Department of Mathematics
Division of Mathematical, Life, and Physical Sciences
South Hall 6607
Telephone: (805) 893-2171
Undergraduate e-mail: ugrad@math.ucsb.edu
Graduate e-mail: math-gradinfo@math.ucsb.edu
Website: www.math.ucsb.edu (will open in a new browser window)
Department Chair: Jeffrey Stopple
Contents:
Adebisi Agboola, Ph.D., Columbia University, Professor (number theory)
Charles A. Akemann, Ph.D., UC Berkeley, Professor (functional analysis)
Stephen Bigelow, Ph.D., UC Berkeley, Associate Professor (low-dimensional topology)
Bjorn Birnir, Ph.D., Courant Institute, Professor (nonlinear partial differential equations)
Maria Isabel Bueno Cachadina, Ph.D., Universidad Carlos III de Madrid, Lecturer (numerical linear algebra)
Paolo Cascini, Ph.D., Courant Institute, Assisant Professor (algebraic geometry)
Hector Ceniceros, Ph.D., Courant Institute, Associate Professor (numerical analysis)
Daryl Cooper, Ph.D., University of Warwick, Professor (topology, group theory)
Xianzhe Dai, Ph.D., State University of New York, Stony Brook, Professor (Geometric Analysis)
Carlos Garcia-Cervera, Ph.D., Courant Institute, Assistant Professor (applied mathematics)
Larry J. Gerstein, Ph.D., University of Notre Dame, Professor (quadratic forms, number theory)
Kenneth R. Goodearl, Ph.D., University of Washington, Professor (algebra, functional analysis)
Sergei Gukov, Ph.D., Princeton University, Associate Professor (geometry, topology, string theory)
Birge Huisgen-Zimmerman, Ph.D., University of Munich, Professor (algebra, representation theory)
William Jacob, Ph.D., Princeton University, Professor (quadratic forms, division algebras)
Denis Labutin, Ph.D., Australian National University, Associate Professor (partial differential equations)
Darren Long, Ph.D., Cambridge University, Professor (low-dimensional topology)
James McKernan, Ph.D., Harvard University, Professor (algebraic geometry)
Jon McCammond, Ph.D., UC Berkeley, Associate Professor (geometric group theory, low-dimensional topology )
Kenneth C. Millett, Ph.D., University of Wisconsin, Professor (algebraic and geometric topology)
John Douglas Moore, Ph.D., UC Berkeley, Professor (differential geometry)
David Morrison, Ph.D., Harvard University, Professor (algebraic geometry, string theory)
Gustavo Ponce, Ph.D., Courant Institute, Professor (nonlinear partial differential equations)
Mihai Putinar, Ph.D., University of Bucharest, Professor (operator theory, complex analysis)
Martin Scharlemann, Ph.D., UC Berkeley, Professor (topology)
Thomas Sideris, Ph.D., Indiana University, Professor (partial differential equations, nonlinear wave equations)
Isadore Singer, Ph.D., University of Chicago, Professor (index theory, mathematical physics)
Jeffrey Stopple, Ph.D., UC San Diego, Professor (number theory)
Guofang Wei, Ph.D., State University of New York, Stony Brook, Professor (differential geometry)
Milen Yakimov, Ph.D., UC Berkeley, Associate Professor (Poisson geometry, representation theory)
Adil Yaqub, Ph.D., UC Berkeley, Professor (ring theory, universal algebras)
Rugang Ye, Ph.D., Bonn University, Professor (differential geometry)
Seymour Bachmuth, Ph.D., New York University, Professor Emeritus (group theory)
Thomas K. Boehme, Ph.D., California Institute of Technology, Professor Emeritus (function analysis)
Andrew M. Bruckner, Ph.D., UC Los Angeles, Professor Emeritus (real analysis)
Michael J. Cambern, Ph.D., UC Berkeley, Professor Emeritus (functional analysis)
Jack G. Ceder, Ph.D., University of Washington, Professor Emeritus (real analysis)
Michael G. Crandall, Ph.D., UC Berkeley, Professor Emeritus (nonlinear differential equations)
John E. Doner, Ph.D., UC Berkeley, Associate Professor Emeritus (logic, computer science)
John A. Ernest, Ph.D., University of Illinois, Professor Emeritus (functional analysis)
Ky Fan, D.Sc., University of Paris, Professor Emeritus (topology, functional analysis)
Eugene C. Johnsen, Ph.D., Ohio State University, Professor Emeritus (combinatorial analysis)
Henryk Minc, Ph.D., University of Edinburgh, Professor Emeritus (linear analysis)
James B. Robertson, Ph.D., Indiana University, Professor Emeritus (probability, ergodic theory)
Alex Rosenberg, Ph.D., University of Chicago, Professor Emeritus (quatratic form, Witt rings)
Melvin Rosenfeld, Ph.D., UC Los Angeles, Associate Professor Emeritus (functional analysis)
Stephen Simons, Ph.D., Cambridge University, Professor Emeritus (functional analysis)
James M. Sloss, Ph.D., UC Berkeley, Professor Emeritus (partial differential equations)
David A. Sprecher, Ph.D., University of Maryland, Professor Emeritus (real analysis)
Max L. Weiss, Ph.D., University of Washington, Professor Emeritus (complex analysis)
Raymond Y. Wong, Ph.D., Louisiana State University, Professor Emeritus (topology)
Julius Zelmanowitz, Ph.D., University of Wisconsin, Professor Emeritus (rings, modules)
Frederick Gibon, Ph.D., (Mechanical Engineering)
Igor Mezic, Ph.D., (Mechanical Engineering)
Jeff Moehlis, Ph.D., (Mechanical Engineering)
Linda R. Petzold, Ph.D., (Computer Science and Mechanical and Environmental Engineering)
Mathematics has been called the queen and the servant of the sciences. As an independent discipline, it was first developed by the ancient Greeks, to whom we owe the notion of “mathematical proof.” In the late seventeenth century, Newton developed calculus to serve as a tool in his treatment of mechanics, allowing him to correctly predict the motion of the planets. This astonishing success definitively demonstrated that mathematics is the ideal language for constructing exact quantitative theories. Today mathematics plays an absolutely fundamental role in physics, economics, and engineering, and plays an ever greater role in fields such as astronomy, chemistry, geology, finance, meteorology, cryptology, ecology, computer science, the social sciences, and a host of other areas. Yet mathematics is also vibrant as a study in its own right, alive with beautiful problems and ongoing developments. These may not be initially motivated by applications, but history indicates that many of the purely mathematical developments of today will become essential to the sciences of the future.
