MATHEMATICS

Subject abbreviation: MATH

Reinhard Schultz, Ph.D., Chair  

Department Office, 2208 Sproul Hall

Professors:

John C. Baez, Ph.D.

Bruce L. Chalmers, Ph.D.

Mei-Chu Chang, Ph.D.

Vyjayanthi Chari, Ph.D.

Gerhard Gierz, Ph.D.

Lawrence H. Harper, Ph.D.

Michel L. Lapidus, Ph.D.

Xiao-Song Lin, Ph.D.

Yat Sun Poon, Ph.D.

Ziv Ran, Ph.D.

Malempati M. Rao, Ph.D.

David E. Rush, Ph.D.

Reinhard Schultz, Ph.D.

Albert R. Stralka, Ph.D.

Bun Wong, Ph.D.

Professors Emeriti:

Theodore J. Barth, Ph.D.

Richard E. Block, Ph.D.

John E. de Pillis, Ph.D.

Charles J. A. Halberg, Jr., Ph.D.

F. Burton Jones, Ph.D.

Frederic T. Metcalf, Ph.D.

Louis J. Ratliff, Jr., Ph.D.

Victor L. Shapiro, Ph.D.

James D. Stafney, Ph.D.

Associate Professors:

Le Baron O. Ferguson, Ph.D.

Neil E. Gretsky, Ph.D.

J. Keith Oddson, Ph.D.

Ivan B. Penkov, Ph.D.

Assistant Professor:

Frederick H. Wilhelm, Jr., Ph.D.

**

Cooperating Faculty:

Marek Chrobak, Ph.D.
(Computer Science and Engineering)

Thomas H. Payne, Ph.D.
(Computer Science and Engineering)

MAJOR  

The Department of Mathematics offers a B.A. and a B.S. degree in programs which share a common, solid mathematical foundation but which 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, are dependent upon 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. degree in Mathematics is more technical and contains a greater concentration of work in the major field. The Pure Mathematics program is directed toward those students who may wish to continue on to graduate work in mathematics. The Applied Mathematics programs, with options in Physics, Statistics, and Economics, 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 as well as 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 who monitors the student's subsequent progress in an academic program. Each quarter a study list must be approved by this advisor.

TEACHING

In order to teach mathematics in the California public schools, students must have completed a baccalaureate program and a graduate credential program. Prior to admission to a credential program, candidates must have demonstrated proficiency in the field in which they will teach. Proficiency can be demonstrated by passing state examinations. For additional information about the preparation required for a teaching credential, please refer to the School of Education section of the catalog.

DEGREE REQUIREMENTS  

UNIVERSITY REQUIREMENTS

General University requirements are Universitywide requirements which all undergraduates must satisfy. See the Undergraduate Studies section for a complete listing.

COLLEGE REQUIREMENTS

Students must fulfill all breadth requirements of the College of Natural and Agricultural Sciences. For a detailed list of requirements and a summary of units, see Degree Requirements under College of Natural and Agricultural Sciences in the Undergraduate Studies section of this catalog.

MAJOR REQUIREMENTS

To fulfill the Natural Sciences requirement, the Department of Mathematics requires the following:

1. One of the year sequences

• a) BIOL 002, BIOL 003, BIOL 005C
• b) CHEM 001A-CHEM 001B-CHEM 001C
• c) PHYS 002A, PHYS 002B, PHYS 002C or PHYS 040A, PHYS 040B, PHYS 040C

2. Either one course in the physical sciences if (a) above is completed or one course in the biological sciences if (b) or (c) above is completed

The major requirements for the B.A. and B.S. degrees in Mathematics are as follows:

For the Bachelor of Arts

1. Lower-division requirements: MATH 009A-MATH 009B-MATH 009C, MATH 010A-MATH 010B, MATH 046

2. Four (4) units of either one course in Computer Science or one upper-division course in Statistics.

3. Thirty-six (36) units of upper-division mathematics, excluding courses in the MATH 190-199 series.

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)

• a) Thirty-six (36) units of upper-division mathematics to include at least 24 units from MATH 131, MATH 132, MATH 145A-MATH 145B, MATH 151A-MATH 151B-MATH 151C, MATH 171, MATH 172
• b) At least three 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:

• a) Economics option
  • (1) MATH 120, MATH 121, MATH 149A-MATH 149B-MATH 149C
  • (2) Twenty (20) units of upper-division economics, including ECON 102A, ECON 107, ECON 108, ECON 110

• b) Physics option
  • (1) MATH 135A, MATH 165A-MATH 165B
  • (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

