Mathematics
4111 McHenry
(831) 4592969
https://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 ways of applying mathematics to their fields. A strong mathematics background is vital to the advanced study of many disciplines including the physical and biological sciences, engineering and the social sciences.
The UCSC mathematics program offers a wide variety of undergraduate mathematics courses:
 Students interested in studying mathematics are strongly encouraged to take algebra, geometry, and trigonometry before entering the university. Students needing mathematics courses for their intended major are strongly encouraged to consider their options, and take the necessary steps for mathematics assessment and placement as early as possible. Progress in some majors could be delayed if the calculus series is not begun upon arrival at UCSC. Students concerned about their ability to place into courses above Mathematics 3 should consider taking Mathematics 2 or its equivalent prior to entering UCSC.
 Lowerdivision courses with numbers in the range 11 through 24 (calculus, linear algebra, vector calculus, and differential equations) prepare students for further study in mathematics, the physical and biological sciences, engineering, or quantitative areas of the social sciences. Science and engineering majors take some or all 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, vector calculus, and proof and problem solving are prerequisite to most of these advanced courses.
Undergraduate Program
Undergraduate Major
Within the major, there are three concentrations leading to the Bachelor of Arts (B.A.) in mathematics: 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 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, an essential skill in all professions.
Program Learning Outcomes
Learning outcomes summarize the most important knowledge, skills, abilities and attitudes that students are expected to develop over the course of their studies. The program learning outcomes clearly communicate the faculty’s expectations to students, provide a framework for faculty evaluation of the curriculum based on empirical data, and help improve and measure the impact of implemented changes.
Mathematics Undergraduate Student Learning Objectives
The mathematics program promotes mathematical skills and knowledge for their intrinsic beauty, effectiveness in developing proficiency in analytical reasoning, and utility in modeling and solving real world problems. To responsibly live within and participate in the transformation of a rapidly changing, complex, and interdependent society, students must develop and unceasingly exercise their analytical abilities. Students who have learned to logically question assertions, recognize patterns, and distinguish the essential and irrelevant aspects of problems can think deeply and precisely, nurture the products of their imagination to fruition in reality, and share their ideas and insights while seeking and benefiting from the knowledge and insights of others.
Students majoring in mathematics attain proficiency in:
Critical thinking. The ability to identify, reflect upon, evaluate, integrate, and apply different types of information and knowledge to form independent judgments including analytical and logical thinking and the habit of drawing conclusions based on quantitative information.
Problem solving. The ability to assess and interpret complex situations, choose among several potentially appropriate mathematical methods of solution, persist in the face of difficulty, and present full and cogent solutions that include appropriate justification for their reasoning.
Effective communication. The ability to communicate and interact effectively with different audiences, collaborate intellectually and creatively in diverse contexts, and appreciate ambiguity and nuance, while emphasizing the importance of clarity and precision in communication and reasoning.
Students acquire and enhance these abilities in mathematical contexts, but the acquired habits of rigorous thought and creative problem solving are invaluable in all aspects of life. These skills are acquired through experience in the context of studying specific mathematical topics and exploring problems chosen to challenge students’ abilities, spurring them on to acquire new techniques and to abandon familiar but restrictive habits of thought. The overarching objectives can be realized in terms of more focused, appraisable objectives specific to mathematics described on the Mathematics Department website.
The Mathematics Department offers three tracks within the mathematics major:
Pure Mathematics
Students in the Pure Mathematics track often go on to graduate study in mathematics; the pathway emphasizes the importance of a wellrounded, indepth mathematical education, and includes advanced coursework in algebra, analysis, and geometry.
Computational Mathematics
Students in the Computational Mathematics track explore applications of mathematics in other fields and gain experience in mathematical modeling of realworld phenomena using ordinary and partial differential equations, approximation and optimization techniques, programming, or game theory.
Mathematics Education
Students in the Mathematics Education track prepare for a career in K–12 mathematics education; students acquire indepth knowledge of subjects covered at an introductory level in the classroom, including number theory, classical geometry, and the history of mathematics, and gain experience in teaching mathematics in an accessible and intuitive, but precise, manner.
Curriculum Matrix