The Department of Mathematics offers five undergraduate programs; B.S. and B.A. degrees in mathematics, a B.S. degree in mathematical sciences; in conjunction with the Department of Economics, a B.A. in economics/mathematics; and in conjunction with the Program in Applied Statistics and Probability, a B.S. in financial mathematics and statistics.
The Department of Mathematics offers two distinct minor programs. These programs allow non-majors to supplement their majors with cohesive course of study that reflects their interests. To ensure appropriate advising and planning, students who are considering a minor in mathematics should consult the department as soon as possible.
The department offers graduate programs leading to the M.A. and Ph.D. degrees. In addition, it offers a wide variety of service courses needed as a foundation for study in the sciences, in engineering, and in other fields.
Undergraduate advisors are available in the department office to answer questions about the department and other academic matters. Detailed information about the majors and about career options in mathematics is available in several publications, including Professional Opportunities in the Mathematical Sciences, which is available in the Department of Mathematics office, 6607 South Hall. The mathematics website (www.math.ucsb.edu) is designed to keep students and faculty informed about current seminars, colloquia, and special events.
Various prizes and awards are offered each year to outstanding majors in mathematics. These include the Raymond L. Wilder award and student memberships in the Mathematical Association of America. Each award is given on the basis of academic excellence in the mathematics program.
Students with a bachelor’s degree in mathematics who are interested in pursuing a California Teaching Credential should contact the credential advisor in the Gevirtz Graduate School of Education as soon as possible.
Diagnostic and placement examination. Students who do not have AP credit must take the Algebra Diagnostic Test (ADT) which is offered online at www.math.ucsb.edu/ugrad/adt. Minimum scores on the ADT are required for enrollment in Mathematics 15 and 3A. The exam is not required for Math 34A.
Results on the Algebra Diagnostic Test are substantially improved by reviewing algebra and trigonometry prior to taking the exam. A copy of Precalculus Review Topics may be obtained from the UCSB Bookstore, (805) 893-2961. Allow two to three weeks for delivery.
The department strictly enforces the requirement of a C grade or better in any course prerequisite to Mathematics 3B-C, 5A-B-C, 8, and 34B.
To enter the honors program in mathematics, a student must have completed 120 units of coursework with an overall grade-point average of at least 3.5 and at least 24 upper-division mathematics and statistics units with a grade-point average of at least 3.5 (excluding Mathematics 100A-B, 193, 195A-B, and PSTAT 133A-B-C and 193). To complete the honors program, the student must maintain a grade-point average of at least 3.5 in all upper-division and graduate mathematics and statistics courses (excluding Mathematics 100A-B, 193, 195A-B, and PSTAT 133A-B-C and 193) and complete one of the following: (a) a senior thesis, Math 197A-B; (b) a two-quarter graduate sequence; or (c) together with an advisor, submit a Distinction in the Major proposal for an interdisciplinary program of three mathematically oriented courses outside the math department to the undergraduate committee for its approval. Option C does not apply to economics/mathematics or financial mathematics majors. Distinction in the Major for each option will be awarded at graduation pending final approval by the Department of Mathematics Undergraduate Committee. Written projects will be submitted to the committee, and grades will be evaluated for coursework options.
Undergraduate Program
As preparation for entering any of the undergraduate mathematics programs, students should have completed two years of algebra and courses in plane geometry and trigonometry in high school. Students lacking this background should take Mathematics 15. In the first two years at UCSB, all students who major in mathematics must complete the appropriate pre-major requirements. All prospective majors and pre-majors must meet with a faculty advisor, prior to admission to full major status, to discuss career opportunities and degree options and to design an upper-division course program. Admission to full major status will be granted only after this meeting has been documented. Samples of recommended programs for each degree option are available in the Department of Mathematics Undergraduate Handbook.
Bachelor of Science - Mathematics
The bachelor of science degree is especially suitable for students who want a rigorous program with an emphasis on theory or who plan to go on to graduate work in mathematics.
Pre-major requirements. Students must complete all pre-major courses with a 2.5 or higher grade-point average. Physics 1 or 6A or 21, Engineering 3, and Computer Science 10 or 5 (any section) are excluded as part of the pre-major grade-point average computation but do apply to the overall GPA. Entry into the pre-major does not guarantee admission to full major status. Upon satisfactory completion of the following pre-major requirements, students may petition to be accepted into full major status: Mathematics 3A-B-C, 5A-B-C, 8; Physics 1, 6A or 21; Computer Science 10 or one course from 5AA-ZZ or Engineering 3.
Upper-division major. Fifty-two upper-division units in mathematics are required, excluding Mathematics 100A-B, 193, 195A-B; PSTAT 133A-B-C and 193. At least 40 of these 52 units must be in Mathematics. These 52 units must include Mathematics 108A-B, 111A-B, 117, 118A-B, 122A, either 111C or 118C, and either 145 or 147A. With an advisor’s approval, 4 of the 52 units may be non-mathematics courses taken as part of a coherent mathematics program.
Bachelor of Science - Mathematical Sciences
This is an applied mathematics degree intended for students interested in computational aspects of mathematics, systems analysis, decision sciences, physical sciences, and operations research. It is suitable as preparation for advanced training in applied mathematics, management science, business administration, or operations research.
Pre-major requirements. Students must complete all pre-major courses with a 2.5 or higher grade-point average. Computer Science 10 or 5 (any section), Engineering 3, and Physics 1 or 6A or 21, are excluded as part of the pre-major grade-point average computation but do apply to the overall major GPA. Entry into the pre-major does not guarantee admission to full major status. Upon satisfactory completion of the following pre-major requirements, students may petition to be accepted into full major status: Mathematics 3A-B-C, 5A-B-C, 8; Physics 1, 6A, or 21; Computer Science 10 or one course from 5AA-ZZ or Engineering 3.
Upper-division major. Fifty-two upper-division units in mathematics are required, excluding Mathematics 100A-B, 193, 195A-B; PSTAT 133A-B-C and 193. At least 40 of these 52 units must be in Mathematics. The 52 units must include: Mathematics 104A-B, 108A, and two two-quarter sequences chosen from Mathematics 119A-B, either 118A-B or 122A-B, 124A-B, 132A-B, 137A-B and PSTAT 120A-B. With an advisor’s approval, up to 4 of the 52 required units may be non-mathematics courses taken as part of a coherent mathematics program.
Bachelor of Arts - Mathematics
This degree provides the student with a broad, liberal education in pure mathematics and is flexible enough to allow a wide variety of upper-division programs that may be created by the student in consultation with a faculty advisor. The B.A. in mathematics contains a special concentration designed specifically as preparation for high-school teaching. However, completion of a concentration will not be formally acknowledged on the student’s official transcripts or diploma.
