Mathematics

2011-12 General Catalog

194 Baskin Engineering
(831) 459-2969
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 cutting-edge 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.
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 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 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 problem-solving, 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 (http://undergrad.pbsci.ucsc.edu) 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.

UCSC Mathematics Placement Examination

Mathematics placement examination scores are valid for one year. Students may take the examination a maximum of two times during the course of their academic career. 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. The placement examination is given at the beginning of each quarter and at prospective-student orientations. Bring photo identification for entry into the placement examination. Calculators are not permitted. See http://undergrad.pbsci.ucsc.edu/advising/exams/mpe.html for more information.

If your MPE score is	May enroll in this course
1-19	2
20-30	3
31-39	11A*
40-45	19A
46 or higher	19A or 20A

* 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 should take courses 3 and 19A-B rather than courses 11A-B.

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. Non-mathematics 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. Non-mathematics majors should consult their major departments before enrolling in a mathematics course.

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, but they must verify eligibility of the course and course completion with the Mathematics Department staff.

Introductory Requirements

Introductory requirements for all concentrations in the major are courses 20A-B, 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 introductory requirement, Applied Mathematics and Statistics (AMS) 5, Statistics. For some 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 an introductory requirement, course 100 is a prerequisite for most upper-division mathematics courses.

It should be emphasized that the nature of mathematics changes dramatically between lower-division and upper-division 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. Therefore, it is strongly recommended that only students who earn grades of B- or better in Mathematics 100 consider applying to the major in mathematics.

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 nine courses (with laboratories, if appropriate) from among those numbered 100 or higher.

Seven of these courses must be:

Mathematics 100, Introduction to Proof and Problem Solving;
Mathematics 103, 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 Non-Euclidean;
and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis.

The remaining two courses are selected by the student from among Mathematics 24 and any mathematics course numbered above 100 (excluding Mathematics 188 or Mathematics 189).

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

1st year	Mathematics 20A-B or 19A-B, 21, 23A
2nd year	Mathematics 23B, 24, 100, 103, 110 or 128A
3rd year	Mathematics 105A-B, 111A-B, 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, 19A-B
2nd year	Mathematics 21, 23A-B, 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 seven mathematics courses (with laboratories, if appropriate) as follows:

Mathematics 24, Ordinary Differential Equations;
Mathematics 100, Introduction to Proof and Problem Solving;
Mathematics 103, Complex Analysis, or Mathematics 105A, Real Analysis;
Mathematics 110, Introduction to Number Theory;
Mathematics 111A, Algebra;
Mathematics 145, Introductory Chaos Theory, or Applied Mathematics and Statistics 114, Introduction to Dynamical Systems, or Applied Mathematics and Statistics 147, Computational Methods and Applications;
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 113, 131, 147, 162
Biomolecular Engineering 110
Computer Engineering 107, 108, 117, 153, 177
Computer Science 101, 102, 104A, 109, 112, 122, 130, 132, 142
Economics 113
Electrical Engineering 103, 130, 135, 151, 154

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	19A-B, 23A, CMPS 12A/L and 12B/M
2nd year	21, 23B, 24, 100, 110, CMPE 16
3rd year	103; 105A; 145 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 (K-12) mathematics. In addition to the pre-major requirements (which for this track include Applied Mathematics and Statistics 5, Statistics), students are required to complete the following nine courses:

Mathematics 100, Introduction to Proof and Problem Solving;
either Mathematics 103, Complex Analysis, or 105A, Real Analysis;
Mathematics 110, Introduction to Number Theory;
Mathematics 111A, Algebra;
Mathematics 128A, Classical Geometry: Euclidean and Non-Euclidean;
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 single-subject 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 teaching-credential 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 19A-B, 23A
2nd year	Mathematics 21, 23B, 100; Applied Mathematics and Statistics 5
3rd year	Mathematics 30, 103, 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 upper-division 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 review of departmental narrative evaluations for all students considered for honors. Highest Honors will be awarded to those students whose performance demonstrates the highest level of excellence in the major as reflected in their narrative evaluations.

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 and 23 B, Multivariable Calculus;
and any five courses numbered 100 and above.

No senior seminar or thesis is required.

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

1st year	Mathematics 19A-B, 23A
2nd year	Mathematics 21, 23B, 24, 100
3rd year	Mathematics 103, 105A, 106A, 121A or 124
4th year	Mathematics 107, 145 or Applied Mathematics and Statistics 114

Disqualification from the Major or Minor

The Mathematics Department has adopted a major disqualification policy that is intended to encourage students to take their performance in the introductory requirements seriously and to make a strong effort to pass these courses.

Students who receive more than one No Pass, D, and/or F in the following introductory requirements will not be permitted to major in any concentration of the mathematics major.

Introductory Requirements

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 will be assessed for disqualification after grades are submitted each quarter and at the end of each summer session.

Students may appeal their disqualification within the appeal period by writing a letter to the department chair. This appeal must be submitted to the advising office no later than 15 days from the date the disqualification notification was mailed, or the 10th day of classes in the quarter of their disqualification, whichever is later. The advising office will subsequently notify the student, the college, and the Office of the Registrar of the decision, no later than 15 days after the submission of the appeal.

A student who has been disqualified from the major may, no earlier than three months from the date of the disqualification, petition to be reinstated. This application will be considered only if there is substantial new evidence that the student is capable of making normal progress in the major.

Combined Majors

The combined major, requiring fewer courses than a double major, is administered through the Economics Department.

Economics and Mathematics

The 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.

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 Ph.D.-level pass on two of the three written preliminary examinations, or a Ph.D.-level pass on one and a master's-level pass on the remaining two. Students who opt for the Ph.D.-level pass on two of the three preliminary examinations must complete the full sequence in the track associated with the preliminary examination they did not pass;
satisfy the foreign-language 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, 203, 204, 205, and 206. 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 required preliminary examinations at the Ph.D. level.

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. The prerequisite for course 2 is a minimum placement examination score of 12.

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 40 or more points on the mathematics placement examination are strongly encouraged to take the 19A-B sequence, which is required for most upper-division 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 23A-B and not 22. Laboratory sections are strongly advised.

Mathematics 23A and 23B, Multivariable 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 upper-division courses.

Mathematics 200+, 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.

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Revised: 8/13/12