All of the key objectives are addressed to some extent in all courses. For example, the ability to formulate precise mathematical statements and to reason logically are essential skills that are progressively developed throughout the curriculum. However, some skills are more heavily emphasized and utilized in some courses than in others. Some courses are specifically intended to help students move to a new level of proficiency with a particular portfolio of skills, while others are accessible only to students who have already reached a given level; the latter courses make heavy use of particular skills, and thus enhance and reinforce the student’s mastery of them, but the skills themselves are not the primary focus of such courses. Some connections between the key objectives, main subjectspecific areas, and courses are indicated in the tables of lower and upperdivision mathematics courses at the Mathematics Department’s website.
Academic Advising
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 Mathematics Department website 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 with requirements, course curriculum, and departmental policy.
All students should review the requirements for their major or intended major and possibly consult with the department sponsoring their major (or expected major) before deciding which mathematics courses to take. More information on what courses are intended for the various types of students may be found here at the Mathematics Department website.
Enrollment Requirements
Students who plan to take a mathematics course at UCSC must first demonstrate sufficient preparation for that course by completing mathematics placement, the College Entrance Examination Board Advanced Placement (AP) calculus examination, the International Baccalaureate (IB) 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 (on Assist.org) and course completion with the mathematics adviser.
UCSC Mathematics Placement
The mathematics placement process assesses student readiness for their first UCSC mathematics class. Students whose areas of study require precalculus or calculus courses are strongly advised to complete placement and any required courses early in their academic careers. Students intending to take one or more mathematics courses at UCSC should begin the placement process as early as possible to fully benefit from the process.
Students completing placement by assessing using ALEKS PPL should familiarize themselves with the assessment instructions and guidelines, course eligibility cutoffs, and score posting schedule.
Advanced Placement (AP) Calculus Examinations
Students completing placement requirements by using their scores from the College Board Advanced Placement Calculus Exam should refer to the Admissions Department Chart for assistance in deciding which course to enroll.
International Baccalaureate (IB) Examination in Mathematics
Students completing placement requirements by using their scores from the International Baccalaureate Exam should refer to the Admissions Department Chart for assistance in deciding which course to enroll.
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, or Mathematics 20A, Honors Calculus
 Mathematics 19B, Calculus for Science, Engineering, and Mathematics, or Mathematics 20B, Honors Calculus
 Mathematics 21, Linear Algebra
 Mathematics 23A, Vector Calculus
 Mathematics 23B, Vector 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, C, D+, D, 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. Students are strongly encouraged to file an appeal as soon as a student is no longer qualified to declare. The mathematics adviser will subsequently notify the student, and their college, of the decision, no later than 15 business days after the submission of the appeal. An appeal decision may be in the form of an approval, denial or conditional approval. For students who have not completed all of the major qualification courses, conditional approvals are based on subsequent performance in the remainder of the qualification courses.
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.
Transfer Students
The Mathematics Department welcomes applications from community college students who have completed the necessary coursework to transfer to our program.
To be considered for admission to UCSC as a participant in any of the mathematics majors, transfer students must pass equivalents of the following courses:
 Mathematics 19A, Calculus for Science, Engineering, and Mathematics
 Mathematics 19B, Calculus for Science, Engineering, and Mathematics
 Mathematics 21, Linear Algebra
 Mathematics 23A, Vector Calculus
The above major preparation courses are highly recommended, and will be required beginning with applications to transfer in Fall 2020. More information about qualifying for the major as a transfer applicant is here.
Students planning to transfer to UCSC from a California community college should reference the assist website to determine which courses are equivalent to these required courses.
To obtain equivalency for MATH 23A, transfer students will have taken a course that may also be equivalent to MATH 23B. Students are encouraged to contact the undergraduate adviser to determine if this applies to their situation.
Major Requirements
Pure Mathematics
This concentration is intended for students who desire a comprehensive understanding of mathematics, including those considering graduate studies in the physical sciences.
LowerDivision Requirements
 Mathematics 19A, Calculus for Science, Engineering, and Mathematics, or Mathematics 20A, Honors Calculus
 Mathematics 19B, Calculus for Science, Engineering, and Mathematics, or Mathematics 20B, Honors Calculus
 Mathematics 21, Linear Algebra
 Mathematics 23A, Vector Calculus
 Mathematics 23B, Vector Calculus
 Mathematics 24, Ordinary Differential Equations
UpperDivision Requirements
 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; Mathematics 128A, Classical Geometry: Euclidean and NonEuclidean; or Mathematics 129, Algebraic Geometry;
 and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis
Electives
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.
Pure Mathematics B.A.: Sample Freshmen Academic Plan