Pre-major requirements. Students must complete all pre-major courses with a 2.5 or higher grade-point average. Computer Science 10 or 5 (any section), Engineering 3, and Physics 1 or 6A or 21 are excluded as part of the pre-major grade-point average computation but do apply to the overall major GPA. Entry into the pre-major does not guarantee admission to full major status. Upon satisfactory completion of the following pre-major requirements, students may petition to be accepted into full major status: Mathematics 3A-B-C, 5A, 8; Physics 1, 6A, or 21; Computer Science 10 or one course from 5AA-ZZ or Engineering 3.
Upper-division major. Forty upper-division units in mathematics are required, excluding Mathematics 100A-B, 193, 195A-B; PSTAT 133A-B-C and 193. At least 28 of these 40 units must be in Mathematics. The 40 units must include the specific requirements for one of the following concentrations, which will not be formally acknowledged on the student’s official transcript or diploma:
Concentration 1 requirements: Mathematics 108A and three two-quarter sequences, chosen from Mathematics 104A-B, 108B-C, 109A-B, 111A-B, 115A-B, 118A-B, 119A-B, 122A-B, 124A-B, 132A-B, 137A-B, either Mathematics 145-147A, or 147A-B, PSTAT 120A-B. With an advisor’s approval, 4 units of the 40 units may be non-mathematics courses taken as part of a coherent mathematics program.
Concentration 2 requirements: Mathematics 101A-B; 102A-B; 103 and 108A. With an advisor’s approval, 4 units of the 40 units may be non-mathematics courses taken as part of a coherent mathematics program.
Bachelor of Arts - Economics/Mathematics
This program is offered jointly with the Department of Economics. It provides a theoretical foundation for advanced study in economics, business administration, law, or management science.
Pre-major requirements. Students must complete all pre-major courses with a 2.7 or higher grade-point average and no individual grade below C-. Entry into the pre-major does not guarantee admission to full major status. Upon satisfactory completion of the following pre-major requirements, students may petition to be accepted into full major status: Economics 1 and 2; PSTAT 120A; Mathematics 3A-B-C, 5A-B-C, and 8.
Upper-division major. Forty-four upper-division units in economics and mathematics are required, excluding Economics 109. The 44 units must include Economics 104A-B, 105, and 140A-B; Mathematics 108A-B and 117; and 12 units of upper-division economics electives. Selected from Economics 100C, 106, 115, 116A-B-C, 117A, 120, 122, 133, 134A-B, 135, 143, 150A-B, 152, 155, 170, 171, 175A-B, 180, 181, 184. For breadth, further elective courses concerning optimization and modeling, such as Mathematics 132A-B-C, are recommended. Students should consult closely with their advisors in the Departments of Economics and Mathematics regarding their upper-division programs, particularly if they intend to pursue graduate study in a closely related area such as mathematical economics, applied mathematics, statistics, or operations research.
Bachelor of Science - Financial Mathematics and Statistics
This is a joint major between the Department of Mathematics and the Department of Statistics and Applied Probability; in cooperation with the Department of Economics. This degree is intended for students who would like to learn how mathematics, probability and statistics play a key role in pricing and hedging securities in the financial markets.
Pre-major requirements. In order to be admitted into the Financial Mathematics and Statistics major, students must complete all of the following pre-major courses with a grade-point average of 2.5 or higher, Mathematics 3A-B-C, 5A-B-C, 8, Economics 1 and 2. In addition, one course is required from the following: Computer Science 5AA-ZZ, 10 or Engineering 3. The computer science and engineering courses are excluded from the pre-major GPA calculation but will apply to the overall major GPA.
Entry into pre-major does not guarantee admission into full major status. Upon satisfactory completion of the pre-major requirements, and after meeting with a faculty advisor to discuss career opportunities and upper-division course electives, students may petition to be accepted to full major status.
Upper-division major. Fifty-two upper-division units in mathematics, statistics, and economics are required, excluding Mathematics 100A-B, 193, and 195A-B and PSTAT 133A-B-C. The 52 units must include Economics 104A, Mathematics 104A-B, 124A-B; PSTAT 120AB, PSTAT 120C, 130; and either PSTAT 170 or Mathematics 170. The remaining 12 elective upper-division units can be chosen from: Economics 104B, 105, 134A-B, 140B; Mathematics 104C, 108A-B, 117; PSTAT 160A-B, 171, 173,174.
All courses to be applied to the minor must be completed on a letter-grade basis. This includes both courses offered in mathematics and those offered by other departments and applied to the minor.
Preparation for the minor. Mathematics 3A-B-C, 5A, and 8.
Upper-division minor. Twenty-four upper-division units in mathematics are required excluding the following: Math 100A-B, 193, 195A-B.
Note: Please see "Academic Minors" for special conditions governing minors in the College of Letters and Science.
Minor - Mathematics for High School Teaching
All courses to be applied to the minor must be completed on a letter-grade basis. This includes both courses offered in mathematics and those offered by other departments and applied to the minor.
Preparation for the minor. Mathematics 3A-B-C, 5A, and 8.
Upper-division minor. Twenty-four upper-division units in mathematics and PSTAT are required. The required courses are: Mathematics 101A-B, 102A-B, 103, and 4 upper-division units of mathematics or PSTAT elective. The following courses will not apply to the minor: Mathematics 100A-B, 193, 195A-B; PSTAT 133A-B-C and 193.
Note: Please see "Academic Minors" for special conditions governing minors in the College of Letters and Science.
Graduate Program
Candidates for admission to graduate programs offered by the Department of Mathematics are required to submit Graduate Record Examination (GRE) general and mathematics subject test scores. Applicants whose native language is not English, are required to take either the Test of English as Foreign Language (TOEFL) or the International English Language Testing System (ISLTS) exam. Excemptions to this requirement will be considered for those students who have completed an undergraduate or graduate education at an institution whose primary language of instruction is English. The minimum TOEFL score for consideration is 550 when taking the paper-based test (PBT), 213 when taking the computer-based test (CBT), and 80 when taking the internet-based test (IBT). The minimum IELTS score for consideration is an Overall Band Score of 7 or higher. TOEFL or IELTS scores must not be more than two years old at the time of application to UCSB.
Foreign students must have a score of 575 (or 231 on the computer-based test) for teaching assistantship consideration. Applicants for teaching assistant positions are encouraged to submit scores for the Test of Spoken English (TSE) at the time of application.
