ARCHIVED CATALOG: Visit catalog.ucsb.edu to view the 2023-2024 General Catalog.

# UC Santa Barbara General Catalog ## Mathematics

Division of Mathematical, Life, and Physical Sciences
South Hall 6607
Website: www.math.ucsb.edu
Department Chair: Stephen Bigelow Some courses displayed may not be offered every year. For actual course offerings by quarter, please consult the Quarterly Class Search or GOLD (for current students). To see the historical record of when a particular course has been taught in the past, please visit the Course Enrollment Histories.

Mathematics
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) Lower Division
MATH 2A. Calculus with Algebra and Trigonometry
(5) STAFF
Prerequisite: Qualifying score on the Mathematics Placement Exam.
Enrollment Comments: Students who have completed Math 34A will only receive 3 units for Math 2A. Not open for credit to students who have completed Math 3A or 3AS or have passed the AP Calculus AB or BC exams.
Math 3A with precalculus: A function approach integrating algebra, trigonometry, and differential calculus. Topics include: one-on-one and onto functions; inverse functions; properties and graphs of polynomial, rational, exponential, and logarithmic functions; properties and graphs of trigonometric functions; analytic geometry; functions and limits; derivatives; techniques and applications of differentiation; introduction to integration; logarithmic and trigonometric functions.
MATH 2B. Calculus with Algebra and Trigonometry
(5) STAFF
Prerequisite: Mathematics 2A with a minimum grade of C.
Enrollment Comments: Students who have completed Math 34B will only receive 3 units for Math 2B. Not open for credit to students who have completed Math 3B or 3BS or have passed the AP Calculus BC exam.
Math 3B with precalculus: A continued review of relevant topics from precalculus (see Math 2A); integral calculus, including definite and indefinite integrals; techniques of integration, introduction to sequences and series; with applications in mathematics and physics.
MATH 3A. Calculus with Applications, First Course
(4) STAFF
Prerequisite: Qualifying score on the Mathematics Placement Exam.
Recommended Preparation: Students with Advanced Placement Credit should contact the department.
Enrollment Comments: Students who have completed Math 34A will only receive 2 units for Math 3A. Not open for credit to students who have completed Math 2A or 3AS or have passed the AP Calculus AB or BC exams.
Differential Calculus including analytic geometry, functions and limits, derivatives, techniques and applications of differentiation; introduction to integration; logarithmic and trigonometric functions.
MATH 3B. Calculus with Applications, Second Course
(4) STAFF
Prerequisite: Mathematics 3A with a minimum grade of C.
Recommended Preparation: Students with Advanced Placement Credit should contact the department.
Enrollment Comments: Students who have completed Math 34B will only receive 2 units for Math 3B. Not open for credit to students who have completed Math 2B or 3BS or have passed the AP Calculus BC exam.
Integral calculus including definite and indefinite integrals, techniques of integration; introduction to sequences and series; with applications in mathematics and physics.
MATH 4A. Linear Algebra with Applications
(4) STAFF
Prerequisite: Math 2B or 3B or 3BI with a minimum grade of C.
Systems of linear equations, matrix algebra, determinants, vector spaces and subspaces, basis and dimension, linear transformations, eigenvalues and eigenvectors, diagonalization, and orthogonality.
MATH 4AI. Inquiry Based Linear Algebra
(4) STAFF
Prerequisite: Mathematics 3B or Mathematics 3BI with a minimum grade of C.
Enrollment Comments: Misc: Not open for credit to students who have completed Math 3C or Math 4A.
Honors version of Mathematics 4A. Mathematical inquiry course is developed through problem solving and discovery.
MATH 4B. Differential Equations
(4) STAFF
Prerequisite: Math 4A or Math 4AI with a minimum grade of C.
First and second order differential equations, separation of variables, linear differential equations, systems of first order equations, nonlinear differential equations and stability.
MATH 4BI. Inquiry Based Differential Equations
(4) STAFF
Prerequisite: Mathematics 3C, 3CI, 4A, or 4AI with a minimum grade of C.
Enrollment Comments: Misc: Not open for credit to students who have completed Math 4B or 5A.