Fall 
Winter 
Spring 
1st 
MATH 19A or 20A 
MATH 19B 
MATH 21 


MATH 23A 

2nd 
MATH 23B 
MATH 103A 
Elective 
MATH 100 



3rd 
MATH 24 
MATH 105A 
Elective 
MATH 128A 



4th 
MATH 117 
MATH 111A 
MATH 194 


Elective 
Pure Mathematics B.A.: Sample Transfer Academic Plan
For students who have completed Math 19A, 19B, 21, and 23A equivalents.

Fall 
Winter 
Spring 
1st 
MATH 23B 
MATH 103A 
MATH 121A 
MATH 100 
Elective 
Elective 

2nd 
MATH 24 
MATH 105A 
MATH 194 
MATH 111A 
Elective 
MATH 117 
Computational Mathematics
This concentration is intended to prepare students for technical careers in industry or government while providing a solid mathematical background.
LowerDivision Requirements
 Mathematics 19A, Calculus for Science, Engineering, and Mathematics, or Mathematics 20A, Honors Calculus
 Mathematics 19B, Calculus for Science, Engineering, and Mathematics, or Mathematics 20B, Honors Calculus
 Mathematics 21, Linear Algebra
 Mathematics 23A, Vector Calculus
 Mathematics 23B, Vector Calculus
 Mathematics 24, Ordinary Differential Equations
UpperDivision Requirements
 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 148/L, Numerical Analysis or Mathematics 145, Introductory Chaos Theory, or Applied Mathematics and Statistics 114, Introduction to Dynamical Systems, Applied Mathematics and Statistics 147, Computational Methods and Applications;
 and either Mathematics 194, Senior Seminar, or Mathematics 195, Senior Thesis
Electives
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, 130, 132, 142
 Earth and Planetary Sciences 172
 Economics 113
 Electrical Engineering 103, 130, 135, 151, 154
 Mathematics 115, 116, 120, 134, 145, 148, 152, 160
 Physics 107, 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.
Computational Mathematics B.A.: Sample Freshmen Academic Plan

Fall 
Winter 
Spring 
1st 
MATH 19A 
MATH 19B 
MATH 21 


MATH 23A 

2nd 
MATH 23B 
MATH 110 
MATH 145/L 
MATH 100 



3rd 
MATH 24 
MATH 106 
Elective 
Elective 



4th 
MATH 111A 
MATH 103A 
MATH 194 


Elective 
Computational Mathematics B.A: Sample Transfer Academic Plan
For students who have completed Math 19A, 19B, 21 and 23A equivalents.

Fall 
Winter 
Spring 
1st 
MATH 23B 
MATH 103A 
MATH 145/L 
MATH 100 
MATH 110 


2nd 
MATH 24 
MATH 106 
MATH 194 
MATH 111A 
Elective 
Elective 
Mathematics Education
This concentration is intended to prepare students for teaching kindergarten through high school (K12) mathematics.
LowerDivision Requirements
 Applied Mathematics and Statistics 5, Statistics
 Mathematics 19A, Calculus for Science, Engineering, and Mathematics, or Mathematics 20A, Honors Calculus
 Mathematics 19B, Calculus for Science, Engineering, and Mathematics, or Mathematics 20B, Honors Calculus
 Mathematics 21, Linear Algebra
 Mathematics 23A, Vector Calculus
 Mathematics 23B, Vector Calculus
UpperDivision Requirements
 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;
 Either Mathematics 188, Supervised Teaching Experience; or Education 50B, CalTeach 1: Mathematics, plus Education 100B, CalTeach 2: Mathematics
 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, the Mathematics Department’s website and the Education Department advising office have more information about the additional required courses for the Subject Matter Program.
Mathematics Education B.A.: Sample Freshmen Academic Plan

Fall 
Winter 
Spring 
1st 
MATH 19A 
MATH 19B 
AMS 5 


MATH 23A 

2nd 
MATH 21 
MATH 100 
MATH 110 
MATH 23B 



3rd 
MATH 128A 
MATH 181 
AMS 131 
EDUC 50B 
EDUC 100B 


4th 
MATH 111A 
MATH 103A 
MATH 194 
Mathematics Education B.A.: Sample Transfer Academic Plan
For students who have completed Math 19A, 19B, 21 and 23A equivalents.

