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UC Riverside
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2002-2003 General Catalog
University of California, Riverside
MATHEMATICS
Subject abbreviation: MATH
Faculty | Program | Minor
Undergraduate Curricula | Graduate Curricula Undergraduate Courses | Graduate Courses | Professional Course Reinhard Schultz, Ph.D., Chair
Professors
The Department of Mathematics offers a B.A. and a B.S. degree in programs that share a common, solid mathematical foundation but differ in their specializations in the pure and applied areas of mathematics. These programs can provide the basis for careers in mathematics itself or within the many scientific and business fields, which, in today's technological society, depend on a basic knowledge of mathematical methods. The B.A. in Mathematics, following the liberal arts tradition, combines a broad coverage of the humanities and social sciences with a moderate amount of advanced mathematics in the major. It is selected most often either by students who intend to obtain a teaching credential with a specialty in mathematics or by students who wish to pursue graduate work in business or the social sciences. The B.S. in Mathematics is more technical and contains a greater concentration of work in the major field. The Pure Mathematics program is directed toward students who may wish to pursue graduate work in mathematics. The Applied Mathematics programs, with options in Biology, Chemistry, Economics, Environmental Sciences, Physics, and Statistics, are designed to provide a rigorous training in mathematics together with a substantial background in the discipline of the option. The Computational Mathematics program is designed to prepare the student for professional work with computers and computer systems and for graduate work in computer science. Academic Advising Each Mathematics major is assigned a departmental advisor who assists the student in formulating educational goals and monitors the student's subsequent progress in an academic program. Each quarter a study list must be approved by this advisor. Teaching Credential Teachers in the public schools in California must have a credential approved by the State Commission on Teacher Credentialing. The credential requires an undergraduate major, baccalaureate degree, and completion of a graduate credential program such as that offered by the Graduate School of Education at UCR. The latter usually requires three quarters and includes education courses and supervised teaching. Before admission and student teaching in a graduate credential program, the candidate must pass the California Basic Education Skills Test (CBEST) and demonstrate subject-matter proficiency in the fields which the candidate will teach. The candidate can demonstrate proficiency either by passing the commission's subject-matter assessment examination, or preferably, completing an undergraduate program that is state approved for teacher preparation. UCR has an approved undergraduate program (Subject Matter Preparation Program) for mathematics majors who plan to teach secondary level grades (7–12). A breadth of course work is necessary, in addition to the specified requirements for the major. Students are urged to start early, preferably as freshmen, selecting courses most helpful for this career. Further information about courses, requirements, and examinations can be obtained at the Student Affairs Office (202 Surge Building) and the Graduate School of Education (1124 Sproul Hall). Degree Requirements University Requirements See the Undergraduate Studies section for requirements that all students must satisfy. College Requirements See Degree Requirements, College of Natural and Agricultural Sciences, in the Undergraduate Studies Section, for requirements that students must satisfy. To fulfill the Natural Sciences requirement, the Department of Mathematics requires the following: 1. One of the year sequences
b) CHEM 001A, CHEM 001B, CHEM 001C c) PHYS 002A, PHYS 002B, PHYS 002C or PHYS 040A, PHYS 040B, PHYS 040C The major requirements for the B.A. and B.S. degrees in Mathematics are as follows: For the Bachelor of Arts
For the Bachelor of Science Lower-division requirements for all programs are MATH 009A, MATH 009B, MATH 009C, MATH 010A, MATH 010B, MATH 046, CS 010 (CS 012 is recommended). 1. Pure Mathematics program (56 units)
b) At least three courses from (a) above must be from MATH 145A, MATH 145B, MATH 151A, MATH 151B, MATH 151C c) Courses in the MATH 190–199 series are excluded d) Twenty (20) additional units of upper-division mathematics, upper-division computer science, or other related courses approved by the undergraduate advisor (For students who wish to pursue graduate work, courses in complex variables, differential equations, and probability may be particularly useful.) 2. Applied Mathematics programs MATH 113 or MATH 131, MATH 132, MATH 146A, MATH 146B, MATH 146C and the courses in one of the following options:
(2) MATH 149A (3) Three courses from MATH 120, MATH 121, MATH 135A, MATH 135B, MATH 149B, MATH 149C (4) BIOL 102, BIOL 105, BIOL 108, BIOL 117 (5) Four (4) additional units of upper-division biology