Honors version of Mathematics 4B. Mathematical inquiry course is developed through problem solving and discovery.
MATH 6A. Vector Calculus with Applications, First Course
(4) STAFF
Prerequisite: Mathematics 4A (or 4AI) with a letter grade of C or better.
Calculus of functions of several variables, vector valued functions of one variable, derivative and integrals of vector functions, double and triple integrals, properties and applications of integrals, change of variables.
MATH 6B. Vector calculus with Applications, Second Course
(4) STAFF
Prerequisite: Mathematics 4B (or 4BI) and Mathematics 6A, each with a minimum grade of C.
Scalar and vector fields, integration along paths, integration over surfaces and solid regions, integral theorems of vector calculus, Fourier series, partial differential equations.
MATH 7H. Honors Seminar-Calculus
(1) STAFF
Prerequisite: Concurrent enrollment in Mathematics 3A or 3B or 3BI or 3C or 4A or 4AI or 4B or 4BI or 5A or 5AI or 6A or 6AI or 6B.
Repeat Comments: May be repeated for credit to a maximum of 6 units.
Emphasizing fundamental concepts and applications. Intended for highly motivated and well prepared students.
MATH 8. Transition to Higher Mathematics
(5) STAFF
Prerequisite: Mathematics 3C or 3CI or 4A or 4AI or 4B or 4BI or 5A or 5AI or 5B or 5BI or 5C or 6A or 6AI or 6B with a grade of B or better.
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.
MATH 34A. Calculus for Social and Life Sciences
(4) STAFF
Enrollment Comments: Not open for credit to students who have completed Math 2A or 3A or 3AS, or are simultaneously enrolled in 2A or 3A, or have passed the AP Calculus AB or BC exams.
Introduction to differential and integral calculus with applications to modeling in the biological sciences.
MATH 34B. Calculus for Social and Life Sciences
(4) STAFF
Prerequisite: Mathematics 2A or 3A or 34A with a grade of C or better.
Enrollment Comments: Not open for credit to students who have completed Math 2B or 3B or are simultaneously enrolled in 2B or 3B, or have passed the AP Calculus BC exam.
Continued study of differential and integral calculus with differential andintegral calculus with applications. Introduction to mathematical modeling with differential equations. Calculus of several variables including an introduction to partial derivatives.
MATH 94. Group Studies in Mathematics
(1) STAFF
Prerequisite: Mathematics 3C or 3CI or 4A or 4AI or 4B or 4BI or 5A or 5AI or 5B or 5BI or 5C or 6A or 6AI or 6B with a grade of B or better.
Enrollment Comments: Requires concurrent enrollment in MATH 190
Lectures and discussions on special topics. Designed for transfer students only. Upper Division
MATH 100A. Mathematics for Elementary Teaching, I
(3) STAFF
Enrollment Comments: 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.
MATH 100B. Mathematics for Elementary Teaching, II
(3) STAFF
Prerequisite: Mathematics 100A.
Enrollment Comments: 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".
MATH 101A. Classical Number Systems
(4) STAFF
Prerequisite: Math 8 with a grade of "C" or better.
Enrollment Comments: Not open for credit to students who have completed Mathematics 118A.
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. Especially suitable for prospective middle and high school teachers.
MATH 101B. Mathematical Systems
(4) STAFF
Prerequisite: Math 101A with a minimum grade of C.
Enrollment Comments: Not open for credit to students who have completed Mathematics 118A.
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. Especially suitable for prospective middle and high school teachers.
MATH 102A. Modern Euclidean and Noneuclidean Geometry
(4) STAFF
Prerequisite: Math 8 with a minimum grade of C.
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. Especially suitable for prospective middle and high school teachers.
MATH 102B. Modern Euclidean and Noneuclidean Geometry
(4) STAFF
Prerequisite: Math 102A with a minimum grade of C.
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. Especially suitable for prospective middle and high school teachers.
MATH 103. Introduction to Group Theory
(4) STAFF
Prerequisite: Mathematics 8 with a grade of "C" or better.
Enrollment Comments: Not open for credit to students who have completed Mathematics 111A.
Permutation groups, cyclic groups, theory of finite groups, group homomorphisms and isomorphisms, and Abelian groups. Applications to number theory and geometry.