Fall 
Winter 
Spring 
1st 
MATH 23B 
MATH 103A 
MATH 110 
MATH 100 
MATH 181 
AMS 5 

2nd 
MATH 128A 
MATH 111A 
MATH 194 
EDUC 50B 
EDUC 100B 
AMS 131 
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.
Comprehensive Requirement
The comprehensive exit requirement in mathematics is satisfied by either MATH 194, Senior Seminar, or MATH 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 19A, Calculus for Science, Engineering, and Mathematics, or Mathematics 20A, Honors Calculus
 Mathematics 19B, Calculus for Science, Engineering, and Mathematics, or Mathematics 20B, Honors Calculus
 Mathematics 21, Linear Algebra;
 Mathematics 23A, Vector Calculus; and
 Mathematics 23B, Vector Calculus;
 Mathematics 100
 The remaining four courses are selected by the student from among any mathematics course numbered above 100 (excluding Mathematics 188 and Mathematics 189), any Applied Mathematics and Statistics (AMS) course numbered 100 or above, or, subject to approval of the undergraduate vice chair, a course from another department. Only one of the four courses can be from the AMS series or another outside department. Under exceptional circumstances, MATH 100 may be substituted by another upperdivision mathematics course. The undergraduate vice chair must review and approve requests on an individual basis.
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
Program Description
The Mathematics Department offers programs leading to the Master of Arts (M.A.) and Doctor of Philosophy (Ph.D.) degrees. Students admitted to the Ph.D. program may receive a master's degree en route to the Ph.D.; students admitted to the M.A. program may apply to the department to transfer to the Ph.D. program upon passing the required preliminary examinations at the Ph.D. level.
The Mathematics Department at U.C. Santa Cruz is small but dynamic, with an ongoing commitment to both research and teaching. The department has leading research programs in several actively developing areas on the frontiers of pure and applied mathematics, interacting strongly with theoretical physics and mechanics. The extraordinary level of National Science Foundation support received by our faculty reflects the high caliber of the research carried out in the department. The current areas of research include:
 Vertex operator algebras, higher genus conformal field theory, modular forms, quasiHopf algebras, infinitedimensional Lie algebras, mathematical physics
 Representations of Lie and padic groups, applications to number theory, Bessel functions, RankinSelberg integrals, GelfandGraev models
 Algebra, group theory, finite groups and their representations, conjectures of Alperin, Dade and Broué, Mackey functors, modular representation theory, fusion systems, blocks of finite groups, bisets, biset functors, Burnside rings, representations of algebras, ring theory, module theory
 Algebraic topology, elliptic cohomology, quantum field theory, automorphic forms, string topology, topology of Lie groups, loop spaces
 Symplectic geometry and topology, Floer homology, Poisson Lie groups
 Dynamical systems, celestial mechanics, geometric mechanics, bifurcation theory, control theory
 Fluid and continuum mechanics, the NavierStokes equation, long time behavior of solutions of PDEs.
 Geometric integration schemes, numerical methods on manifolds
Algebraic geometry  Differential geometry, nonlinear analysis, harmonic maps, GinzburgLandau problem.
 General relativity, Einstein's equations, positive mass conjecture, Teichmuller theory
 Galois and incidence geometry
 Algebraic number theory, elliptic curves, Lfunctions, padic Lfunctions, special values of Lfunctions, GrossStark conjecture, Heegner points
 Graph theory, expander graphs, prime number distribution
 Functional analysis, random matrix theory, spectral gap, operator theory, Banach algebras, harmonic analysis, WienerHopf factorization, statistical physics
Contiguous Bachelor’s/Master’s Pathway
The 4+1 pathway into the mathematics master’s program is an option that allows undergraduates at U.C. Santa Cruz to:
 Take two required graduate courses during their undergraduate degree in preparation to finish the master’s degree in just one additional year.
 Apply to the mathematics master’s program through a streamlined application process.
Undergraduate students currently enrolled in the mathematics B.A. have the opportunity, any time after the start of their junior year until December 1st of their senior year, to apply to be admitted to the 4+1 contiguous pathway leading to the mathematics master’s degree. Qualified undergraduates from other undergraduate majors may also apply to the pathway and their applications will be considered on a casebycase basis.
The requirements for admission into the 4+1 pathway are:
 A GPA in the major of 3.5 or more;
 to have taken MATH 105B or MATH 111B; and
 to have taken, to be currently enrolled, or have the plan to enroll by fall of the senior year in one of the required graduate courses.
Interested students should set up a meeting with the mathematics undergraduate adviser to discuss their curriculum plan and complete the application forms. The deadline for application to the pathway is December 1st of the senior year, although students are encouraged to apply earlier.
