### Courses in Mathematics (MAT) #### Lower Division Courses **B. Elementary Algebra (no credit)** Lecture—3 hours. Basic concepts of algebra, including polynomials, factoring, equations, graphs, and inequalities. Offered only if sufficient number of students enroll. Not open to Concurrent student enrollment. (P/NP grading only.) (There is a fee of $45.)—I. (I.) **C. Trigonometry (no credit)** Lecture—2 hours. Basic concepts of trigonometry, including trigonometric functions, identities, inverse functions, and applications. Offered only if sufficient number of students enroll. Not open to Concurrent student enrollment. (P/NP grading only.) (There is a fee of $30.)—I, II. (I, II.) **D. Intermediate Algebra (no credit)** Lecture—3 hours. Basic concepts of algebra, designed to prepare the student for college work in mathematics, such as course 16A or 21A. Functions, equations, graphs, logarithms, and systems of equations. Offered only if sufficient number of students enroll. Not open to Concurrent student enrollment. (P/NP grading only.) (There is a fee of $15.)—I, II. (I, II.) **12. Precalculus (3)** Lecture—3 hours. Prerequisite: Two years of high school algebra, plane geometry, plane trigonometry; and obtaining required score on the Precalculus Qualifying Examination. Topics selected for their use in calculus, including functions and their graphs, slope, zeroes of polynomials, exponential, logarithmic and trigonometric functions, sketching surfaces and solids. Not open for credit to students who have completed any of courses 16A, 16B, 16C, 21A, 21B, or 21C with a C– or better.—I, II, III. (I, II, III.) (Note: Mathematics 16A, 16B, and 16C are intended for students who will take no more Mathematics courses.) **16A. Short Calculus (3)** Lecture—3 hours. Prerequisite: one and one half years of high school algebra, plane geometry, plane trigonometry, and satisfaction of the Mathematics Placement Requirement. Limits; differentiation of algebraic functions; analytic geometry; applications, in particular to maxima and minima problems. Not open for credit to students who have completed course 21A. GE credit: SciEng.—I, II, III. (I, II, III.) **16B. Short Calculus (3)** Lecture—3 hours. Prerequisite: course 16A or 21A. Integration; calculus for trigonometric, exponential, and logarithmic functions; applications. Not open for credit to students who have received credit for course 21B. GE credit: SciEng.—I, II, III. (I, II, III.) **16C. Short Calculus (3)** Lecture—3 hours. Prerequisite: course 16B or 21B. Differential equations; partial derivatives; double integrals; applications; series. Not open for credit to students who have received credit for course 21C. GE credit: SciEng.—I, II, III. (I, II, III.) **17A. Calculus for Biology and Medicine (4)** Lecture—3 hours; discussion—1 hour. Prerequisite: two years of high school algebra, plane geometry, plane trigonometry, and analytical geometry, and satisfaction of the Mathematics Placement Requirement. Introduction to differential calculus via applications in biology and medicine. Limits, derivatives of polynomials, trigonometric, and exponential functions, graphing, applications of the derivative to biology and medicine. Only 2 units of credit to students who have completed course 16A. Not open for credit to students who have completed course 21A.—I. **17B. Calculus for Biology and Medicine (4)** Lecture—3 hours; discussion—1 hour. Prerequisite: course 17A or 21A. Introduction to integral calculus and elementary differential equations via applications to biology and medicine. Fundamental theorem of calculus, techniques of integration including integral tables and numerical methods, improper integrals, elementary first order differential equations, applications in biology and medicine. Only 2 units of credit to students who have completed course 16B. Not open for credit to students who have completed course 21B.—II. **17C. Calculus for Biology and Medicine (4)** Lecture—3 hours; discussion—1 hour. Prerequisite: course 17B or 21B. Matrix algebra, functions of several variables, partial derivatives, systems of differential equations, and applications to biology and medicine. Only 2 units of credit to students who have completed course 21C.—III. **21A. Calculus (4)** Lecture—3 hours; discussion—1 hour. Prerequisite: two years of high school algebra, plane geometry, plane trigonometry, and analytic geometry or course 12 and satisfaction of the Mathematics Placement Requirement. Functions, limits, continuity. Slope and derivative. Differentiation of algebraic and transcendental functions. Applications to motion, natural growth, graphing, extrema of a function. Differentials. L’Hopital’s rule. Two units of credit to students who have completed course 16A. GE credit: SciEng.—I, II, III. (I, II, III.) **21AH. Honors Calculus (4)** Lecture/discussion—4 hours. Prerequisite: a Precalculus Qualifying Examination score significantly higher than the minimum for course 21A is required. More intensive treatment of material covered in course 21A. GE credit: SciEng.—I. (I.) **21AL. Emerging Scholars Program Calculus Workshop (2)** Workshop—6 hours. Prerequisite: concurrent enrollment in course 21A. Functions, limits, continuity. Slope and derivative. Differentiation of algebraic and transcendental functions. Applications to motion, natural growth, graphing, extrema of a function. Differentials, L’Hopital’s rule. Enrollment for students in the Emerging Scholars Program by instructor’s invitation only. (P/NP grading only.)—I. (I.) **21B. Calculus (4)** Lecture—3 hours; discussion; 1 hour. Prerequisite: course 21A or 21AH. Continuation of course 21A. Definition of definite integral, fundamental theorem of calculus, techniques of integration. Application to area, volume, arc length, average of a function, improper integral, surface of revolution. Only two units of credit will be allowed to students who have received credit for course 16B or 16C. GE credit: SciEng.