MATH 104A. Introduction Into Numerical Analysis
(4) STAFF
Prerequisite: Mathematics 4B or 4BI, 6A or 6AI, and 6B; or 5A or 5AI, 5B or 5BI and 5C; and Math 117; and, Computer Science 5AA-ZZ or 10 or 8 or 16 or Engineering 3. A grade of C or above is required in all prerequisite courses.
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).
MATH 104B. Numerical Analysis
(4) STAFF
Prerequisite: Math 104A with a minimum grade of C.
Numerical methods for the solution of systems of linear equations (direct and iteractive methods), and the finite difference methods for boundary value problems for (ordinary and partial) differential equations.
MATH 104C. Advanced Topics in Numerical Analysis
(4) STAFF
Prerequisite: Math 104B with a minimum grade of C.
Topics in approximation theory; numerical methods for finding eigenvalues of a matrix; and advanced topics in numerical methods for ordinary and partial differential equations.
MATH 108A. Introduction to Linear Algebra
(4) STAFF
Prerequisite: Mathematics 3C or 3CI or 4A or 4AI, 4B or 4BI or 5A or 5AI; and Math 8 with a grade of "C" or better.
Abstract vector spaces and subspaces. Span and linear independence. Basis and dimension. Linear maps. Eigenvalues and eigenvectors.
MATH 108B. Introduction to Linear Algebra
(4) STAFF
Prerequisite: Math 108A with a minimum grade of C.
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.
MATH 108C. Matrix Analysis
(4) STAFF
Prerequisite: Math 108A with a minimum grade of C
Eigenvalues, eigenvectors and similarity; Unitary equivalence and normal matrices; QR factorization; Least squares; Singular Value decomposition; Moore-Penrose generalized inverse; Hermitian and symmetric matrices; Variational characterization of eigenvalues; Positive definite matrices; Nonnegative matrices; Perron-Frobenius Theorem.
MATH 109A. Introduction to Mathematical Logic
(4) STAFF
Prerequisite: Mathematics 8 or Computer Science 40.
An introduction to mathematical logic with applications in computer science and mathematics. Topics include propositional and predicate calculi; models; proof systems, decidability and undecidability, automated theorem-proving, unification, logic programming, and program verification.
MATH 109B. Introduction to Mathematical Logic
(4) STAFF
Prerequisite: Mathematics 109A.
An introduction to mathematical logic with applications in computer science and mathematics. topics include propositional and predicate calculi; models; proof systems, decidability and undecidability, automated theorem-proving, unification, logic programming, and program verification.
MATH 109C. Introduction to Mathematical Logic
(4) STAFF
Prerequisite: Mathematics 109B.
An introduction to mathematical logic with applications in computer science and mathematics. Topics include propositional and predicate calculi; models; proof systems, decidability and undecidability, automated theorem-proving, unification, logic programming, and program verification.
MATH 111A. Introduction to Abstract Algebra
(4) STAFF
Prerequisite: Mathematics 8 and 108A with a grade of "C" or better.
An introduction to algebraic structures with an emphasis on groups.
MATH 111B. Abstract Algebra
(4) STAFF
Prerequisite: Math 111A with a minimum grade of C.
Rings, fields, Galois theory.
MATH 111C. Abstract Algebra
(4) STAFF
Prerequisite: Math 111B with a minimum grade of C.
Rings, fields, Galois theory.
MATH 113. Non-euclidean Geometry
(4) STAFF
Prerequisite: Mathematics 8 with a grade of "C" or better.
An introduction to hyperbolic geometry with some discussion of other non- euclidean systems.
MATH 114. Computational Tools for Numerical Analysis
(4) STAFF
Prerequisite: Math 6B, Math 8, and Math 108A or Math 117 each with a letter grade of "C" or higher.
Recommended Preparation: At least one quarter of programming experience.
Introduction to programming with specific examples in scientific computing. The algorithms used will be analyzed in detail in Math 104A-B-C. The emphasis will be the translation of numerical algorithms into actual working code.
MATH 115A. Introduction to Number Theory
(4) STAFF
Prerequisite: Mathematics 8 with a grade of "C" or better.