Students in the pathway who apply through the streamlined application process to the master’s program are not guaranteed admission. The Mathematics Department expects to admit students who have passed two of the required graduate courses and have maintained a GPA in the major of 3.5 or more. Once accepted into the master’s program, students from the pathway will follow the same requirements as any other students in the twoyear track with expected graduation in the fifth year.
Preparation for Graduate Work
In order to be prepared for the master’s or Ph.D. program, it is recommended to have a B.A. or B.S. in mathematics. Having taken more than the bare minimum of required upperdivision classes in the mathematics major will be most helpful.
Admission to the Graduate Program
Applications to the graduate program can be submitted through the Graduate Division. The deadline is usually during the first half of January. Admission is decided by a faculty committee and is based on a combination of factors including: GRE scores (in particular the GRE Math Subject Score), letters of recommendation, GPA, and classes taken.
Financial Support
The Mathematics Department is strongly committed to the financial support of graduate students who are making good progress toward either the master's or the Ph.D. degree. For the purpose of financial support, a student’s progress is measured against the degree programs and timetables.
A teaching assistantship (TA) is the most common form of financial support for graduate students in good academic standing. TA appointments are usually made at 50 percent time (an assigned workload of approximately 220 hours for the quarter). Teaching assistants are under the supervision of the faculty member responsible for the course.
All students are strongly urged to complete a Free Application for Financial Student Aid (FAFSA) each year by the start of fall quarter to determine eligibility for needbased awards. Students are also encouraged to apply for support from the Financial Aid Office as well as from the Mathematics Department.
No needbased fellowship can be awarded to a student who does not have a current FAFSA on file. Students facing special financial hardship are urged to make this known to the department in a timely manner.
The Mathematics Department will do everything in its power to ensure that all students in good standing are granted sufficient financial aid to continue their study of mathematics.
Relationship of Master’s and Doctoral Programs
Students in the master’s and doctoral program take the same classes in the core sequences and the same preliminary examinations. Ph.D. and master’s students have the same passing requirements in the core classes. However, the preliminary examination requirements for Ph.D. and master’s students are different and are outlined below.
Master of Arts (M.A.) Degree in Mathematics
The objectives of the mathematics M.A. program give students advanced fundamental knowledge in the areas of algebra, analysis, and geometry in order to prepare them for admission in top Ph.D. programs, for work in industry, or for a teaching career at community colleges. Students will possess the ability to solve problems and communicate solutions and concepts clearly and in rigorous mathematical language.
Master's students are expected to complete their degree within two years. Students admitted to the M.A. program may apply to the Mathematics Department to transfer to the Ph.D. program upon passing the required preliminary examinations at the Ph.D. level.
Requirements
Students are required to complete four of the following courses from the three core sequences:
 MATH 200, Algebra I
 MATH 201, Algebra II
 MATH 202, Algebra III
 MATH 204, Analysis I
 MATH 205, Analysis II
 MATH 206, Analysis III
 MATH 208, Manifolds I
 MATH 209, Manifolds II
 MATH 210, Manifolds III
Students are also required to complete five additional courses in mathematics. Courses in a related subject may be substituted by approval from the graduate vice chair. Sample courses include:
 MATH 203, Algebra IV
 MATH 207, Complex Analysis
 MATH 211, Algebraic Topology
 MATH 212, Differential Geometry
 MATH 213A/B, Partial Differential Equations I/II
 MATH 214, Theory of Finite Groups
 MATH 215, Operator Theory
 MATH 216, Advanced Analysis
 MATH 217, Operator Theory
 MATH 218, Advanced Analysis
 MATH 217, Advanced Elliptic Partial Differential Equations
 MATH 218, Advanced Parabolic and Hyperbolic Partial Differential Equations
 MATH 219, Nonlinear Functional Analysis
 MATH 220A/B, Representation Theory I/II
 MATH 222A/B, Algebraic Number Theory
 MATH 223A/B, Algebraic Geometry I/II
 MATH 225A, Lie Algebras
 MATH 225B, Infinite Dimensional Lie Algebras
 MATH 226A/B, Infinite Dimensional Lie Algebras and Quantum Field Theory I/II
 MATH 227, Lie Groups
 MATH 228, Lie Incidence Geometries
 MATH 229, KacMoody Algebras
 MATH 232, Morse Theory
 MATH 233, Random Matrix Theory
 MATH 234, Riemann Surfaces
 MATH 235, Dynamical Systems Theory
 MATH 238, Elliptic Functions and Modular Forms
 MATH 239, Homological Algebra
 MATH 240A/B, Representations of Finite Groups I/II