• c) Statistics option
  • (1) MATH 120, MATH 149A-MATH 149B-MATH 149C
  • (2) Either STAT 130 or STAT 146
  • (3) STAT 161, STAT 170A-STAT 170B, STAT 171

3. Computational Mathematics program

• a) MATH 112, MATH 113 or MATH 131, MATH 120, MATH 132, MATH 135A-MATH 135B
• b) CS 012, CS 140A-CS 140B, CS 150
• c) Twenty-four (24) units of technical electives to be chosen from
  • (1) MATH 121, MATH 125A-MATH 125B, MATH 146A-MATH 146B-MATH 146C, MATH 149A-MATH 149B-MATH 149C, MATH 171
  • (2) CS 130, CS 166, CS 170, CS 175, CS 177

MATHEMATICS HONORS PROGRAM

Candidates for the Honors Program in Mathematics must

1. Complete 9 units of upper-division mathematics in addition to the requirements of the major

2. Complete MATH 145B, MATH 151A-MATH 151B-MATH 151C, and MATH 171 with a grade of "B" or better in each course and have an overall grade point average of at least 3.50 in mathematics

3. Complete one of the following:

• a) A paper based on an approved plan of independent study
• b) Three one-quarter graduate courses in mathematics with a grade of "B" or better.

It is the responsibility of the honors candidates to notify the department of their eligibility.

MINOR

The following are the requirements for a minor in Mathematics.

1. Lower-division requirements (20 units): MATH 009A-MATH 009B-MATH 009C, MATH 010A-MATH 010B

2. Upper-division requirements: Twenty-four (24) units of upper-division mathematics courses

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. Because strategy in choosing courses to be taken here and courses to be taken abroad varies depending on personal goals and the country visited, early planning is advised. Consult the departmental Student Affairs Officer for assistance. For further details 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.

GRADUATE PROGRAMS  

Domestic applicants to these graduate programs must supply Graduate Record Examination scores for the General Test (verbal, quantitative, and analytical).

MASTER OF ARTS OR SCIENCE IN MATHEMATICS

General University requirements are listed in the Graduate Studies section of this catalog. Specific requirements of the Department are (1) completion of two of the following sequences, MATH 201A-MATH 201B-MATH 201C, MATH 205A-MATH 205B-MATH 205C, MATH 209A-MATH 209B-MATH 209C, and MATH 210A-MATH 210B, with a grade of "C" or better in each course and a G.P.A. of 3.00 in each chosen sequence; (2) as a substitute for one or more course sequences in (1), passing a Ph.D. qualifying examination fulfills the course requirement of the corresponding sequence; (3) taking 36 units of approved courses of which at least 18 must be in the 200 series courses in mathematics; (4) completion of the courses MATH 131, MATH 132, MATH 151A, and MATH 151B, or their equivalents.

MASTER OF SCIENCE IN MATHEMATICS (APPLIED)

General University requirements are listed in the Graduate Studies section of this catalog. Specific requirements of the Department are (1) passing written qualifying examinations at the master's level (or higher) in two of the following fields: Advanced Ordinary Differential Equations, Partial Differential Equations, Advanced Statistical Inference, Calculus of Variations, Combinatorial Theory, Real Analysis, and Advanced Numerical Analysis; (2) 36 units of approved courses, of which 18 must be in the 200 series; (3) completion of the courses MATH 131, MATH 132 , MATH 151A-MATH 151B, MATH 146A, MATH 149A, or their equivalent. Also, MATH 165A is recommended, but not required.

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Specific requirements of the Department are (1) passing the introductory courses in algebra (MATH 201A-MATH 201B-MATH 201C), complex analysis (MATH 210A-MATH 210B), real analysis (MATH 209A-MATH 209B-MATH 209C), and topology/differentiable manifolds (MATH 205A-MATH 205B-MATH 205C); (2) passing at least three of the four qualifying examinations in algebra, complex analysis, real analysis and topology/differentiable manifolds with a grade of "A." The fourth of the above qualifying examinations must be passed with a grade of "B" or better; a student is allowed to take the qualifying examination at most twice in each area; (3) completing four quarter-courses in mathematics numbered between 211 and 259.

The normative time to the Ph.D. is 15 quarters.