Recommended Preparation: Students are encouraged to take both 115A and 115B in the same academic year as topics may very from year to year.
Divisibility, congruences, primitive roots an indices, quadratic residues and the quadratic reciprocity law, number-theoretic functions, Diophantine equations, the distribution of primes, number-theorhetic methods in cryptography, quadratic forms, continued fractions, and the approximation of real numbers, algebraic number theory, partitions.
MATH 115B. Introduction to Number Theory
(4) STAFF
Prerequisite: Math 115A with a minimum grade of C.
Recommended Preparation: Students are encouraged to take both 115A and 115B in the same academic year as topics may vary from year to year.
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.
MATH 115C. Topics in Number Theory
(4) STAFF
Prerequisite: Math 115B with a minimum grade of C or consent of instructor.
Recommended Preparation: Completion of additional courses may prove useful, depending on the topics to be considered. Consult the department or instructor for details.
Selected topics in number theory at the direction of the instructor.
MATH 116. Combinatorial Analysis
(4) STAFF
Prerequisite: Mathematics 8 with a grade of "C" or better.
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.
MATH 117. Methods of Analysis
(4) STAFF
Prerequisite: Mathematics 8 or PSTAT 8 with a grade of "C" or better.
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.
MATH 118A. Introduction to Real Analysis
(4) STAFF
Prerequisite: Mathematics 5A or 5AI and 5B or 5BI; or 4B or 4BI and 6A or 6AI; and 108A and 117 each with a grade of "C" or better.
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
MATH 118B. Introduction to Real Analysis
(4) STAFF
Prerequisite: Math 118A and 108B with a minimum grade of C.
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
MATH 118C. Introduction to Real Analysis
(4) STAFF
Prerequisite: Math 118B with a minimum grade of C.
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
MATH 119A. Ordinary Differential Equations
(4) STAFF
Prerequisite: Mathematics 5A or 5AI and 5B or 5BI with a minimum grade of C; or 4B or 4BI and 6A or 6AI with a minimum grade of C; and Math 8 with a minimum grade of C.
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.
MATH 119B. Chaotic Dynamics and Bifurcation Theory
(4) STAFF
Prerequisite: Mathematics 6B and Math 119A, each with a minimum grade of C
Hyperbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.
MATH 122A. Introduction to Theory of Complex Variables
(4) STAFF
Prerequisite: Mathematics 4B or 4BI and 6A with a minimum grade of C; and Math 8 with a minimum grade of C.
Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.
MATH 122B. Introduction to Theory of Complex Variables
(4) STAFF
Prerequisite: Math 122A with a minimum grade of C.
Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.
MATH 124A. Partial Differential Equations
(4) STAFF
Prerequisite: Math 4B or 4BI, 6A, and 6B with a minimum grade of C; and Math 8 with a minimum grade of C.
Wave, heat, and potential equations.
MATH 124B. Fourier Series and Numerical Methods
(4) STAFF
Prerequisite: Math 4B or 4BI, and 6A, and 6B with a minimum grade of C; and 124A with a minimum grade of C.
Fourier series; generalized functions; and numerical methods.
MATH 132A. Introduction to Operations Research
(4) STAFF
Prerequisite: Math 6A and Math 108A, both with a grade of C or above. Computer Science 8 or 16, or Engineering 3 with a minimum grade of C.
Linear programming, the simplex method, duality, applications to the transportation and assignment problems, sensitivity analysis, problem formulation.
MATH 132B. Introduction to Operations Research
(4) STAFF
Prerequisite: Mathematics 117 and 132A, each with a minimum grade of C.
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.
MATH 137A. Graph and Network Theory
(4) STAFF
Prerequisite: Mathematics 4B or 4BI; and Math 8 with a grade of "C" or better.
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.
MATH 137B. Graph and Network Theory
(4) STAFF
Prerequisite: Math 137A with a minimum grade of C.
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.
MATH 145. Introduction to Topology
(4) STAFF
Prerequisite: Math 8 with a C or better. Either Math 108A or Math 117 with a minimum grade of C.
Metric spaces, continuity, compactness, classification of surfaces, Euler characteristics, and fundamental groups. Additional topics at the discretion of the instructor.