 MATH 246, Representations of Algebras
 MATH 248, Symplectic Geometry
 MATH 249A/B/C, Mechanics I/II/III
 MATH 252, Fluid Mechanics
 MATH 254, Geometric Analysis
 MATH 256, Algebraic Curves
 MATH 260, Combinatorics
 MATH 280, Topics in Analysis
 MATH 281, Topics in Algebra
 MATH 282, Topics in Geometry
 MATH 283, Topics in Combinatorial Theory
 MATH 284, Topics in Dynamics
 MATH 285, Topics in Partial Differential Equations
 MATH 286, Topics in Number Theory
 MATH 287, Topics in Topology
Additional requirements for the M.A. degree are dependent on the student’s chosen track: the thesis track or the comprehensive examination track.
Thesis Track
Students are required to complete a master’s thesis. A master’s thesis does not have to consist of original research results. At the minimum, it should show mastery of a specific subject area that goes beyond the knowledge taught in the core sequences in algebra, analysis, or geometry. This track is recommended for students that want to transfer into a top Ph.D. program.
The student, in consultation with the graduate vice chair, is responsible for selecting a master’s thesis reading committee. The majority of the membership of a thesis reading committee shall be members of the Santa Cruz Division of the Academic Senate. The Graduate Division must approve the committee.
The Nominations for Master’s Thesis Reading Committee Form must be completed and submitted by the end of the second week of the quarter in which the degree will be granted. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate adviser and program coordinator for review and submission to the Graduate Division.
More information about thesis submission can be found at the Graduate Division website.
Comprehensive Examination Track
Students are required to obtain a secondlevel pass on one of three written preliminary examinations: algebra, analysis, or geometry. A secondlevel pass signifies that the student has a very good understanding of the basic concepts, but not necessarily enough to conduct independent research.
Applying for Graduation
M.A. students must complete the Application for the Master’s Degree form by the appropriate quarter’s deadline listed in the current Academic calendar.
The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the Graduate Adviser and Program Coordinator for review and submission to the Graduate Division.
Ph.D. Degree in Mathematics
The objectives of the mathematics Ph.D. program are to prepare students for a career in academia, industry, or teaching. At the end of their studies, students will possess the ability to solve problems and communicate solutions in rigorous mathematical language, to communicate mathematical concepts effectively, and to conduct independent research.
Entering graduate students are advised initially by an assigned faculty mentor. Within the first two years, and typically after passing the preliminary examinations, the student selects a Ph.D. adviser in the area of the student's research interest.
Each graduate student is expected to consult with their adviser to formulate a plan of study and research. The student's adviser ultimately will be the student's thesis adviser.
Ph.D. students are expected to obtain their Ph.D. degree within six years. Students admitted to the Ph.D. program may receive a master's degree en route to the Ph.D.
Preliminary Examinations
Preliminary examinations are given for each core sequence in the fields of algebra, analysis, and geometrytopology at the beginning, middle, and end of each academic year. The exams will be designed and graded by a committee of three members.
A firstlevel pass signifies that the student has the basic knowledge to start research with a thesis adviser in this particular area. A secondlevel pass signifies that the student has a very good understanding of the basic concepts, but not necessarily enough to conduct independent research.
Ph.D. students must 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 threecourse sequence in the field associated with the preliminary examination in which they did not achieve a firstlevel pass. Students may take the preliminary examinations as often as they wish.
Ph.D. students should complete the preliminary examinations and core sequence requirements by the end of their second year in order to make satisfactory progress. If a graduate student does not fulfill these requirements by the end of their second year, they may be placed on academic probation, depending on their progress in the program. If a graduate student has not fulfilled these requirements by the end of their third year, they are subject to dismissal from the program.
Topics for the preliminary examinations include:
 Algebra
 Linear algebra
 Group theory
 Ring and module theory
 Field theory
 Galois theory
 Analysis
 Basic analysis
 General topology
 Metric spaces
 Measure and integration
 Complex analysis
 Functional analysis
 Geometrytopology (manifolds)
 Manifold and tangent bundle
 Differential forms and integration on manifolds
 Fundamental group and covering space
 (Co)homology
 Differential geometry
Required Coursework