In addition to departmental requirements, candidates for graduate degrees must fulfill the university degree requirements found in the section "Graduate Education at UCSB.”
In the following description of the M.A. and Ph.D. programs in mathematics, frequent mention will be made of “area requirements.” Area requirements exist in the disciplines of algebra, analysis, applied mathematics, geometry/topology, and other areas in probability and statistics. Students whose primary interest is in the area of statistics or probability should apply for admission to the Department of Statistics and Applied Probability, not to the Department of Mathematics. The area requirements are fulfilled by satisfactorily completing an examination and a one-year graduate course within the discipline. Complete descriptions of various area requirements and how they may be satisfied can be found in the publication Graduate Program Handbook, which is available on our website. Contact the staff graduate advisor at math-gradinfo@math.ucsb.edu, or at the following address: Department of Mathematics, University of California, Santa Barbara, CA 93106. This information can also be obtained via our website at www.math.ucsb.edu/grad.
Admission
The applicant must (1) fulfill the scholarship requirements for graduate study, and (2) hold a bachelor’s degree in mathematics or a closely related field. Evaluation of the candidate’s past work will be made by the admissions committee, and supplemental undergraduate courses will be required when necessary.
Degree Requirements
The department offers two plans for completing the degree: Plan 1 (thesis) and Plan 2 (examination option).
Both plans require completion of 42 units with the grade of at least B in each course, 24 of which must be in selected graduate courses offered by the Department of Mathematics. The remaining 18 units may be chosen from upper-division or graduate courses in mathematics, or in appropriate related fields with the approval of the Mathematics Graduate Committee. No more than 8 of the 24 graduate units may be in Mathematics 596/598 combined.
Plan 1, Thesis: In addition to the above, Plan 1 requires demonstration of adequate knowledge in linear algebra, modern algebra, real analysis, and complex analysis, and preparation of an acceptable thesis and oral defense of the thesis before a faculty committee. The 24 graduate units in mathematics must include at least one full-year course sequence that satisfies one of the area requirements.
Plan 2, Examination Option: Students must satisfy the area requirements in algebra and analysis. A student who wishes to substitute a different area requirement for one of the above areas must petition the departmental graduate committee.
Master of Arts - Applied Mathematics
Admission
The candidate must (1) fulfill the scholarship requirements for graduate study; (2) hold a bachelor’s degree in mathematics or a closely related field; and (3) have had undergraduate coursework in linear algebra, differential equations, advanced calculus, and in some area in which mathematics is applied. Evaluation of the candidate’s past work will be made by the admissions committee, and supplemental undergraduate courses will be required when necessary.
Degree Requirements
The department offers two plans for completing the degree: Plan 1 (thesis), and Plan 2 (examination option). All candidates must complete 42 units with the grade of B or better in each course, 24 units of which must be in graduate course sequences approved and offered by the Department of Mathematics. No more than 8 of the 24 graduate units may be in Mathematics 596/598 combined.
The remaining 18 units may be in upper-division or graduate-level courses in mathematics or, with the approval of the graduate committee, outside of mathematics, with a limit of 9 units outside the department.
Plan 1, Thesis: Students must prepare an acceptable thesis under the supervision of a faculty member and do an oral defense of it before a faculty committee. The 24 graduate units in mathematics must include at least one full-year course sequence that satisfies one of the area requirements.
Plan 2, Examination Option: Students must satisfy the area requirements in Applied Mathematics and Analysis. Students may petition the graduate committee to substitute a different area for Analysis.
Students interested in continuing to the Ph.D. normally follow Plan 2 for the master’s degree. To be invited to continue to the Ph.D. level, students are expected to complete their coursework and comprehensive examinations at a higher level than is expected of terminal master’s degree candidates.
Doctor of Philosophy - Mathematics
Admission
A candidate for admission to the Ph.D. program in mathematics must fulfill the scholarship requirements for graduate study presented in the section of this catalog on graduate education and should have a strong undergraduate background in the mathematical sciences.
Degree Requirements
A student advances to candidacy for the degree by doing the following:
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Passing 72 units of 200-level graduate mathematics courses with a grade of at least B or S in each course (grades for coursework satisfying area requirements must meet the minimum required A- average). These 72 units must include at least one further full-year graduate sequence not being used to satisfy requirement (b).
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Satisfying three area requirements, normally algebra and analysis, plus a third area to be determined in consultation with the graduate advisor. S/U grading is not allowed in coursework used to satisfy area requirements.
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Passing an oral qualifying examination on the proposed plan and subject matter for the doctoral dissertation and on mathematical topics related to the student’s research.
After advancing to candidacy, the student completes the requirements for the degree by submitting an acceptable dissertation representing an original mathematical contribution, and defending this dissertation before a faculty committee.
Optional Graduate Degree Emphasis in Computational Science and Engineering
The Departments of Chemical Engineering, Computer Science, Electrical and Computer Engineering, Earth Science, Mathematics, and Mechanical and Environmental Engineering offer an interdisciplinary master's and Ph.D. degree emphasis in Computational Science and Engineering (CSE). Detailed program information can be found at www.cse.ucsb.edu.
CSE is a rapidly growing multi-disciplinary area with connections to the sciences, engineering, mathematics, and computer science. Computer models and simulations have become an important part of the research repetoire, supplementing (and in some cases replacing) experimentation. Going from application area to computational results requires domain expertise, mathematical modeling, numerical analysis, algorithm development, software implementation, program execution, analysis, validation, and visualization of results. CSE addresses these issues.
Although CSE includes elements from computer science, applied mathematics, engineering and science, it focuses on the integration of knowledge and methodologies from all of these disciplines and, as such, is a subject distinct from any of them.
All students pursuing an emphasis in CSE must complete the following:
- Numerical Methods: Mathematics 206A-B-C-D (students must take at least three)
- Parallel Computing: Computer Science 240A-B (students must take at least one)
- Applied Mathematics: Students must take a two course sequence from either the Mathematics 243A-B or the Mathematics 246A-B sequence
The specific requirements for the M.A. in Mathematics (thesis option only) with the CSE emphasis are as follows:
- The completion of the above requirements for an M.A. in mathematics
- A master’s thesis in CSE
The thesis must be written under the supervision of a CSE ladder faculty member. The thesis committee must include a minimum of three permanent ladder faculty members, at least two from Mathematics and one from CSE (may be CSE faculty member from another department).
Students pursuing a Ph.D. with an emphasis in CSE must:
- Complete the above requirements for a Ph.D. in mathematics
- Write and defend a dissertation in CSE
The student’s dissertation must be written under the supervision of a Mathematics CSE ladder faculty member. The doctoral examination committee must include at least one CSE ladder faculty member and at least one ladder faculty member from another department.