MATH 147A. Introductory Differential Geometry
(4) STAFF
Prerequisite: Math 6A with a minimum grade of C; and Math 108A or 117 with a minimum grade of C.
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.
MATH 147B. Introductory Differential Geometry
(4) STAFF
Prerequisite: Mathematics 147A with a minimum grade of C.
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.
MATH 170. Introduction to Mathematical Finance
(4) STAFF
Prerequisite: PSTAT 120A-B and 160A, all completed with a minimum grade of C or better.
Recommended Preparation: PSTAT 160B and 171.
Enrollment Comments: Same course as PSTAT 170.
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.
MATH 181A. Advanced Problem Solving in Mathematical, Historical and Pedagogical Contexts.
(4) STAFF
Prerequisite: Math 8 with a minimum grade of C
Focuses on the representations, strategies, and language learners use to conceptualize and develop fundamental ideas of mathematics. Includes advanced mathematical problem solving and its implications for teaching and learning at the secondary level. Especially suitable for prospective middle and high school teachers.
MATH 181B. Advanced Problem Solving in Mathematical, Historical, and Pedagogical Contexts
(4) STAFF
Prerequisite: Math 181A or ED 134 with a minimum grade of C
Continuation of Math 181A or ED 134. Focuses on the representations, strategies, and language learners use to conceptualize and develop fundamental ideas of mathematics. Includes advanced mathematical problem solving and its implications for teaching and learning at the secondary level. Especially suitable for prospective middle and high school teachers.
MATH 182. History of Mathematics
(4) STAFF
Prerequisite: Mathematics 8 with a grade of "C" or better.
An examination of the major achievements in mathematical thinking throughout history. Topics may include the history of numerical systems in early civilizations, the development of formal proof, mathematical contributions from diverse populations and the impact of technological innovations on mathematics. Especially suitable for prospective middle and high school teachers.
MATH 190. Special Topics in Mathematics
(4) STAFF
Prerequisite: Mathematics 3C or 3CI or 4A or 4AI or 4B or 4BI or 5A or 5AI or 5B or 5BI or 5C or 6A or 6AI or 6B with a grade of B or better. Concurrent enrollment in Math 94.
Enrollment Comments: 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. Designed for transfer students only.
MATH 193. Internship in Mathematics
(1-4)
Prerequisite: Consent of instructor and department.
Enrollment Comments: May be repeated for credit up to a maximum of four units. Credit will not be given toward upper-division Mathematics major requirements.
MATH 194GS. Group Studies for Advanced Students
(1) STAFF
Prerequisite: Consent of instructor.
Enrollment Comments: Enrollment normally limited to 12 or fewer students.
Participants will select a math-related book or papers, read the section before the next meeting and discuss reading at the meeting. Readings may include biographies of mathematicians, histories or popularizations of mathematics, textbooks, and readings in mathematical physics or biology.
MATH 197A. Senior Thesis
(1-4)
Prerequisite: Open to senior majors only; consent of department and instructor.
Enrollment Comments: Up to four units may apply to the major. Up to eight units total in all Mathematics 197/199RA courses may apply toward the major. Students must have a minimum overall grade-point average of 3.0 and a 3.5 or better grade-point-average in the major.
Independent research under the supervision of a faculty member which will result in a senior thesis. Student will concentrate on reading and gathering material for a thesis.
MATH 197B. Senior Thesis
(1-4)
Prerequisite: Mathematics 197A. Open to senior majors only; consent of department and instructor.
Enrollment Comments: 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 four units may apply towards the major.
Independent research under the supervision of a faculty member which will result in a senior thesis. Student will concentrate on writing a thesis.
MATH 199. Independent Studies in Mathematics
(1-5) STAFF
Prerequisite: Upper-division standing; completion of 2 upper-division courses in math; consent of instructor and department.
Enrollment Comments: Students must have a cumulative 3.0 for the proceeding 3 quarter(s). Limit of 5 units per quarter and 30 units total in all independent studies courses (98/99/99RA/198/199/199AA-ZZ) combined. Only 8 units in Math 197/199AA-ZZ courses may apply to the major.