LOWER-DIVISION COURSES  

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): none. Basic algebra, linear functions and equations, quadratic functions and equations, operations with functions. This course is not intended to meet any mathematics or physical science requirement and is intended for students who plan to take MATH 005 but are not prepared to take that course. Carries workload credit equivalent to four 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-MATH 009B-MATH 009C.
First-Year Calculus. (4-4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): for MATH 009A: MATH 005 with a grade of "C-" or better or equivalent; for MATH 009B: MATH 009A or MATH 09HA with a grade of "C-" or better; for MATH 009C: MATH 009B or MATH 09HB with a grade of "C-" or better. 009A. Introduction to the differential calculus of functions of one variable; 009B. Introduction to the integral calculus of functions of one variable; 009C. Further topics from integral calculus, improper integrals, infinite series, Taylor's series, and Taylor's theorem. Credit is awarded for only one of MATH 009A or MATH 09HA, only one of MATH 009B or MATH 09HB, and only one of MATH 009C or MATH 09HC.

MATH 09HA-MATH 09HB-MATH 09HC.
First Year Honors Calculus. (4-4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): for MATH 09HA: a score of 47 or higher on the Mathematics Placement Precalculus Examination; for MATH 09HB: either a score of 4 or higher on the AB Advanced Placement Test in Mathematics or a grade of "A-" or above in MATH 009A or MATH 09HA; for MATH 09HC: a grade of "A-" or above in MATH 009B or MATH 09HB. Honors course corresponding to MATH 009A-MATH 009B-MATH 009C for students with strong mathematical backgrounds. Emphasis is on theory and rigor. Students may not receive credit for more than one of MATH 009A or MATH 09HA or more than one of MATH 009B or MATH 09HB or more than one of MATH 009C or MATH 09HC.

MATH 010A-MATH 010B.
Calculus of Several Variables. (4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): for 010A: MATH 009B; for 010B: MATH 009C and MATH 010A. Euclidean geometry, matrices and linear functions, determinants, partial derivatives, directional derivatives, Jacobians, gradients, chain rule, Taylor's theorem for several variables. 010B: Vector, differential calculus continued, implicit differentiation, extreme values, multiple integration, line integrals, vector field theory, 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 or equivalent. 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 will not be given for MATH 022 if it has already been given for MATH 009A or MATH 09HA.

MATH 023.
Applied Matrix Algebra. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 005 or equivalent. Matrix operations, linear dependence and independence, ranks and inverses, systems of linear equations, determinants, eigenvalues, and eigenvectors with business and economic applications. This course is designed for students who are not mathematics majors and does not count toward fulfillment of the mathematics major requirement.

MATH 042.
Differential Equations with Biological Applications. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009B. First order equations with applications to population biology, first order difference equation with applications to plant propagation and harvesting, first order systems and second order equations with applications to predator-prey problems. Only one of MATH 042 or MATH 046 may be taken for credit.

MATH 046.
Introduction to Ordinary Differential Equations. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009C. First order equations, linear second order equations, series solutions. Laplace transforms, applications to the physical and biological sciences.

UPPER-DIVISION COURSES

MATH 112.
Finite Mathematics. (4)

Year. 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): MATH 010A. Matrices and systems of linear equations, determinants, Gaussian elimination and pivoting, vector spaces, linear independence and linear transformation, orthogonality, eigenvalues and eigenvectors. Selected topics and applications. Numerical linear algebra and extensive computer use are integrated with these topics. Credit is awarded for only one of MATH 113 or MATH 131.

MATH 120.
Optimization Techniques. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 113 or MATH 131 (may be taken concurrently). Steepest descent and direct search methods for minima and maxima. The techniques of linear programming. Problems having applications in both the physical and behavioral sciences are used for illustrative purposes.

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-MATH 125B.
Introduction to Combinatorics. (4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 009C, MATH 112. Elements of graph theory, Polya's theory of counting, principle of inclusion-exclusion, Hall matching theorem, combinatorial designs.

MATH 131.
Linear Algebra I. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A. An introduction to vector spaces, matrices, and linear transformations. Only one of the MATH 113 or MATH 131 maybe taken for credit.

MATH 132.
Linear Algebra II. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 113 or MATH 131. Further topics in linear algebra including eigenvalues, 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-MATH 135B.
Numerical Analysis. (4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): CS 010 or CS 018 or equivalent; MATH 113 or MATH 131 (may be taken concurrently). Numerical methods for the determination of solutions to nonlinear equations and simultaneous linear equations. Interpolation, numerical integration, and the numerical solution of ordinary differential equations. Techniques of error analysis. Computer applications.

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-MATH 137B.
Plane Curves. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B; MATH 132; for MATH 137B: MATH 137A. The complex projective plane, homogeneous polynomials, plane curves; intersection multiplicities and Bezout's theorem; simple and singular points, tangents, duality; structure of cubic curves; birational transformations and 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)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010B, MATH 138A. Gaussian curvature; geodesics; Gauss-Bonnet Theorem.