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Mathematics Courses
3A. Calculus with Applications, First Course
(4) Staff
Prerequisite: Algebra Diagnostic Test.
Reduced credit of 2 units will be given to students who have received credit for Mathematics 2A or 34A. Not open for credit to students who have completed Mathematics 3AS.
Students with Advanced Placement credit should contact the department.
Differential Calculus including analytic geometry, functions and limits, derivatives, techniques and applications of differentiation, logarithmic and trigonometric functions.
3B. Calculus with Applications, Second Course
(4) Staff
Prerequisite: Mathematics 3A with a minimum grade of C.
Not open for credit to students who have completed Mathematics 3BS. Reduced credit of 2 units will be given to students who have received credit for Mathematics 34B.
Students with Advanced Placement credit should contact the department.
Integral calculus including definite and indefinite integrals, techniques of integration, with applications in mathematics and physics.
3BI. Inquiry Based Calculus I
(4) Staff
Prerequisite: AP score of 3 or higher; consent of instructor.
Not open for credit to students who have completed Mathematics 3B. Reduced credit of 2 units will be given to students who have received credit for Mathematics 34B.
Honors version of Mathematics 3B. Mathematical inquiry course is developed through problem solving and discovery.
3C. Differential Equations and Linear Algebra, First Course
(4) Staff
Prerequisite: Mathematics 3B with a minimum grade of C.
Not open for credit to students who have completed Mathematics 3CS or 3CI.
First order ODEs including direction fields, separation of variables, first order linear equations, growth and decay, nonlinear models. Linear algebra including systems of linear equations, matrix inverses, determinants, vector spaces and subspaces, basis and dimension.
3CI. Inquiry Based Calculus II
(4) Staff
Prerequisite: Mathematics 3B or Mathematics 3BI with a minimum grade of C.
Not open for credit to students who have completed Mathematics 3C.
Honors version of Mathematics 3C. Mathematical inquiry course is developed through problem solving and discovery.
3H. Honors Seminar, Calculus
(1) Staff
Prerequisites: concurrent enrollment in Mathematics 3A or 3B or 3C.
May be repeated for credit to a maximum of 3 units.
A supplement to the Mathematics 3 sequence emphasizing fundamental concepts and applications. Intended for highly motivated and well prepared students.
5A. Differential Equations and Linear Algebra, Second Course
(4) Staff
Prerequisite: Mathematics 3C or 3CI with a grade of C or better.
Second order linear ODEs, linear transformations including eigenvalues, eigenvectors and diagonalization. Linear systems of ODEs. Nonlinear systems and linearization.
5AI. Inquiry Based Calculus III
(4) Staff
Prerequisite: Mathematics 3C or 3CI with a minimum grade of C.
Not open for credit to students who have completed Mathematics 5A.
Honors version of Mathematics 5A. Mathematical inquiry course is developed through problem solving and discovery.
5B. Vector Calculus with Applications, First Course
(4) Staff
Prerequisites: Mathematics 5A or 5AI with a grade of C or better.
Differential Calculus of Functions of Several Variables. Gradient, Divergence, Curl. Double Integrals and Triple Integrals. Line Integrals in the Plane. Green’s Theorem and Independence of Path.
5BI. Inquiry Based Calculus IV
(4) Staff
Prerequisite: Mathematics 5A or 5AI with a minimum grade of C.
Not open for credit to students who have completed Mathematics 5b.
Honors version of Mathematics 5B. Mathematical inquiry course is developed through problem solving and discovery.
5C. Vector Calculus with Applications, Second Course
(4) Staff
Prerequisites: Mathematics 5B or 5BI with a grade of C or better.
Line Integrals in Space, Surface Integrals. Divergence Theorem, Stokes’s Theorem. Infinite Series, Fourier Series. Introduction to PDE.
5H. Honors Seminar, Advanced Calculus and Linear Algebra
(1) Staff
Prerequisites: concurrent enrollment in Mathematics 5A or 5B or 5C.
May be repeated for credit to a maximum of 3 units.
A supplement to the Mathematics 5 sequence emphasizing fundamental concepts and applications. Intended for highly motivated and well prepared students.
8. A Transition to Higher Mathematics
(5) Staff
Prerequisite: Mathematics 3B with a minimum grade of C.
Introduction to the elements of propositional logic, techniques of mathematical proof, and fundamental mathematical structures including sets, functions, relations, and other topics as time permits. Mastery of this material is essential for students planning to major in mathematics.
15. Precalculus
(4) Staff
Prerequisite: a score at the required level on the Algebra Diagnostic Test.
Students who have earned a grade of C or better in a course with a prerequisite including algebra or trigonometry may not receive credit for this course.
A functional approach integrating algebra and trigonometry. Topics include: one-to-one and onto functions; inverse functions; properties and graphs of polynomial, rational, exponential, and logarithmic functions; properties and graphs of inverse trigonometric identities; and trigonometric equations.
34A. Calculus for Social and Life Sciences
(4) Staff
Not open for credit to students who have completed Mathematics 3A.
Introduction to differential and integral calculus with applications to modeling in the biological sciences.
34B. Calculus for Social and Life Sciences
(4) Staff
Prerequisite: Mathematics 3A or 3AS or 34A with a grade of C or better.
Not open for credit to students who have completed Mathematics 3B or 3BS.
Continued study of differential and integral calculus with applications. Introduction to mathematical modeling with differential equations. Calculus of several variables including an introduction to partial derivatives.
91. Workshops in Mathematics
(1) Staff
May be repeated for credit to a maximum of 4 units.
Group workshops affiliated with selected lower-division mathematics courses.
94. Group Studies in Mathematics
(1-4) Staff
Prerequisite: consent of instructor.
Lectures and discussions on special topics.
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100A. Mathematics for Elementary Teaching, I
(3) Staff
Prerequisite: upper-division standing.
Course cannot be used to satisfy any mathematics major or minor requirements.
This class teaches ways to think about and explain elementary school mathematics. Topics include: cultural and base-n number systems, algorithms, elementary number theory, probability, and graphing.
100B. Mathematics for Elementary Teaching, II
(3) Staff
Prerequisite: Mathematics 100A.
Course cannot be used to satisfy any mathematics major or minor requirements.
Completes the explanation of elementary school mathematics by discussing geometry and algebra. Discusses the pedagogy with the California Mathematics Framework, the NCTM Standards, and “replacement units.”
101A. Classical Number Systems
(4) Staff
Prerequisites: Mathematics 3A and 8.