Coursework consists of academic research supervised by a faculty member on a topic not available in established course offerings.
MATH 199RA. Independent Research Assistance
(1-4) STAFF
Prerequisite: Upper-division standing; completion of 2 upper-division courses in math; consent of instructor and department.
Enrollment Comments: Students must have a cumulative 3.0 for the proceeding 3 quarter(s). Limit of 5 units per quarter and 30 units total allowed in all independent studies courses (98/99/99RA/198/199/199AA-ZZ) combined. Only 8 units in Math 197/199AA-ZZ courses may apply to the major.
Coursework consists of faculty supervised research assistance. Graduate
MATH 201A. Real Analysis
(4)
Prerequisite: 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.
MATH 201B. Real Analysis
(4)
Prerequisite: 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.
MATH 201C. Real Analysis
(4)
Prerequisite: 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.
MATH 202A. Complex Analysis
(4)
Prerequisite: 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.
MATH 202B. Complex Analysis
(4)
Prerequisite: 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.
MATH 202C. Complex Analysis
(4)
Prerequisite: 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.
MATH 206A. Matrix Analysis and Computation
(4) STAFF
Prerequisite: Consent of instructor.
Enrollment Comments: Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language. Same course as Computer Science 211A, ME 210A, ECE 210A, Geology 251A, and Chemical Engineering 211A.
Graduate level-matrix theory with introduction to matrix computations. SVD's, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and iterative methods for matrix computations.
MATH 206B. Numerical Simulation
(4) STAFF
Prerequisite: Consent of instructor.
Enrollment Comments: Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language. Same course as Computer Science 211B, ME 210B, ECE 210B, Geology 251B, and Chemical Engineering 211B.
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.
MATH 206C. Numerical Solution of Partial Differential Equations--Finite Difference Methods
(4) STAFF
Prerequisite: Consent of instructor.
Enrollment Comments: Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language. Same course as Computer Science 211C, ME 210C, ECE 210C, Geology 251C, and Chemical Engineering 211C.
Finite difference methods for hyperbolic, parabolic and elliptic PDE's, 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.
MATH 206D. Numerical Solution of Partial Differential Equations - Finite Element Methods
(4) STAFF
Prerequisite: Consent of instructor.
Enrollment Comments: Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language. Same course as Computer Science 211D, ME 210D, ECE 210D, Geology 251D, and Chemical Engineering 211D.
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.
MATH 209. Set Theory
(4)
Prerequisite: Consent of instructor.
Study of axiomic set theory; topics include relations and functions, orderings, ordinal and cardinal numbers and their arithmetic, transfinite constructible sets, consistency and independence results of Godel and Cohen.
MATH 214A. Ordinary Differential Equations
(4)
Prerequisite: Not open to mathematics majors.
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.
MATH 214B. Chaotic Dynamics and Bifurcation Theory
(4)
Prerequisite: Not open to mathematics majors.
Hyberbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.
MATH 215A. Partial Differential Equations
(4)
Prerequisite: Not open to mathematics majors.
Wave, heat, and potential equations.
MATH 215B. Fourier Series and Numerical Methods
(4)
Prerequisite: Not open to mathematics majors.
Fourier series; generalized functions; and numerical methods.
MATH 220A. Modern Algebra
(4)
Prerequisite: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 220B. Modern Algebra
(4)
Prerequisite: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 220C. Modern Algebra
(4)
Prerequisite: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 221A. Foundations of Topology
(4)
Prerequisite: Mathematics 118A or equivalent.
Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Other topics as time allows.
MATH 221B. Homotopy Theory
(4)
Prerequisite: Mathematics 221A.
Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.
MATH 221C. Differential Topology
(4)
Prerequisite: Mathematics 221A.
Topological manifolds, differentiable manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, vector fields.
MATH 225A. Topics in Number Theory
(4)
Prerequisite: Mathematics 220A-B-C.
Enrollment Comments: May be repeated for credit with instructor and department approval.
Selected topics in number theory.
MATH 225B. Topics in Number Theory
(4)
Prerequisite: Mathematics 220A-B-C.
Enrollment Comments: May be repeated for credit with instructor and department approval.
Selected topics in number theory.