(2) Either PHYS 040A, PHYS 040B, PHYS 040C (preferred); or PHYS 002A, PHYS 002B, PHYS 002C (3) Four courses from MATH 120, MATH 135A, MATH 135B, MATH 149A, MATH 149B, MATH 149C, MATH 165A, MATH 165B (4) CHEM 110A, CHEM 110B, CHEM 111, CHEM 113 (5) Four (4) additional units of upper-division chemistry
(2) Twenty (20) units of upper-division economics to consist of ECON 102A and four courses to be chosen from ECON 102B, ECON 103A, ECON 103B, ECON 107, ECON 108, ECON 110, ECON 111, ECON 134/BSAD 134, ECON 135, ECON 143A/ENSC 143A, ECON 143B/ENSC 143B, ECON 143C/ENSC 143C, ECON 156, ECON 206
(2) ECON 006/ENSC 006 (3) GEO 001 is recommended (4) MATH 149A (5) Three courses from MATH 120, MATH 121, MATH 135A, MATH 135B, MATH 149B, MATH 149C, CS 177, STAT 155 (6) ENSC 100, ENSC 100L, ENSC 101, ENSC 102 (7) Eight (8) additional units of upper-division environmental sciences
(2) Either MATH 120 or MATH 171 (3) PHYS 130A, PHYS 130B (4) Either PHYS 135A, PHYS 135B, PHYS 136 or PHYS 156A, PHYS 156B
(2) Either STAT 130 or STAT 146 (3) STAT 161, STAT 170A, STAT 170B, STAT 171
b) CS 012, CS 014, CS 141, CS 150 c) One additional CS course to be chosen from the list of approved technical elective courses. d) Twenty-four (24) units of technical electives to be chosen from
(2) CS 130, CS 133, CS 166, CS 170, CS 171, CS 177 Mathematics Honors Program Candidates for the Honors Program in Mathematics must complete
It is the responsibility of the honors candidates to notify the department of their eligibility. The following are the requirements for a minor in Mathematics.
Students with a minor in Mathematics should consult with a faculty advisor in Mathematics to construct a specific program consistent with their goals. See Minors under the College of Natural and Agricultural Sciences in the Undergraduate Studies section of this catalog for additional information on minors. Education Abroad Program The Mathematics Department encourages eligible students to participate in the Education Abroad Program (EAP). The EAP is an excellent opportunity to travel and learn more about another country and its culture while taking courses which earn units toward graduation. In addition to year-long programs, a wide range of shorter options is available. While on EAP, students are still eligible for financial assistance. Students are advised to plan study abroad well in advance to ensure that the courses taken fit with their overall program at UCR. Consult the departmental student affairs officer for assistance. For further details see the University of California's EAP Web site at www.uoeap.ucsb.edu or contact UCR's International Services Center at (909) 787-4113. See Education Abroad Program under International Services Center in the Student Services section of this catalog. A list of participating countries is found under Education Abroad Program in the Curricula and Courses section. Domestic applicants to graduate programs in the Department of Mathematics must supply GRE scores for the General Test (verbal, quantitative, and analytical). M.A. or M.S. in Mathematics General university requirements are listed in the Graduate Studies section of this catalog. Specific requirements of the department are as follows:
M.S. in Mathematics (Applied) General university requirements are listed in the Graduate Studies section of this catalog. Specific requirements of the department are as follows:
Doctoral Degree in Mathematics Specific requirements of the department are as follows:
Normative Time to Degree 15 quarters Mathematics placement examinations are scheduled each year before the fall quarter begins. They are mandatory for entering freshmen and recommended for advanced standing students who wish to enroll in MATH 003, MATH 005, MATH 009A, MATH 014, MATH 015, MATH 022, or MATH 023. To qualify for MATH 009A, MATH 022, and MATH 023, a student must score at least 36 (60%) on the Pre-Calculus Examination. To qualify for MATH 005, MATH 014, and MATH 015, a student must score at least 18 (30%) on the Pre-Calculus Examination or at least 27 (60%) on the Mathematical Analysis Examination. MATH 003. Basic Algebra. (0) Lecture, two hours; laboratory, four hours. Prerequisite(s): an appropriate score on the Pre-calculus Exam or the Math Analysis Readiness Exam as determined by the Mathematics Department. Covers basic algebra, including linear functions and equations, quadratic functions and equations, and operations with functions. Designed to prepare students for MATH 005. Does not meet any mathematics or physical science requirement. Carries workload credit equivalent to 4 units but does not count towards graduation units. MATH 005. Introduction to College Mathematics. (5) Lecture, four hours; discussion, one hour. Prerequisite(s): MATH 003 with a grade of "C-" or better or equivalent, or a sufficiently high test score on the Mathematical Analysis Examination, as determined by the Mathematics Department. A study of inequalities, absolute value, functions, graphing, logarithms, trigonometry, roots of polynomials, and other elementary concepts of mathematics. MATH 009A. First-Year Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 005 with a grade of "C-" or better or equivalent. Introduction to the differential calculus of functions of one variable. Credit is awarded for only one of