A threecourse sequence in each of the three fields of algebra, analysis, and geometrytopology (manifolds) will be offered each year. Preliminary examinations will be given for each core sequence at the beginning, middle, and end of each academic year.
Firstlevel passage of a preliminary examination satisfies the core sequence requirement for that field. Ph.D. students are required to complete the full core sequence in the field associated with the preliminary examination in which they do not achieve a firstlevel pass. The core sequences are as follows:
 MATH 200, Algebra I
 MATH 201, Algebra II
 MATH 202, Algebra III
 MATH 204, Analysis I
 MATH 205, Analysis II
 MATH 206, Analysis III
 MATH 208, Manifolds I
 MATH 209, Manifolds II
 MATH 210, Manifolds III
Students are also required to complete six additional courses in mathematics. No more than three courses may be independent study or thesis research courses. Sample courses include:
 MATH 203, Algebra IV
 MATH 207, Complex Analysis
 MATH 211, Algebraic Topology
 MATH 212, Differential Geometry
 MATH 213A/B, Partial Differential Equations I/II
 MATH 214, Theory of Finite Groups
 MATH 215, Operator Theory
 MATH 216, Advanced Analysis
 MATH 217, Operator Theory
 MATH 218, Advanced Analysis
 MATH 217, Advanced Elliptic Partial Differential Equations
 MATH 218, Advanced Parabolic and Hyperbolic Partial Differential Equations
 MATH 219, Nonlinear Functional Analysis
 MATH 220A/B, Representation Theory I/II
 MATH 222A/B, Algebraic Number Theory
 MATH 223A/B, Algebraic Geometry I/II
 MATH 225A, Lie Algebras
 MATH 225B, Infinite Dimensional Lie Algebras
 MATH 226A/B, Infinite Dimensional Lie Algebras and Quantum Field Theory I/II
 MATH 227, Lie Groups
 MATH 228, Lie Incidence Geometries
 MATH 229, KacMoody Algebras
 MATH 232, Morse Theory
 MATH 233, Random Matrix Theory
 MATH 234, Riemann Surfaces
 MATH 235, Dynamical Systems Theory
 MATH 238, Elliptic Functions and Modular Forms
 MATH 239, Homological Algebra
 MATH 240A/B, Representations of Finite Groups I/II
 MATH 246, Representations of Algebras
 MATH 248, Symplectic Geometry
 MATH 249A/B/C, Mechanics I/II/III
 MATH 252, Fluid Mechanics
 MATH 254, Geometric Analysis
 MATH 256, Algebraic Curves
 MATH 260, Combinatorics
 MATH 280, Topics in Analysis
 MATH 281, Topics in Algebra
 MATH 282, Topics in Geometry
 MATH 283, Topics in Combinatorial Theory
 MATH 284, Topics in Dynamics
 MATH 285, Topics in Partial Differential Equations
 MATH 286, Topics in Number Theory
 MATH 287, Topics in Topology
Foreign Language Requirement
The foreign language requirement must be satisfied before taking the oral qualifying examination. Graduate students in the Ph.D. program are required to demonstrate knowledge of French, German, or Russian, sufficient to read the mathematical literature in the language. Any member of the mathematics faculty may administer a foreign language examination.
The examination can be either oral or written. It typically requires translation of a text in one of the three foreign languages into English.
The Report on Language Requirement Form must be filled out by the student and the faculty member administering the examination. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate adviser and program coordinator for review and submission to the Graduate Division.
Oral Qualifying Examination
All graduate students in the Ph.D. program are required to take an oral examination, called the oral qualifying examination, for advancement to candidacy for the Ph.D. Degree. Students typically complete this examination between their 7th and 12th quarter in residence.
Students will demonstrate that they have a sufficient understanding of their Ph.D. thesis problem. Any student who has not passed their oral exam by the end of the fourth year may be subject to academic probation or dismissal from the program.
The Report on Qualifying Examination Form must be filled out by the qualifying examination committee immediately following the examination. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate adviser and program coordinator for review and submission to the Graduate Division. The student may request to see a copy of the report.
If the student fails the examination, a reexamination can be given within the next three months. The membership of the examining committee usually remains fixed.
Qualifying Examination Committee Composition
The examining committee consists of the student’s faculty adviser, at least two other faculty members from the Mathematics Department, and at least one outside tenured faculty member from either another discipline at UCSC or another academic institution (involved in research and graduate education of the same or different discipline). The student, in consultation with the student’s faculty adviser, selects the committee. The chair of the committee must be someone other than the student’s faculty adviser.
The Graduate Division must approve the committee. The Committee Nomination of Ph.D. Qualifying Examination Form must be completed and submitted at least one month prior to the requested exam date. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate adviser and program coordinator for review and submission to the Graduate Division.