MATH 225C. Topics in Number Theory
(4)
Prerequisite: Mathematics 220A-B-C.
Enrollment Comments: May be repeated for credit with instructor and department approval.
Selected topics in number theory.
MATH 227A. Advanced Topics in Geometric and Algebraic Topology
(4)
Prerequisite: Consent of instructor.
Enrollment Comments: 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.
MATH 227B. Advanced Topics in Geometric and Algebraic Topology
(4)
Prerequisite: Consent of instructor.
Enrollment Comments: 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.
MATH 227C. Advanced Topics in Geometric and Algebraic Topology
(4)
Prerequisite: Consent of instructor.
Enrollment Comments: 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.
MATH 228A. Functional Analysis
(4)
Prerequisite: Mathematics 201A-B-C.
Topics in functional analysis such as operators on Hilbert space, convex analysis, fixed point theorems, distribution theory, unbounded operators.
MATH 228B. Functional Analysis
(4)
Prerequisite: Mathematics 201A-B-C.
Topics in functional analysis such as operators on Hilbert space, convex analysis, fixed point theorems, distribution theory, unbounded operators.
MATH 228C. Functional Analysis
(4)
Prerequisite: Mathematics 201A-B-C.
Topics in functional analysis such as operators on Hilbert space, convex analysis, fixed point theorems, distribution theory, unbounded operators.
MATH 231A. Lie Groups and Lie Algebras
(4)
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.
MATH 231B. Lie Groups and Lie Algebras
(4)
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.
MATH 232A. Algebraic Topology
(4)
Prerequisite: Mathematics 108A-B and 145.
Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.
MATH 232B. Algebraic Topology
(4)
Prerequisite: Mathematics 108A-B and 145.
Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.
MATH 236A. Homological Algebra
(4)
Prerequisite: 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.
MATH 236B. Homological Algebra
(4)
Prerequisite: 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.
MATH 237A. Algebraic Geometry
(4)
Prerequisite: 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, Riemann-Roch theorem. Special topics may include: elliptic curves, intersection numbers, Bezout's theorem, Max Noether's theorem.
MATH 237B. Algebraic Geometry
(4)
Prerequisite: 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, Riemann-Roch theorem. Special topics may include: elliptic curves, intersection numbers,Bezout's theorem, Max Noether's theorem.
MATH 237C. Algebraic Geometry
(4) STAFF
Prerequisite: 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 Riemann-Roch theorem. Special topics may include: elliptic curves, intersection numbers,Bezout's theorem, Max Noether's theorem.
MATH 240A. Introduction to Differential Geometry and Riemannian Geometry
(4)
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.
MATH 240B. Introduction to Differential Geometry and Riemannian Geometry
(4)
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.
MATH 240C. Introduction to Differential Geometry and Riemannian Geometry
(4)
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.
MATH 241A. Topics in Differential Geometry
(4)
Prerequisite: Mathematics 240A-B-C..
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifold, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
MATH 241B. Topics in Differential Geometry
(4)
Prerequisite: Mathematics 240A-B-C.
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifold, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
MATH 241C. Topics in Differential Geometry
(4)
Prerequisite: Mathematics 240A-B-C.
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifold, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
MATH 243A. Ordinary Differential Equations
(4)
Prerequisite: 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.
MATH 243B. Ordinary Differential Equations
(4)
Prerequisite: Mathematics 118A-B-C.
Existance and stability of solutions, Floq uet 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.
MATH 243C. Ordinary Differential Equations
(4)
Prerequisite: Mathematics 118A-B-C.
Existance 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.
MATH 246A. Partial Differential Equations
(4)
Prerequisite: Mathematics 201A-B-C.
First-order nonlinear equations; the Cauchy problem, elements of distribution theory an Sobolev spaces; the heat, wave, and Laplace equations; additional topics such as quasilinear symmetric hyperbolic systems, elliptic regularity theory.
MATH 246B. Partial Differential Equations
(4)
Prerequisite: 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.
MATH 246C. Partial Differential Equations
(4)
Prerequisite: 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, ellipitic regularity theory.
MATH 260AAZZ. Seminars in Mathematics
(1-6)
Prerequisite: Consent of instructor.