Not open for credit to students who have completed Mathematics 118A.
Especially suitable for prospective teachers. A conceptual rather than an axiomatic development starting with the natural numbers and progressing through the integral, rational, real, and complex number systems. The historical implications of these developments in number systems.
101B. Mathematical Systems
(4) Staff
Prerequisite: Mathematics 101A.
Not open for credit to students who have completed Mathematics 118A.
Especially suitable for prospective teachers. The theory of operations within rings and fields and the foundations of the real number system. Ideals, quotient rings, and factorization theorems. The history and the historical implications of these developments in mathematical systems.
102A-B. Modern Euclidean and Noneuclidean Geometry
(4-4) Staff
Prerequisites: Mathematics 3B (for 102A): Mathematics 102A (for 102B).
Especially suitable for prospective teachers. Topics in plane and solid geometry. The axioms of pure, Euclidean, projective, and noneuclidean geometry. Transformational geometry (isometries, dilitations, involutions, perspectivities, and projectivities). The history and the historical implications of these developments in geometry.
103. Introduction to Group Theory
(4) Staff
Prerequisite: Mathematics 8.
Not open for credit to students who have completed Mathematics 111A.
Intended primarily for prospective teachers. Introduction to group theory. Permutation groups, cyclic groups, theory of finite groups, group homomorphisms and isomorphisms, and Abelian groups. Applications to number theory and geometry.
104A. Introduction Into Numerical Analysis
(4) Staff
Prerequisites: Mathematics 5A-B-C; and, Computer Science 5AA-ZZ or 10 or 11AA-ZZ or 12 or Engineering 3.
Numerical methods for the solution of nonlinear equations (Newton method), for integration (quadrature formulas and composite integration), and for the initial value problem for ordinary differential equations (Euler and Kutta methods).
104B. Numerical Analysis
(4) Staff
Prerequisite: Mathematics 104A.
Numerical methods for the solution of systems of linear equations (direct and interactive methods), and the finite difference methods for boundary value problems for (ordinary and partial) differential equations.
104C. Advanced Topics in Numerical Analysis
(4) Staff
Prerequisite: Mathematics 104B
Topics in approximation theory; numerical methods for finding eigenvalues of a matrix; and advanced topics in numerical methods for ordinary and partial differential equations.
108A. Introduction to Linear Algebra
(4) Staff
Prerequisites: Mathematics 5A and 8.
Abstract Vector spaces and subspaces. Span and linear independence. Basis and dimension. Linear maps. Eigenvalues and eigenvectors.
108B. Advanced Linear Algebra
(4) Staff
Prerequisite: Mathematics 108A.
Diagonalization, inner product spaces, projections, least-squares approximations, invariant factors and elementary divisors, canonical forms, topics from advanced matrix theory, applied linear algebra, and group representation theory.
111A. Introduction to Abstract Algebra
(4) Staff
Prerequisite: Mathematics 108A.
An introduction to algebraic structures with an emphasis on groups.
111B-C. Abstract Algebra
(4-4) Staff
Prerequisite: Mathematics 111A (for Mathematics 111B): Mathematics 111B (for Mathematics 111C).
Rings, fields, Galois theory.
113. Non-Euclidean Geometry
(4) Staff
Prerequisite: Mathematics 8.
An introduction to hyperbolic geometry with some discussion of other non-Euclidean systems.
115A-B. Introduction to Number Theory
(4-4) Staff
Prerequisite: Mathematics 8 (for 115A): Mathematics 115A (for 115B).
Divisibility, congruences, primitive roots and indices, quadratic residues and the quadratic reciprocity law, number-theoretic functions. Diophantine equations, the distribution of primes, number-theoretic methods in cryptography, quadratic forms, continued fractions and the approximation of real numbers, algebraic number theory, partitions.
115C. Topics in Number Theory
(4) Staff
Prerequisite: consent of instructor.
Recommended preparation: Mathematics 115A-B; consult the department or instructor for details.
Selected topics in number theory at the direction of the instructor.
116. Combinatorial Analysis
(4) Staff
Prerequisite: Mathematics 8.
Elementary counting principles, binomial coefficients, generating functions, recurrence relations, the principle of inclusion and exclusion, distributions and partitions, systems of distinct representatives, applications to computation.
117. Methods of Analysis
(4) Staff
Prerequisite: Mathematics 8.
Introduction to methods of proof in analysis. Topics include limits, sequences and series, continuity, compactness, as well as other topics. This course is intended to follow Mathematics 8 and to introduce students to the level of sophistication of upper-division mathematics.
118A-B-C. Introduction to Real Analysis
(4-4-4) Staff
Prerequisites: Mathematics 5A-B and 108A-B and 117 (for Mathematics 118A): Mathematics 118A (for Mathematics 118B): Mathematics 118B (for Mathematics 118C).
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
119A. Ordinary Differential Equations
(4) Staff
Prerequisites: Mathematics 5A-B.
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.
119B. Chaotic Dynamics and Bifurcation Theory
(4) Staff
Prerequisites: Mathematics 5A-B-C.
Recommended preparation: Mathematics 119A.
Hyperbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.
122A-B. Introduction to Theory of Complex Variables
(4-4) Staff
Prerequisites: Mathematics 5A-B (for Mathematics 122A): Mathematics 122A (for Mathematics 122B).
Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.
124A. Partial Differential Equations
(4) Staff
Prerequisites: Mathematics 5A-B-C.
Wave, heat, and potential equations.
124B. Fourier Series and Numerical Methods
(4) Staff
Prerequisites: Mathematics 5A-B-C.
Recommended preparation: Mathematics 124A.
Fourier series; generalized functions; and numerical methods.
130. Introduction to Mathematical Modeling
(4) Staff
Prerequisites: Mathematics 5A-B.
Introduction to the principles of mathematical modeling, both discrete and continuous.
132A. Introduction to Operations Research
(4) Staff
Prerequisite: Mathematics 5A.
Linear programming, the simplex method, duality, applications to the transportation and assignment problems, sensitivity analysis, problem formulation.
132B. Introduction to Operations Research
(4) Staff
Prerequisites: Mathematics 5B and 132A.
Network analysis: shortest route, minimal spanning tree and maximal flow problems; PERT including the critical path method; dynamic programming; game theory; integer programming, nonlinear programming.
137A-B. Graph and Network Theory
(4-4) Staff
Prerequisites: Mathematics 5A and 8 (for Mathematics 137A): Mathematics 137A (for Mathematics 137B).