The committee decides on the topics for the examination, which should be broad enough to encompass a substantial body of knowledge in the area of the student’s interest. The written list of topics to be included in the examination, along with a short bibliography, is prepared by the student. A copy is given to each committee member and a copy is put into the student’s permanent records.
Advancement to Candidacy
To make satisfactory progress, a Ph.D. student should advance to candidacy by the end of their fourth year. A Ph.D. student who has not advanced to candidacy by the end of the fourth year will be placed on academic probation or be subject to dismissal from the program.
Students must complete the following in order to advance to candidacy:
 Complete the preliminary examinations and core sequences in accordance with the requirements outlined above;
 Satisfy the language requirement;
 Pass the qualifying examination;
 Have a dissertation reading committee approved by the Mathematics Department and the Graduate Division;
 Have no incomplete grades (I) on their record.
An advancement to candidacy fee will be billed to the student’s account. The student will be officially advanced the following term after all of these requirements are met.
Dissertation Reading Committee Composition
A Ph.D. student, in consultation with the graduate vice chair, is responsible for selecting a dissertation reading committee. The committee consists of the student’s adviser and at least two other members of the mathematics faculty. In special circumstances, a committee member may be chosen from another department and/or from another institution. The student’s adviser is the chair of the committee.
The Graduate Division must approve the committee. The Nominations for Dissertation Reading Committee Form must be completed and submitted prior to advancement to candidacy. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate adviser and program coordinator for review and submission to the Graduate Division.
A new form must be submitted for approval if changes to the dissertation reading committee must be made.
Dissertation/Dissertation Defense
Each graduate student in the Ph.D. program is required to write a Ph.D. dissertation or thesis on a research topic in mathematics. The Ph.D. dissertation should contain original research results that are publishable in a peerreviewed journal. All members of the student’s dissertation committee must read and approve the dissertation.
After the dissertation has been approved, the student has an option of making a public oral presentation of the mathematical results contained in the dissertation—the “thesis defense.” A recommendation by the dissertation committee will be made to the Mathematics Department and to the Graduate Council on the granting of the Ph.D. degree.
More information about dissertation submission can be found at the Graduate Division website.
Teaching Requirement
Ph.D. students must complete a minimum of three quarters as a teaching assistant (TA). All TAs are required to participate in the department's teaching assistant training program.
TA appointments are usually made at 50 percent time (an assigned workload of approximately 220 hours per quarter). TAs are under the supervision of the faculty member responsible for the course. TAs are covered by a collective bargaining agreement between the University of California and the United Auto Workers (UAW).
Instructors and their TA(s) will meet at the beginning of the quarter to complete the Notification of TA Duties form in order to identify the agreed upon tasks. The performance of these tasks will form the basis of the endofquarter performance evaluation and will use the following criteria: quality of work; accuracy and attention to detail; interaction with students, peers, and instructor; knowledge of subject; and dependability. The specific allocation of TA duties is subject to change, depending on enrollments and the number of teaching assistantships in the department allocation. The general duties vary, depending on the course assigned and level of the course.
Review of Progress
Ph.D. students are expected to adhere to the below degree timetable:
 Preliminary examinations and course sequence requirements
Completed by the end of the student’s 2nd year  Language examination
Completed by the end of the student’s 3rd year  Oral qualifying examination (and advancement to candidacy)
Completed no later than student’s 12th quarter  Dissertation defense
Completed no later than the end of the 6th year
Annual meetings with the graduate vice chair and the graduate adviser and program coordinator are conducted with each student on a oneonone basis. These meetings serve to notify the student of their current progress within the program and outline expectations for the continuation of normative progress toward the Ph.D. degree.
Applying for Graduation
Ph.D. students must complete the Application for the Doctor of Philosophy Degree form by the appropriate quarter’s deadline listed in the current Academic calendar.
The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate adviser and program coordinator for review and submission to the Graduate Division.
Revised: 07/15/18