciding which mathematics pathway to take.
College Board Advanced Placement Calculus Examinations
Students who have received 4 credits for the College Entrance Examination Board Advanced Placement (AP) calculus examination should normally enroll in course 19B, and those with 8 credits should normally enroll in course 23A. However, students who received a score of 3 on the calculus AB or BC AP examination, should enroll in course 19A or 19B, respectively, to improve their knowledge of calculus before continuing their studies. Students who wish to challenge themselves, and who received a score of 4 or 5 on the AB or a score of 3, 4, or 5 on the BC examination may choose course 20A, Honors Calculus. Nonmathematics majors should consult their major departments before enrolling in a mathematics course.
If your AP AB score is 
May enroll in this course 

3 
Mathematics 11A or 19A 
4 or 5 
Mathematics 20A or 11B or 19B 
If your AP BC score is 
May enroll in this course 

3 
Mathematics 11B or 19B or 20A 
4 or 5 
Mathematics 20A or 22 or 23A 
International Baccalaureate Higher Level Examination in Mathematics
Students who have received a score of 5, 6, or 7 on the International Baccalaureate (IB) Higher Level Examination in Mathematics may enroll in course 20A, Honors Calculus; 22, Calculus of Several Variables; or 23A, Multivariable Calculus. Nonmathematics majors should consult their major departments before enrolling in a mathematics course.
Declaration of the Mathematics Major
Admission to the mathematics major (all concentrations) is contingent on students successfully passing the following introductory courses or their equivalents:

Mathematics 19A, Calculus for Science, Engineering, and Mathematics

Mathematics 19B, Calculus for Science, Engineering, and Mathematics

Mathematics 21, Linear Algebra

Mathematics 23A, Multivariable Calculus

Mathematics 23B, Multivariable Calculus

Mathematics 100, Introduction to Proof and Problem Solving
Students may only declare once they have passed all introductory courses or their equivalent courses with a grade of C or better. Students who receive two grades of NP, D, or F in the introductory courses are not eligible to declare in the major. Students who are not eligible to declare may submit an appeal to the department's undergraduate vice chair. The advising office will subsequently notify the student, and the college of the decision, no later than 15 business days after the submission of the appeal.
It should be emphasized that the nature of mathematics changes dramatically between lowerdivision and upperdivision courses. Students often find that the material becomes far more abstract and theoretical. In addition, the role of computation in assignments diminishes and a greater weight is placed on deductive reasoning and the integral role of mathematical proofs. The Mathematics Department recommends that students interested in a mathematics major enroll in Mathematics 100 as early as prerequisites allow in order to decide whether they are interested in upperdivision mathematics courses. It is strongly recommended that only students who earn grades of B or better in Mathematics 100 consider applying to the major in mathematics. Students with a grade less than B in Mathematics 100 are urged to take Mathematics 101.
Major Requirements
Pure Mathematics
This concentration is intended for students who desire a comprehensive understanding of mathematics, including those considering graduate studies in the natural sciences. Students are required to complete at least 11 courses (with laboratories, if appropriate).
Eight of these courses must be:

Mathematics 24, Ordinary Differential Equations;

Mathematics 100, Introduction to Proof and Problem Solving;

Mathematics 103A, Complex Analysis;

Mathematics 105A, Real Analysis;

Mathematics 111A, Algebra;

Mathematics 117, Advanced Linear Algebra;

one of Mathematics 121A, Differential Geometry, Mathematics 124, Introduction to Topology, or Mathematics 128A, Classical Geometry: Euclidean and NonEuclidean, or Mathematics 129, Algebraic Geometry;

and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.
The remaining three courses are selected by the student from among any mathematics course numbered above 100 (excluding Mathematics 188 and Mathematics 189) and Applied Mathematics and Statistics (AMS) 100 or above. Only one of the three courses can be from the AMS series.
A typical program for a pure mathematics major might include the following:
1st year 
Mathematics 20AB or 19AB, 21, 23A 
2nd year 
Mathematics 23B, 24, 100, 103A, 110 or 128A 
3rd year 
Mathematics 105AB, 111AB, 106 
4th year 
Mathematics 107, 117, 121A, 194 
The first two years of a typical program for a pure mathematics major who begins mathematics studies with precalculus might include the following:
1st year 
Mathematics 3, 19AB 
2nd year 
Mathematics 21, 23AB, 24, 100 
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 eight mathematics courses (with laboratories, if appropriate) as follows:

Mathematics 24, Ordinary Differential Equations;

Mathematics 100, Introduction to Proof and Problem Solving;

Mathematics 103A, Complex Analysis, or Mathematics 105A, Real Analysis;
 Mathematics 106, Systems of Ordinary Differential Equations, or Mathematics 107, Partial Differential Equations;

Mathematics 110, Introduction to Number Theory;

Mathematics 111A, Algebra;
 or Mathematics 117, Advanced Linear Algebra;

Mathematics 106, Systems of Ordinary Differential Equations, or Mathematics 145, Introductory Chaos Theory, or Applied Mathematics and Statistics 114, Introduction to Dynamical Systems;

and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.
In addition, students must complete two courses selected from the following:

Applied Mathematics and Statistics, 100 or above

Biomolecular Engineering 110

Computer Engineering 107, 108, 153, 177

Computer Science 101, 102, 104A, 109, 112, 122, 130, 132, 142
 Earth and Planetary Sciences 172

Economics 113

Electrical Engineering 103, 130, 135, 151, 154
 Physics 115
Some of these courses have prerequisites within their departments. Students are encouraged to plan their computational electives early, so that all prerequisites can be satisfied in a timely manner. Other upperdivision courses with heavy emphasis on computational mathematics may occasionally be accepted with permission of the Mathematics Department.
Mathematics majors who wish to enroll in Computer Science 101 or Computer Science 122 should contact the instructor to request a permission code.
A typical program for a computational mathematics major might include the following:
1st year 
19AB, 23A, CMPS 12A/L and 12B/M 
2nd year 
21, 23B, 24, 100, 110, CMPE 16 
3rd year 
103A; 105A; 145/L or AMS 147; CMPS 101 
4th year 
106, 111A, CMPS 109, 194 
Mathematics Education
This concentration is intended to prepare students for teaching kindergarten through high school (K12) mathematics. Students are required to complete the following 10 courses:

Applied Mathematics and Statistics 5, Statistics

Mathematics 100, Introduction to Proof and Problem Solving;

either Mathematics 103A, Complex Analysis, or 105A, Real Analysis;

Mathematics 110, Introduction to Number Theory;

Mathematics 111A, Algebra;

Mathematics 128A, Classical Geometry: Euclidean and NonEuclidean;

Applied Mathematics and Statistics 131, Introduction to Probability Theory;

Mathematics 181, History of Math;

Mathematics 188, Supervised Teaching Experience;

and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.
UCSC students can pursue a degree in mathematics while preparing to teach at the secondary level. In California, students seeking a singlesubject credential (for secondary teaching) in mathematics are required to take the CSET, a series of examinations that must be passed in order to enter a teachingcredential program (formerly The National Teachers Examination). Students who complete the mathematics education track, plus three additional specified courses, qualify for the California Single Subject Program, exempting themselves from the CSET. Both the Mathematics Department undergraduate adviser and the Education Department advising office have more information about the additional required courses.
A typical program for a mathematics education major might include the following:
1st year 
Mathematics 19AB, 23A 
2nd year 
Mathematics 21, 23B, 100; Applied Mathematics and Statistics 5 
3rd year 
Mathematics 101, 103A, 110, 181; Applied Mathematics and Statistics 131 
4th year 
Mathematics 111A, 128A, 188, 194 
Disciplinary Communication (DC) Requirement
Students of every major must satisfy that major’s upperdivision Disciplinary Communication (DC) requirement. The DC requirement in mathematics is satisfied by Mathematics 100, Introduction to Proof and Problem Solving, and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.
Honors
Honors in the Mathematics Department are awarded to graduating students whose academic performance in the major demonstrates excellence at a GPA of 3.5 or above. Highest Honors are determined by a cumulative review of student performance in mathematics courses. They are awarded to students who excel in challenging courses and in their capstone projects.
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 as follows:

Mathematics 21, Linear Algebra;

Mathematics 23A, Vector Calculus; and

Mathematics 23B, Vector Calculus;
 Mathematics 100

and any four courses numbered above 100.
No senior seminar or thesis is required.
A typical mathematics minor program for a physics major might be:
1st year 
Mathematics 19AB, 23A 
2nd year 
Mathematics 21, 23B, 24, 100 
3rd year 
Mathematics 103A, 105A, 106, 121A or 124 
4th year 
Mathematics 107, 145 or Applied Mathematics and Statistics 114 
Combined Majors
Economics and Mathematics
The combined major in economics and mathematics is designed to meet the needs of undergraduate students who plan to pursue doctoral study in economics or business, or who wish to pursue a career as an actuary or other professional requiring a sophisticated understanding of economics and mathematics. The major combines the main undergraduate content of both economics and mathematics within a programmatic structure that joins the two disciplines. It provides a coursework combination required to prepare for a modern economics Ph.D. program, or for technically demanding professional careers. A full description can be found in the economics section of this catalog. The combined major, requiring fewer courses than a double major, is administered through the Economics Department.
Graduate Program
The Mathematics Department offers programs leading to the master of arts (M.A.) and doctor of philosophy (Ph.D.) degrees. Contact the Division of Graduate Studies for further information on the M.A. and Ph.D. programs, as well as on university application procedures.
M.A. Degree Requirements
Students are required to complete two of Mathematics 200, 201, 202, 203; two of Mathematics 204, 205, 206; one of Mathematics 208, 209, 210; and complete five additional courses in mathematics or a related subject by approval. In addition, students must do one of the following:

pass an M.A.level preliminary examination;

write a master’s thesis.
Ph.D. Degree Requirements
All of the following are required:

obtain a firstlevel pass on at least one of the three written preliminary examinations and a secondlevel pass on at least one other. Students must complete the full course sequence in the track associated with the preliminary examination in which they did not achieve a firstlevel pass;

satisfy the foreignlanguage requirement;

pass the oral qualifying examination;

complete three quarters as a teaching assistant;

complete six graduate courses in mathematics other than Mathematics 200, 201, 202, 204, 205, 206, 208, 209, and 210. No more than three courses may be independent study or thesis research courses;

write a Ph.D. thesis.
Students admitted to the Ph.D. program may receive an M.A. degree en route to the Ph.D.
Course Information
Mathematics 2, College Algebra for Calculus, is designed for students who do not meet the requirements for admission to Mathematics 3, Precalculus, and who need comprehensive and careful preparation for calculus. Mathematics 2 emphasizes algebra, graphs, and functions.
Mathematics 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.
Mathematics 11A and 11B, Calculus with Applications, are intended for biology and Earth sciences majors. However, students in these majors who score 400 or more points on the mathematics placement examination are strongly encouraged to take the 19AB sequence, which is required for most upperdivision mathematics courses. Laboratory sections are strongly advised.
Mathematics 19A and 19B, 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 strongly advised.
Mathematics 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 strongly advised.
Mathematics 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 Mathematics 111A.
Mathematics 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 Mathematics 23AB and not 22. Laboratory sections are strongly advised.
Mathematics 23A and 23B, Vector Calculus, are intended for mathematics majors and minors and students in computer engineering, computer science, electrical engineering, information systems management, and physics majors which require more rigorous mathematical training. Laboratory sections are strongly advised.
Mathematics 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. This course serves as a prerequisite for nearly all upperdivision courses and partially fulfills the Disciplinary Communication (DC) requirement.
Mathematics 200+, Graduatelevel 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.
Revised: 09/01/14