Elements of graph and network theory including paths, circuits, trees, coloring, planarity, matching theory, Hall’s Theorem, applications to scheduling theory, flows in networks, Menger’s Theorem, and other topics as time permits.
145. Introduction to Topology
(4) Staff
Prerequisite: Mathematics 8.
Metric spaces, continuity, compactness, classification of surfaces, Euler characteristics, and fundamental groups. Additional topics at the discretion of the instructor.
147A-B. Introductory Differential Geometry
(4-4) Staff
Prerequisites: Mathematics 5B; and, Mathematics 108A or 117 (for Mathematics 147A): Mathematics 147A (for Mathematics 147B).
Curves and surfaces in three-dimensional Euclidean space, first and second fundamental forms, Gaussian and mean curvature, geodesics, Gauss-Bonnet theorem, and non-euclidean geometry.
170. Introduction to Mathematical Finance
(4) Staff
Prerequisites: PSTAT 120A-B and 160A.
Same course as PSTAT 170.
Recommended preparation: PSTAT 160B and 171.
Describes mathematical methods for estimating and evaluating asset pricing models, equilibrium and derivative pricing, options, bonds, and the term-structure of interest rates. Also introduces finance optimization models for risk management and financial engineering.
178. Introduction to Cryptography
(4) Staff
Prerequisites: Computer Science 10; and, PSTAT 120A or 121A or equivalent courses.
An introduction to the basic concepts and techniques of cryptography and cryptanalysis. Topics include: The Shannon Theory, classical systems, the enigma machine, the data encryption standard, public key systems, digital signatures, file security.
181A-B. Advanced Problem Solving: Mathematical, Historical, and Pedagogical Contexts
(4) Staff
Prerequisites: Mathematics 5A; and, an upper-division mathematics course (for Mathematics 181A): Consent of instructor (for Mathematics 181B).
Designed for prospective teachers. Problem solving. Problems in number theory, dynamical systems, or other topics, including investigations of mathematics and its historical contexts. The difference between formal mathematics and the process of doing mathematics. Supervised field work on problem solving.
190. Special Topics in Mathematics
(4) Staff
Prerequisite: consent of instructor.
May be repeated for credit to a maximum of 8 units.
Information about the special topics to be presented may be obtained from the office of the Department of Mathematics.
193. Internship in Mathematics
(1-4) Staff
Prerequisites: consent of instructor and department.
May be repeated for credit to maximum of 4 units, but no credit will be applied toward upper-division major.
Faculty-sponsored academic internship in industrial or research firms.
195A-B. Internship in Mathematics Teaching
(4-4)
Prerequisites: upper-division standing in the major; and two upper-division mathematics courses.
No credit allowed toward the major or minor.
Supervised mathematics teaching internship in local schools and participation in the Mathematics Teaching Seminar on mathematics learning and teaching. A paper on mathematics and its teaching required.
197A. Senior Thesis
(1-4) Staff
Prerequisites: open to senior majors only; consent of department and instructor.
Students must have a minimum overall grade-point average of 3.0 and a 3.5 or better grade-point average in the major. Up to 4 units may apply to the major. Up to 8 units total in all Mathematics 197/199RA courses may apply toward the major.
Independent research under the supervision of a faculty member which will result in a senior thesis. Students will concentrate on reading and gathering material for a thesis.
197B. Senior Thesis
(1-4) Staff
Prerequisites: Mathematics 197A; open to senior majors only; consent of department and instructor.
Students must have a minimum overall grade-point average of 3.0 and a 3.5 or better grade-point average in the major. Up to 4 units may apply to the major. Up to 8 units total in all Mathematics 197/199/199RA courses may apply toward the major.
Independent research under the supervision of a faculty member which will result in a senior thesis. Students will concentrate on writing a thesis
199. Independent Studies in Mathematics
(1-5) Staff
Prerequisites: upper-division standing; completion of two upper-division courses in mathematics; and consent of instructor and department.
Students must have a minimum 3.0 grade-point average for the preceding three quarters and are limited to 5 units per quarter and 30 units total in all 98/99/198/199/199AA-ZZ courses combined. Only 8 units total in all Mathematics 197/199/199AA-ZZ courses may apply toward the major.
Coursework shall consist of academic research supervised by a faculty member on a topic not available in established course offerings.
199RA. Independent Research Assistance
(1-4) Staff
Prerequisites: upper-division standing; completion of two upper-division courses in mathematics; consent of instructor and department.
Students must have a minimum 3.0 grade-point average for the preceding three quarters and are limited to 5 units per quarter and 30 units total in all 198/199/199AA-ZZ courses combined. Only 8 units total in all Mathematics 197/199/199AA-ZZ courses may apply toward the major.
Coursework shall consist of faculty supervised research assistance.
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The department does not offer all the courses listed below each year, but does offer the following courses every year: Mathematics 201A-B-C, 206A-B-C-D, 220A-B-C, 221A-B-C, 240A-B-C and an aditonal first-year graduate sequence in applied mathematics. The department offers approximately eight other one-year courses in mathematics each year.
201A-B-C. Real Analysis
(4-4-4) Staff
Prerequisites: Mathematics 118A-B-C.
Measure theory and integration. Point set topology. Principles of functional analysis. Lp-spaces. The Riesz representation theorem. Topics in real and functional analysis.
202A-B-C. Complex Analysis
(4-4-4) Staff
Prerequisites: Mathematics 118A-B-C or 122A.
Analytic functions. Complex integration, Cauchy’s theorem. Series and product developments. Entire functions. Conformal mappings. Topics in complex analysis.
206A. Matrix Analysis and Computation
(4) Staff
Prerequisite: consent of instructor.
Same course as Computer Science 211A, ME 210A, ECE 210A, Geology 251A, and Chemical Engineering 211A. Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language.
Graduate level matrix theory with introduction to matrix computations. SVDs, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and interative methods for matrix computations.
206B. Numerical Simulation
(4) Staff
Prerequisite: consent of instructor.
Same course as Computer Science 211B, ME 210B, ECE 210B, Geology 251B, and Chemical Engineering 211B. Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language.
Linear multistep methods and Runge-Kutta methods for ordinary differential equations: stability, order and convergence. Stiffness. Differential algebraic equations. Numerical solution of boundary value problems.
206C. Numerical Solution of Partial Differential Equations - Finite Difference Methods
(4) Staff
Prerequisite: consent of instructor.
Same course as Computer Science 211C, ME 210C, ECE 210C, Geology 251C, and Chemical Engineering 211C. Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language.
Finite difference methods for hyperbolic, parabolic and elliptic PDEs, with application to problems in science and engineering. Convergence, consistency, order and stability of finite difference methods. Dissipation and dispersion. Finite volume methods. Software design and adaptivity.