MATH 141A-MATH 141B.
Fractal Geometry with Applications. (4)

Lecture,three hours; discussion, one hour. Prerequisite(s): either MATH 010B and MATH 046 or consent of instructor. Classical fractals. Fractal dimensions. Self-similar fractals. Chaotic dynamics of fractals. Dynamical systems and strange attractors. Interation theory. Julia sets. The Mandelbrot set. The beauty of fractals: The mathematics behind the computer graphics. Mathematical description of irregular shapes (clouds, trees, coastlines, mountains, galaxies, lungs, snowflakes). Applications to physics, engineering, biology, and computer science.

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-MATH 146B-MATH 146C.
Probability and Mathematical Statistics. (4-4-4) See Addenda item

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A-MATH 010B, MATH 046. Theory of linear differential equations, orthogonal expansions, boundary value problems for ordinary and partial differential equations, separation of variables, transform methods.

MATH 149A-MATH 149B-MATH 149C.
Probability and Mathematical Statistics. (4-4-4)

Lecture, three hours; laboratory, one hour. Prerequisite(s):MATH 010A-MATH 010B, MATH 046 (may be taken concurrently). An introduction to the mathematical theory of probability and statistics. Discrete and continuous distributions, sampling distributions, tests of hypotheses, estimation, maximum likelihood techniques, regression and correlation. Students may not receive credit for more than one of the sequences MATH 149A-MATH 149B-MATH 149C and STAT 160A-STAT 160B-STAT 160C.

MATH 151A-MATH 151B-MATH 151C.
Advanced Calculus. (4-4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 010A-MATH 010B, MATH 046, MATH 145A, or consent of instructor. A rigorous development of mathematical analysis. Real and complex numbers. Sequences and series. Continuity. Differentiation. The Riemann-Stieltjes integral. Sequences and series of functions. Functions of several variables.

MATH 153.
History of Mathematics. (4)

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-MATH 165B.
Introduction to Complex Variables. (4-4)

Lecture, three hours, discussion, one hour. Prerequisite(s): MATH 010B. An introduction to the theory of analytic functions of a complex variable. Mappings by elementary functions, complex integrals, and Cauchy's theorem. Power series and Laurent series, the theory of residues, and conformal mapping. 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 173.
Introduction to Algebraic Geometry. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 171, MATH 172. Plane curves. The Nullstellensatz, affine and projective varieties; tangent space and dimension; projective curves and surfaces.

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.

MATH 191 (E-Z).
Seminar in Mathematics. (1-4)

Seminar, one to four hours. Prerequisite(s): upper-division standing or consent of instructor. 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)

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.

GRADUATE COURSES  

MATH 201A-MATH 201B-MATH 201C.
Algebra. (4-4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 171, MATH 172, or equivalents. Basic theory of groups, rings, modules and fields. The Sylow theorems, modules over a principal ideal domain, the functors hom and tensor, the tensor and exterior algebras, applications to matrices, algebraic and transcendental extensions of fields and Galois theory. Rush

MATH 205A-MATH 205B-MATH 205C.
Topology. (4-4-4)

Lecture, three hours; research, three hours. Prerequisite(s): MATH 145B or equivalent. Introduction to point-set topology, homotopy theory and homology theory.

MATH 209A-MATH 209B-MATH 209C.
Real Analysis. (4-4-4)

Lecture, three hours. Prerequisite(s): MATH 151C. Measure theory, integration, representation theorems. Complex measures, integration on product spaces, differentiation. Lebesgue spaces, Hilbert space, Banach space.

MATH 210A-MATH 210B.
Complex Analysis. (4-4)

Lecture, three hours. Prerequisite(s): MATH 151C and MATH 165A. Complex functions, Cauchy's theorem and consequences, Taylor and Laurent series, representation theorems for meromorphic and entire functions, residues, harmonic functions, analytic continuation, conformal mapping, and Riemann surfaces.

MATH 211A-MATH 211B.
Ordinary Differential Equations. (4-4)

Lecture,three hours; discussion, one hour. Prerequisite(s): MATH 151C. Existence and uniqueness of solutions, linear differential equations, singularities of the first and second kinds, self-adjoint eigenvalue problems on a finite interval, singular self-adjoint boundary-value problems for second-order equations; method of averaging and numerical integration; autonomous systems. Method of Liapounov; stability for linear systems.

MATH 212A-MATH 212B.
Partial Differential Equations. (4-4)

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 213A-MATH 213B.
Partial Differential Operators. (4-4)

Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 209C. Application of functional analysis to problems of existence, uniqueness, and regularity of solutions of partial differential equations.