Enrollment Comments: 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.
MATH 260A. Seminars in Mathematics
(1-6) STAFF
Enrollment Comments: 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.
MATH 260AA. Seminars in Mathematics
MATH 260B. Seminars in Mathematics
MATH 260BB. Seminars in Mathematics
MATH 260C. Seminars in Mathematics
MATH 260CC. Seminars in Mathematics
MATH 260D. Seminars in Mathematics
MATH 260DD. Seminars in Mathematics
MATH 260E. Seminars in Mathematics
MATH 260EE. Seminars in Mathematics
(1-6) STAFF
Enrollment Comments: 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.
MATH 260ES. Seminars in Mathematics
MATH 260F. Seminars in Mathematics
MATH 260G. Seminars in Mathematics
MATH 260GG. Seminars in Mathematics
MATH 260H. Seminars in Mathematics
MATH 260HH. Seminars in Mathematics
MATH 260I. Seminars in Mathematics
MATH 260II. Seminars in Mathematics
MATH 260J. Seminars in Mathematics
MATH 260JJ. Seminars in Mathematics
MATH 260K. Seminars in Mathematics
MATH 260KK. Seminars in Mathematics
MATH 260L. Seminars in Mathematics
MATH 260LL. Seminars in Mathematics
MATH 260M. Seminars in Mathematics
MATH 260MM. Seminars in Mathematics
MATH 260N. Seminars in Mathematics
MATH 260NN. Seminars in Mathematics
MATH 260O. Seminars in Mathematics
MATH 260OO. Seminars in Mathematics
MATH 260P. Seminars in Mathematics
MATH 260Q. Seminars in Mathematics
MATH 260QQ. Seminars in Mathematics
MATH 260R. Seminars in Mathematics
MATH 260S. Seminars in Mathematics
MATH 260SS. Seminars in Mathematics
MATH 260T. Seminars in Mathematics
MATH 260TT. Seminars in Mathematics
MATH 260U. Foundations in Mathematics
MATH 260UU. Seminars in Mathematics
MATH 260V. Seminars in Mathematics
MATH 260W. Seminars in Mathematics
MATH 260X. Seminars in Mathematics
MATH 260Y. Seminars in Mathematics
MATH 260Z. Seminars in Mathematics
MATH 500. Teaching Assistant Practicum
(1-4)
Prerequisite: Appointment as teaching assistant and departmental approval.
Enrollment Comments: No unit credit allowed toward degree.
Supervised teaching of undergraduate mathematics courses.
MATH 501. Teaching Assistant Training
(1-2)
Prerequisite: Departmental and instructor approval.
Enrollment Comments: No unit credit allowed toward degree.
Consideration of ideas about the process of learning mathematics and discussion of approaches to teaching.
MATH 502. Teaching Associate Practicum
(1-5)
Prerequisite: Appointment as associate; departmental approval.
Enrollment Comments: No unit credit allowed toward degree.
MATH 510. Reading for Area Examinations
(2-6)
Prerequisite: Enrollment in M.A. or Ph.D. program. Consent of instructor.
MATH 596. Directed Reading and Research
(1-6)
Prerequisite: Graduate standing and consent of instructor.
Enrollment Comments: May be repeated for credit. Only 8 units total in all Mathematics 596, 598, 599 courses may be applied toward the degree.
MATH 596AA. Directed Reading and Research
(1-6)
Prerequisite: Graduate standing and consent of instructor.
Enrollment Comments: May be repeated for credit as determined by department chairman.
MATH 598. Master's Thesis Research and Preparation
(1-6)
Prerequisite: Graduate standing and consent of instructor.
Enrollment Comments: No unit credit allowed toward degree.
Master's thesis research and preparation.
MATH 598AA. Master's Thesis Research and Preparation
(1-6)
Prerequisite: Graduate standing and consent of instructor.
Enrollment Comments: No unit credit allowed toward degree.
Master's thesis research and preparation.
MATH 599. Dissertation Preparation
(1-6)
Prerequisite: Consent of instructor.
Enrollment Comments: May be repeated for credit. Only 8 units total in all Mathematics 596, 598,599 courses may apply toward degree.
Dissertation preparation.