206D. Numerical Solution of Partial Differential Equations - Finite Element Methods
(4) Staff
Prerequisite: consent of instructor.
Same course as Computer Science 211D, ME 210D, ECE 210D, Geology 251D, and Chemical Engineering 211D. Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language.
Weighted residual and finite element methods for the solution of hyperbolic, parabolic and elliptic partial differential equations, with application to problems in science and engineering. Error estimates. Standard and discontinuous Galerkin methods.
209. Set Theory
(4) Staff
Prerequisite: consent of instructor.
Study of axiomatic set theory; topics include relations and functions, orderings, ordinal and cardinal numbers and their arithmetic, transfinite constructible sets, consistency and independence results of Gödel and Cohen.
214A. Ordinary Differential Equations
(4) Staff
Prerequisite: Not open to mathematics majors.
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.
214B. Chaotic Dynamics and Bifurcation Theory
(4) Staff
Prerequisite: Not open to mathematics majors.
Hyperbolic structure and chaos; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.
215A. Partial Differential Equations
(4) Staff
Prerequisite: Not open to mathematics majors.
Wave, heat, and potential equations.
215B. Fourier Series and Numerical Methods
(4) Staff
Prerequisite: Not open to mathematics majors.
Fourier series; generalized functions; and numerical methods.
220A-B-C. Modern Algebra
(4-4-4) Staff
Prerequisites: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
221A. Foundations of Topology
(4) Staff
Prerequisite: Mathematics 118A or equivalent.
Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Other topics as time allows.
221B. Homotopy Theory
(4) Staff
Prerequisite: Mathematics 221A.
Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.
221C. Differential Topology
(4) Staff
Prerequisite: Mathematics 221A.
Topological manifolds, differential manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefchetz fixed point theorem, vector fields.
225A-B. Topics in Number Theory
(4-4) Staff
Prerequisites: Mathematics 220A-B-C.
May be repeated for credit with instructor and department approval.
Selected topics in number theory.
227A-B-C. Advanced Topics in Geometric and Algebraic Topology
(4-4-4) Staff
Prerequisite: consent of instructor.
May be repeated for credit with instructor and department approval.
Topics, varying from year to year, include piecewise linear and differential topology, manifolds, fiber bundles and fiber spaces, homotopy theory, and spectral sequences.
229A-B-C. Operator Algebras
(4-4-4) Staff
Prerequisites: Mathematics 201A-B-C.
Bananch algebras. The Gelfand transform.
C*-algebras and von Neumann algebras. Positivity. States. The Gelfand-Naimark-Segal construction, *-representations of C*-algebras. Von Neumann’s bicommutant theorem. Kaplansky’s density theorem. Comparison of projections. Examples and applications. Advanced topics in the theory of operator algebras.
231A. Lie Groups and Lie Algebras
(4) Staff
Prerequisite: consent of instructor.
Differentiable manifolds, definition and examples of Lie groups, Lie group-Lie algebra correspondence, nilpotent and solvable Lie algebras, classification of semi-simple Lie algebras over the complexes, representations of Lie groups and Lie algebras, special topics.
232A-B. Algebraic Topology
(4-4) Staff
Prerequisites: Mathematics 108A-B and 145.
Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.
236A-B. Homological Algebra
(4-4) Staff
Prerequisites: Mathematics 220A-B-C.
Algebraic construction of homology and cohomology theories, aimed at applications to topology, geometry, groups and rings. Special emphasis on hom and tensor functors; projective, injective and flat modules; exact sequences; chain complexes; derived functors, in particular, ext and tor.
237A-B. Algebraic Geometry
(4-4) Staff
Prerequisites: Mathematics 220A-B-C.
Affine/projective varieties, Hilbert’s Nullstellensatz, morphisms of varieties, rational maps, dimension, singular/nonsingular points, blowing up of varieties, tangent spaces, divisors, differentials, Rieman-Roch theorem. Special topics include: elliptic curves, intersection numbers, Bezout’s theorem, Max Noether’s theorem.
240A-B-C. Introduction to Differential Geometry and Riemannian Geometry
(4-4-4) Staff
Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures and Dirac operator, comparison theorems in Riemannian geometry.
241A-B-C. Topics in Differential Geometry
(4-4-4) Staff
Prerequisites: Mathematics 240A-B-C.
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifolds, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
243A-B-C. Ordinary Differential Equations
(4-4-4) Staff
Prerequisites: Mathematics 118A-B-C.
Existence and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant manifolds, existence and stability of periodic solutions, bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.
246A-B-C. Partial Differential Equations
(4-4-4) Staff
Prerequisites: Mathematics 201A-B-C.
First-order nonlinear equations; the Cauchy problem, elements of distribution theory and Sobolev spaces; the heat, wave, and Laplace equations; additional topics such as quasilinear symmetric hyperbolic systems, elliptic regularity theory.
260AA-ZZ. Seminars in Mathematics
(1-6) Staff
Prerequisite: consent of instructor.
May be repeated for credit.
Topics in algebra, analysis, applied mathematics, combinatorial mathematics, functional analysis, geometry, statistics, topology, by means of lectures and informal conferences with members of staff.
500. Teaching Assistant Practicum
(1-4) Staff
Prerequisites: appointment as teaching assistant and departmental approval.
No unit credit allowed toward degree.
Supervised teaching of undergraduate mathematics courses.
501. Teaching Assistant Training
(1-2) Staff
Prerequisites: departmental and instructor approval.
No unit credit allowed toward degree.
Consideration of ideas about the process of learning mathematics and discussion of approaches to teaching.
502. Teaching Associate Practicum
(1-5) Staff
Prerequisite: appointment as associate and departmental approval.
No unit credit allowed toward degree.
Supervised teaching of undergraduate courses.
510. Reading for Area Examinations
(2-6) Staff
Prerequisites: enrollment in M.A. or Ph.D. program; consent of instructor.
596. Directed Reading and Research
(1-6) Staff
Prerequisite: consent of instructor.
May be repeated for credit. Only 8 units total in all Mathematics 596, 598, 599 courses may apply toward the degree.
598. Master’s Thesis Research and Preparation
(1-6) Staff
Prerequisites: graduate standing and consent of instructor.
May be repeated for credit. Only 8 units total in all Mathematics 596, 598, 599 courses may apply toward the degree.
599. Ph.D. Dissertation Preparation
(1-6) Staff
Prerequisite: consent of instructor.
May be repeated for credit. Only 8 units total in all Mathematics 596, 598, 599 courses may apply toward the degree.