MATH 214A-MATH 214B.
Advanced Statistical Inference. (4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 149A-MATH 149B-MATH 149C or consent of instructor. Topics in estimation and testing hypothesis, decision, theory and Bayesian inference, multivariate analysis including time series.

MATH 216A-MATH 216B.
Combinatorial Theory. (4-4)

Lecture, three hours; discussion, one hour. Introduction to Combinatorial Optimization and Combinatorial Geometry including flows on networks, matroids, linear programming, lattices, exchange properties, Mobius function, Galois connection, coordinization.

MATH 217A-MATH 217B.
Theory of Probability. (4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 209A-MATH 209B-MATH 209C. Weak and strong limit theorems including the laws of large numbers, the central limit problem, and infinitely divisible distributions. Martingales and other topics in stochastic processes.

MATH 218.
Nonlinear Analysis. (4)

Lecture, three hours; research, three hours. Prerequisite(s): MATH 209B. Fixed point theorems, minimax methods, variational inequalities, and topological degree theory applied to differential equations, integral equations, game theory, and other parts of mathematics.

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 222.
Group Theory. (4)

Lecture, three hours. Prerequisite(s): MATH 201C. Various topics in group theory such as finite simple groups, permutation groups, group representations, infinite abelian groups, ordered groups, cohomology of groups, linear groups, reflection groups and other related groups.

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-MATH 225B.
Commutative Algebra. (4-4)

Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 201A-MATH 201B-MATH 201C. Basic theory of commutative rings, primary decomposition, integral dependence and valuation rings, intersection theorem of Krull, structure theorems for complete local rings, geometric local rings.

MATH 227A-MATH 227B.
Lie Algebras. (4-4)

Lecture, three hours; outside research,three hours. Prerequisite(s): MATH 201A-MATH 201B. Basic definitions; solvable and nilpotent Lie algebras; structure and classification of semisimple Lie algebras; enveloping algebras and representation theory; representations of semisimple Lie algebras; generalization to Kac-Moody Lie algebras; and modular Lie algebras.

MATH 228A-MATH 228B.
Functional Analysis. (4-4)

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-MATH 229B.
Stochastic Processes. (4-4)

Lecture, three hours; outside research, three hours. Prerequisite(s): either MATH 214A-MATH 214B or MATH 217A-MATH 217B or consent of instructor. Sample path analysis of stochastic processes: separability and regularity properties. Topics from martingale and Markov processes, stochastic integration, semimartingales and stochastic differential equations. Each of these topics has an extensive theory, and so the courses are repeatable.

MATH 230.
Harmonic Analysis. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 209A-MATH 209B-MATH 209C. Haar measure, classical and abstract Fourier analysis on locally compact Abelian groups, introduction to unitary representations, nonabelian harmonic analysis.

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 233.
Linear Operators. (4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 209A-MATH 209B-MATH 209C. General theory of linear operators on Hilbert space and Banach space concentration on the structure and properties of general classes of linear operators: the spectral theorem and structure of certain nonselfadjoint operators.

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 and MATH 228A, 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-MATH 243B.
Algebraic Geometry. (4-4)

Lecture, three hours; outside research, three hours. Prerequisite(s): MATH 201A-MATH 201B-MATH 201C; MATH 205A or concurrent enrollment or equivalent. Algebraic varieties in affine and projective space, mappings and correspondences, sheaves and cohomology, detailed study of curves and special topics.

MATH 246A-MATH 246B.
Algebraic Topology. (4-4)

Lecture, three hours; discussion, one hour. Prerequisite(s): MATH 205B. An introduction to simplices, geometric complexes and polytopes, manifolds, dimension theory, the topological index, homotopy, homology and transformation groups.

MATH 260.
Seminar. (1-4)

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. The course can be repeated and will be graded Satisfactory (S) or No Credit (NC).

MATH 290.
Directed Studies. (1-6)

Prerequisite(s): consent of instructor. Research and special studies in mathematics. Course may be repeated. Course will be graded Satisfactory (S) or No Credit (NC).

MATH 299.
Research for Thesis or Dissertation. (1-12)

Prerequisite(s): consent of department. Course will be graded Satisfactory (S) or No Credit (NC). Course is repeatable.

PROFESSIONAL COURSE

MATH 302.
Apprentice Teaching. (2)

Clinic, two hours. Prerequisite(s): Limited to Teaching Assistants and Associates in Mathematics. Supervised teaching in upper- and lower-division Mathematics courses. Required fall and winter quarters of all Mathematics Teaching Assistants and Associates. Intended to aid in the learning of effective teaching methods such as the handling of Mathematics discussion sections, preparation and grading of examinations, and student relations. Graded Satisfactory (S) or No Credit (NC).