MATH 009A or MATH 09HA. MATH 009B. First-Year Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009A with a grade of "C-" or better or MATH 09HA with a grade of "C-" or better. Introduction to the integral calculus of functions of one variable. Credit is awarded for only one of MATH 009B or MATH 09HB. MATH 009C. First-Year Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009B with a grade of "C-" or better or MATH 09HB with a grade of "C-" or better. Further topics from integral calculus, improper integrals, infinite series, Taylor's series, and Taylor's theorem. Credit is awarded for only one of MATH 009C or MATH 09HC. MATH 09HA. First-Year Honors Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): a score of 47 or higher on the Mathematics Placement Precalculus Examination. Honors course corresponding to MATH 009A for students with strong mathematical backgrounds. Emphasis is on theory and rigor. Credit is awarded for only one of MATH 009A or MATH 09HA. MATH 09HB. First-Year Honors Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): a score of 4 or higher on the AB Advanced Placement Test in Mathematics or MATH 009A with a grade of "A-" or better or MATH 09HA with a grade of "A-" or better. Honors course corresponding to MATH 009B for students with strong mathematical backgrounds. Emphasis is on theory and rigor. Credit is awarded for only one of MATH 009B or MATH 09HB. MATH 09HC. First-Year Honors Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009B with a grade of "A-" or better or MATH 09HB with a grade of "A-" or better. Honors course corresponding to MATH 009C for students with strong mathematical backgrounds. Emphasis is on theory and rigor. Credit is awarded for only one of MATH 009C or MATH 09HC. MATH 010A. Calculus of Several Variables. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009B with a grade of "C-" or better or MATH 09HB with a "C-" or better or equivalent. Topics include Euclidean geometry, matrices and linear functions, determinants, partial derivatives, directional derivatives, Jacobians, gradients, chain rule, and Taylor's theorem for several variables. MATH 010B. Calculus of Several Variables. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009C with a grade of "C-" or better or MATH 09HC with a grade of "C-" or better; MATH 010A with a grade of "C-" or better; or equivalents. Covers vectors; differential calculus, including implicit differentiation and extreme values; multiple integration; line integrals; vector field theory; and theorems of Gauss, Green, and Stokes. MATH 014. Mathematics, A Humanistic Approach. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 003 or equivalent. Intended to fulfill the breadth requirement for students outside the natural and agricultural sciences. A survey of numerical and logical methods illustrating the role of mathematics in the development of civilization. Topics will include integral, rational, and irrational numbers; number systems; infinity; the concept of proof; as well as a glimpse of calculus. Only one of MATH 014 or MATH 015 may be taken for credit. MATH 015. Liberal Arts Mathematics. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 003 or equivalent. Designed to fulfill the breadth requirement for students outside the natural and agricultural sciences. Illustrates the interaction of mathematics with other subject areas through the study of selected topics of contemporary mathematics. Topics are chosen from discrete mathematics, counting and probability, and the interaction between algebra and geometry. MATH 022. Calculus for Business. (5) Lecture, three hours; discussion, two hours. Prerequisite(s): MATH 005 with a grade of "C-" or better or MATH 023. Explores relations and functions (linear, polynomial, logarithmic, and exponential), differential calculus of functions of one and two variables, and integration (indefinite and definite) with applications to business and economic problems. Credit is not awarded for MATH 022 if a grade of "C-" or better has already been awarded for MATH 009A or MATH 09HA. MATH 023. Applied Matrix Algebra. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 005 with a grade of "C-" or better or MATH 022 or MATH 009A or MATH 09HA or equivalent. A study of matrix operations, linear dependence and independence, ranks and inverses, systems of linear equations, determinants, eigenvalues, and eigenvectors with business and economic applications. Designed for students who are not Mathematics majors. MATH 046. Introduction to Ordinary Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009C with a grade of "C-" or better or MATH 09HC with a grade of "C-" or better or equivalent. Introduction to first-order equations, linear second-order equations, series solutions, and Laplace transforms, with applications to the physical and biological sciences. UPPER-DIVISION COURSES MATH 112. Finite Mathematics. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009A, CS 010. Introduction to the basic concepts of finite and structural mathematics with emphasis on applications to computer science. Topics include axiomatic systems, combinatorics, propositional and predicate calculi, graph theory, trees, state diagrams, networks, induction, elementary enumeration, and recurrence relations. MATH 113. Applied Linear Algebra. (5) Lecture, three hours; discussion, two hours. Prerequisite(s): concurrent enrollment in or completion of MATH 010A. Study of matrices and systems of linear equations, determinants, Gaussian elimination and pivoting, vector spaces, linear independence and linear transformation, orthogonality, eigenvalues, and eigenvectors. Also examines selected topics and applications. Integrates numerical linear algebra and extensive computer use with these topics. Credit is awarded for only one of MATH 113 or MATH 131. MATH 120. Optimization. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A; MATH 113 or MATH 131 (may be taken concurrently). Introduction to classical optimization, including unconstrained and constrained problems in several variables, Jacobian and Lagrangian methods, and the Kuhn-Tucker conditions. Covers the basic concepts of linear programming, including the simplex method and duality, with applications to other subjects. MATH 121. Game Theory. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A. Games in extensive, normal, and characteristic form as models of conflict and/or cooperation. Two-person zero-sum games, minimax theorem, relation to linear programming. Non-zero-sum games, Nash equilibrium theorem, bargaining, the core, Shapley value. Economic market games. MATH 125A. Introduction to Combinatorics. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009C or MATH 09HC; MATH 112. Introduction to elements of graph theory and the theory of counting. MATH 125B. Introduction to Combinatorics. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009C or MATH 09HC; MATH 112; MATH 125A. Continuation of MATH 125A. Topics include the principle of inclusion-exclusion, the Hall matching theorem, and combinatorial designs. MATH 131. Linear Algebra I. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): concurrent enrollment in or completion of MATH 010A. An introduction to vector spaces, matrices, and linear transformations. Credit is awarded for only one of MATH 113 or MATH 131. MATH 132. Linear Algebra II. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 113 with a grade of "C-" or better or MATH 131 with a grade of "C-" or better or equivalent. Further study of topics in linear algebra, including eigenvalues. Exploration of Hermitian and unitary matrices, positive definite matrices, and canonical forms. MATH 133. Geometry. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 113 or MATH 131 or consent of instructor. Elementary theory of affine and projective planes, the line at infinity, finite geometries, Euclidean and non-Euclidean geometries, groups of transformations, and other algebraic structures related to geometry. MATH 135A. Numerical Analysis. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): CS 010 or equivalent; MATH 113 or MATH 131 (may be taken concurrently). A study of numerical methods for determining solutions to nonlinear equations and simultaneous linear equations. Topics also include interpolation, techniques of error analysis, and computer applications. MATH 135B. Numerical Analysis. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): CS 010; MATH 113 or MATH 131; MATH 135A. Continuation of MATH 135A. Explores numerical methods, numerical integration, and the numerical solution of ordinary differential equations. MATH 136. Introduction to the Theory of Numbers. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 113 or MATH 131. Prime and composite integers, number theoretic functions, diophantine equations, congruences, quadratic reciprocity, additive arithmetic. MATH 137A. Plane Curves. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B, MATH 132. A study of the complex projective plane, homogeneous polynominals, plane curves, intersection multiplicities, and Bezout's theorem. MATH 137B. Plane Curves. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B, MATH 132, MATH 137A. Topics include simple and singular points, tangents, and duality; the structure of cubic curves; and birational transformations and the resolution of singularities. MATH 138A. Introduction to Differential Geometry. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 113 or MATH 131. Elementary theory of curves and surfaces. First and second fundamental forms. MATH 138B. Introduction to Differential Geometry. (4) S Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B, MATH 138A. Gaussian curvature; geodesics; Gauss-Bonnet Theorem. MATH 144. Introduction to Set Theory. (4) lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A. Algebra of subsets of a set. Algebra of relations and functions. Cardinal and ordinal numbers and their arithmetic operations. The well-ordering theorem, transfinite induction, and Zorn's lemma. MATH 145A. Introduction to Topology. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 144. Elementary topology in metric spaces. MATH 145B. Introduction to Topology. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 145A. Geometric topology, algebra associated with finite complexes and applications. MATH 146A. Ordinary and Partial Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046. Focuses on the theory of linear differential equations and transform methods. MATH 146B. Ordinary and Partial Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 146A. Further study of theory of linear differential equations and problems in valuing ordinary differential equations. MATH 146C. Ordinary and Partial Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 146A, MATH 146B. Explores boundary value problems for partial differential equations, orthogonal expansions, and separation of variables. MATH 149A. Probability and Mathematical Statistics. (4) Lecture, three hours; laboratory, one hour. Prerequisite(s): MATH 010A, MATH 010B, completion of or concurrent enrollment in MATH 046. An introduction to the mathematical theory of probability and discrete and continuous distributions. Credit is awarded for only one of the MATH 149A, MATH 149B, and MATH 149C or STAT 160A, STAT 160B, and STAT 160C sequences. MATH 149B. Probability and Mathematical Statistics. (4) Lecture, three hours; laboratory, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 149A. Continuation of MATH 149A. Topics include sampling and limit distributions. Credit is awarded for only one of the MATH 149A, MATH 149B, and MATH 149C or STAT 160A, STAT 160B, and STAT 160C sequences. MATH 149C. Probability and Mathematical Statistics. (4) Lecture, three hours; laboratory, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 149A, MATH 149B. Continuation of MATH 149B. Topics include tests of hypotheses, estimation, maximum likelihood techniques, regression, and correlation. Credit is awarded for only one of the MATH 149A, MATH 149B, and MATH 149C or STAT 160A, STAT 160B, and STAT 160C sequences. MATH 151A. Advanced Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 145A; or consent of instructor. Involves a rigorous development of mathematical analysis, real and complex numbers, sequences and series, continuity, differentiation, and the Riemann-Stieltjes integral. MATH 151B. Advanced Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 145A, MATH 151A; or consent of instructor. Continuation of MATH 151A. Topics include sequences and series of functions and functions of several variables. MATH 151C. Advanced Calculus. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A, MATH 010B, MATH 046, MATH 145A, MATH 151A, MATH 151B; or consent of instructor. Continuation of MATH 151B. Further study of several variables, integration of differential forms, and Lebesgue integration. MATH 153. History of Mathematics. (4) S Lecture, three hours; discussion, one hour or term paper, three hours. Prerequisite(s): MATH 009C or consent of instructor. A survey from a historical point of view of various developments in mathematics with particular emphasis on the nineteenth and early twentieth centuries. MATH 165A. Introduction to Complex Variables. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B. An introduction to the theory of analytic functions of a complex variable. Includes mappings by elementary functions, complex integrals, as well as Cauchy's theorem, power series, and Laurent series. MATH 165B. Introduction to Complex Variables. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B, MATH 165A. Topics include the theory of residues, conformal mapping, and applications to physical problems. MATH 171. Introduction to Modern Algebra. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 131, MATH 144. An introduction to the fundamental concepts of modern algebra: groups, subgroups, quotient groups, homomorphisms, symmetry groups, fundamental properties of rings, integral domains, ideals, and quotient rings. MATH 172. Modern Algebra. (4) S Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 171. Fundamental concepts of modern algebra: groups, fields, polynomials, geometric constructions, algebraic coding, boolean algebras. MATH 190. Special Studies. (1-5) To be taken with the consent of the chair of the department as a means of meeting special curricular problems. Course is repeatable. MATH 191 (E-Z). Seminar in Mathematics. (1-4) Seminar, one to four hours. Prerequisite(s): upper-division standing or consent of instructor. Additional prerequisites may be required for segments of this course; see Department. Consideration of selected current problems in mathematics. MATH 194. Independent Reading. (1-2) Independent reading in materials not covered in course work. Normally taken in the senior year. Total credit for MATH 194 may not exceed four units. MATH 198-I. Internship in Mathematics. (1-4) Variable hours. Prerequisite(s): upper-division standing, with at least 12 units of upper-division credits toward the major. An academic internship to provide the student with career experience as a mathematician in a governmental, industrial, or research unit under the joint supervision of an off-campus sponsor and a faculty member in Mathematics. Each individual program must have the prior approval of both supervisors and the department chair. A final written report is required. Graded Satisfactory (S) or No Credit (NC). May be repeated for a total of eight units. MATH 201A. Algebra. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 171, MATH 172, or equivalents. Topics include basic theory of groups and rings, the Sylow theorems, solvable groups, the Jordan-Holder theorem. MATH 201B. Algebra. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 201A. Topics include rings, the functors hom and tensor, modules over a principle ideal domain, and applications to matrices. MATH 201C. Algebra. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 201B. Topics include algebraic and transcendental extensions of fields and the Galois theory, and the tensor and exterior algebras. MATH 205A. Topology. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 145B or equivalent. An introduction to pointset topology. MATH 205B. Topology. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 205A or equivalent. Covers homotopy theory and homology theory. MATH 205C. Topology. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 205A, MATH 205B, or equivalents. Covers differential topology. MATH 209A. Real Analysis. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 151C. Topics include Lebesgue measure, integration, and differentiation. MATH 209B. Real Analysis. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 209A. Topics include representation theorems, Hilbert space, Lebesgue spaces, and Banach spaces. MATH 209C. Real Analysis. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 209B. Topics include complex measures, general measure spaces, integration on product spaces, and Lebesgue spaces. MATH 210A. Complex Analysis. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 151C, MATH 165A. Studies include complex analytic functions, Cauchy's theorem, Cauchy's integral formula and the Laurent series, and the residue theorem. MATH 210B. Complex Analysis. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 210A. Studies include entire and meromorphic functions, normal families and the Riemann mapping theorem, and harmonic functions and the Dirichlet problem. MATH 211A. Ordinary Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 151C. Topics include the existence and uniqueness of solutions, linear differential equations, singularities of the first and second kind, self-adjoint eigenvalue problems on a finite interval, and singular self-adjoint boundary-value problems for second-order equations. MATH 211B. Ordinary Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 211A. Topics include the method of averaging and numerical integration, autonomous systems, the method of Liapounov, and stability for linear systems. MATH 212. Partial Differential Equations. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 151C and MATH 165A. Classical theory of initial and boundary value problems for hyperbolic, parabolic and elliptic partial differential equations. MATH 216A. Combinatorial Theory. (4)Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 125B. Introduction to combinatorial optimization and combinatorial geometry including flows on networks, matriods, linear programming, and lattices. MATH 216B. Combinatorial Theory. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 216A. Introduction to combinatorial optimization and combinatorial geometry including optimal programming, exchange properties, Mobius function, Galois connection, and coordinization. MATH 217A. Theory of Probability. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 209A, MATH 209B, MATH 209C. Topics include independence, strong limit theorems including the strong law and the Kolmogorov three series theorem, weak law and the central limit theorem, the Helley-Bray theorem, and Bochner's theorem on positive definite functions. MATH 217B. Theory of Probability. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 217A. Topics include infinitely divisible distributions, the law of the iterated logarithm, and Martingales. MATH 220. Approximation Theory. (4) Lecture, three hours; research, three hours. Prerequisite(s): MATH 209C. The study of the best approximation operator including the classical Chebyshev theory concerning approximations of continuous functions from a fixed finite-dimensional subspace (e.g. nth degree polynomials). Also a study of the minimal projection operator. MATH 221. Several Complex Variables. (4) Lecture, three hours; research, three hours. Prerequisite(s): MATH 151A, MATH 151B, MATH 165A, MATH 165B. Hartog's theorems, domains of holomorphy, pseudoconvexity, Levi's problem, coherent analytic sheaves, Cartan's theorems A and B. MATH 223. Algebraic Number Theory. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 201A. Algebraic number theory, including principal ideal domains, integral independence, algebraic number fields, classical ideal theory in Dedekind domains, classes of ideals, valuations, p-adic number. MATH 224. Introduction to Homological Algebra. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 201C or consent of instructor. Theory of derived functors and its application to rings and associative algebras. MATH 225A. Commutative Algebra. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 201A, MATH 201B, MATH 201C. Studies include basic theory of commutative rings, primary decomposition, integral dependence and valuation rings, and the intersection theorem of Krull. MATH 225B. Commutative Algebra. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 225A. Studies include structure theorems for complete local rings and geometric local rings. MATH 227A. Lie Algebras. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 201A, MATH 201B. Studies include basic definitions, solvable and nilpotent Lie algebras, and structure and classification of semisimple Lie algebras. MATH 227B. Lie Algebras. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 227A. Studies include enveloping algebras and representation theory, representations of semisimple Lie algebras, generalization to Kac-Moody Lie algebras, and modular Lie algebras. MATH 228. Functional Analysis. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 209A, MATH 209B, MATH 209C. Topological linear spaces; function spaces; linear operators; spectral theory; operational calculus; and further selected topics. MATH 229A. Stochastic Processes. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 217A, MATH 217B; or consent of instructor. Topics include sample path analysis of stochastic processes, in particular, separability and regularity properties. Course is repeatable. MATH 229B. Stochastic Processes. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 229A. Topics include martingale and Markov processes, stochastic integration, semimartingales, and stochastic differential equations. Course is repeatable. MATH 232A. Geometry I (Introduction to Manifolds). (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 131 and MATH 151C. Basic notions and examples; vector fields and flows; tensors and vector bundles; differential forms, integration and deRham's theorem. MATH 232B. Geometry II (Introduction to Differential). (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 232A. Local and global theory of curves. Surfaces in R3: the Gauss map, fundamental forms, curvature. Riemannian geometry: the Levi-Civita connection, curvature, geodesics, exponential map, completeness, Gauss-Bonnet theorem for surfaces. MATH 241. Mathematical Physics: Classical Mechanics. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 205A, MATH 205B, MATH 205C; or PHYS 205; or consent of instructor. Hamilton's principle of least action. Variational methods and Lagrange's equations. Hamilton's equations. Introduction to symplectic geometry and its applications to classical mechanics. Poisson brackets. Conserved quantities and Noether's theorem. Examples of Hamiltonian and dissipative dynamical systems. Introduction to classical chaos. MATH 242. Mathematical Physics: Quantum Mechanics. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 209A, MATH 209B, MATH 209C, MATH 228; or consent of instructor. Foundations of quantum theory together with the relevant mathematics. Probabilistic interpretation of quantum mechanics, self-adjoint operators and physical observables, noncommutativity and the uncertainty principle. Spectral theory for (unbounded) self-adjoint operators. Stone's theorem and other topics. MATH 243A. Algebraic Geometry. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 201A, MATH 201B. Topics include algebraic varieties in affine and projective space and their basic attributes such as dimension, degree, tangent space, and singularities; and products, mappings, and correspondences. MATH 243B. Algebraic Geometry. (4) Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 243A. Topics include further study of varieties, sheaves, and cohomology and detailed study of curves and special topics. MATH 246A. Algebraic Topology. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 205A; MATH 205B or equivalent. Topics include simplicial and cell complexes, polyhedra, manifolds, homology and cohomology theory, and homotophy theory. MATH 246B. Algebraic Topology. (4) Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 246A. Covers topics such as topological indices, Lefschetz fixed point theory, Poincaré duality, vector bundles and characteristic classes, and transformation groups. MATH 260. Seminar. (1-4) Variable hours. Prerequisite(s): consent of department. Seminar on special topics of mathematics in preparation for individual research. Course is repeatable. MATH 289. Colloquium in Mathematics. (1) Prerequisite(s): graduate standing. Specialized discussions by staff, students and visiting scientists on current research topics in Mathematics. Graded Satisfactory (S) or No Credit (NC). Course is repeatable. MATH 290. Directed Studies. (1-6) Prerequisite(s): consent of instructor. Research and special studies in mathematics. Graded Satisfactory (S) or No Credit (NC). Course may be repeated. MATH 299. Research for Thesis or Dissertation. (1-12) Prerequisite(s): consent of department. Graded Satisfactory (S) or No Credit (NC). Course is repeatable. MATH 302. Apprentice Teaching. (2-4) Lecture, zero to one hour; seminar, two to four hours; consultation, one to two hours. Prerequisite(s): appointment as a teaching assistant or associate in Mathematics. Supervised training for teaching in lower- and upper-division Mathematics courses. Topics include effective teaching methods, such as those involved in leading mathematics discussion sections, preparing and grading examinations, and relating to students. Required each quarter of all teaching assistants and associates in Mathematics. Units to be decided in consultation with graduate advisor. Graded Satisfactory (S) or No Credit (NC). Course is repeatable. 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