—I, II, III. (I, II, III.) **21BH. Honors Calculus (4)** Lecture/discussion—4 hours. Prerequisite: a grade of B or better in course 21A or 21AH. More intensive treatment of material covered in course 21B. Students completing 21BH can continue with course 21CH or the regular 21C. GE credit: SciEng.—II. (II.) **21BL. Emerging Scholars Program Calculus Workshop (2)** Workshop—6 hours. Prerequisite: course 21A or 21AH, concurrent enrollment in course 21B. Continuation of course 21A. Definition of definite integral, fundamental theorem of calculus, techniques of integration. Application to area, volume, arc length, average of a function, improper integrals, surface of revolution. Enrollment for students in the Emerging Scholars Program by instructor’s invitation only. (P/NP grading only.)—II. (II.) **21C. Calculus (4)** Lecture—3 hours; discussion; 1 hour. Prerequisite: course 21B or 21BH. Continuation of course 21B. Sequences, series, tests for convergence, Taylor expansions. Partial derivatives, total differentials. Applications to maximum and minimum problems in two or more variables. Definite integrals over plane and solid regions in various coordinate systems. Applications to physical systems. GE credit: SciEng.—I, II, III. (I, II, III.) **21CH. Honors Calculus (4)** Lecture/discussion—4 hours. Prerequisite: a grade of B or better in course 21B or 21BH. More intensive treatment of material covered in course 21C. GE credit: SciEng.—III. (III.) **21CL. Emerging Scholars Program Calculus Workshop (2)** Workshop—6 hours. Prerequisite: course 21B or 21BH, concurrent enrollment in course 21C. Continuation of course 21B. Sequences, series, tests for convergence, Taylor expansions. Partial derivatives, total differentials. Applications to maximum and minimum problems in two or more variables. Definite integrals over plane and solid regions in various coordinate systems. Applications to physical systems. Enrollment for students in the Emerging Scholars Program by instructor’s invitation only. (P/NP grading only.)—III. (III.) **21D. Vector Analysis (4)** Lecture—3 hours; discussion; 1 hour. Prerequisite: course 21C or 21CH. Continuation of course 21C. Vector algebra, vector calculus, scalar and vector fields. Line and surface integrals. Green’s theorem, Stokes’ theorem, divergence theorem.—I, II, III. (I, II, III.) **21M. Accelerated Calculus (5)** Lecture/discussion—4 hours; discussion/laboratory—1 hour. Prerequisite: grade of B or higher in both semesters of high school calculus or a score of 4 or higher on the Advanced Placement Calculus AB exam, and obtaining the required score on the Precalculus Qualifying Examination and its trigonometric component. Accelerated treatment of material from courses 21A and 21B, with detailed presentation of theory, definitions, and proofs, and treatment of computational aspects of calculus at a condensed but sophisticated level. Not open for credit to students who have completed course 21A or 21B; only 3 units of credit will be allowed to students who have completed course 16A and only 2 units of credit will be allowed to students who have completed course 16B. GE credit: SciEng.—I. (I.) **22A. Linear Algebra (3)** Lecture—3 hours. Prerequisite: nine units of college mathematics and Engineering 6 or knowledge of Matlab or course 22AL (to be taken concurrently). Matrices and linear transformation, determinants, eigenvalues, eigenvectors, diagonalization, factorization.—I, II, III. (I, II, III.) **22AL. Linear Algebra Computer Laboratory (1)** Laboratory—2-3 hours. Prerequisite: nine units of college mathematics. Introduction to Matlab and its use in linear algebra. (P/NP grading only.)—I, II, III. (I, II, III.) **22B. Differential Equations (3)** Lecture—3 hours. Prerequisite: courses 21C, 22A. Solutions of elementary differential equations.—I, II, III. (I, II, III.) **36. Fundamentals of Mathematics (3)** Lecture—3 hours. Prerequisite: satisfaction of the Mathematics Placement Requirement. Introduction to fundamental mathematical ideas selected from the principal areas of modern mathematics. Properties of the primes, the fundamental theorems of arithmetic, properties of the rationals and irrationals, binary and other number systems. Not open for credit to students who have completed course 108. GE credit: SciEng.—I. **71A-71B. Explorations in Elementary Mathematics (3-3)** Lecture—2 hours; laboratory—3 hours. Prerequisite: two years of high school mathematics. Weekly explorations of mathematical ideas related to the elementary school curriculum will be carried out by cooperative learning groups. Lectures will provide background and synthesize the results of group exploration. (Deferred grading only, pending completion of sequence.)—I-II. (I-II.) **89. Elementary Problem Solving (1)** Lecture—1 hour. Prerequisite: high school mathematics through precalculus. Solve and present solutions to challenging and interesting problems in elementary mathematics. May be repeated once for credit. (P/NP grading only.)—I, II, III. **98. Directed Group Study (1-5)** Prerequisite: consent of instructor. (P/NP grading only.) **99. Special Study for Undergraduates (1-5)** Prerequisite: consent of instructor. (P/NP grading only.) #### Upper Division Courses **108. Introduction to Abstract Mathematics (4)** Lecture/discussion—4 hours. Prerequisite: course 21B or consent of instructor. Rigorous treatment of abstract mathematics with the emphasis on developing ability to understand and present mathematics arguments. GE credit: Wrt.—I, II, III. (I, II, III.) **111. History of Mathematics (4)** Lecture—3 hours; term paper. Prerequisite: 8 units of upper division mathematics including course 108. The history of mathematics from ancient times through the development of calculus. Mathematics from Arab, Hindu, Chinese and other cultures. Selected topics from the history of modern mathematics.—I. (I.) **114. Convex Geometry (4)** Lecture—3 hours; extensive problem solving. Prerequisite: courses 21C, 22A, 108, or consent of instructor. Topics selected from the theory of convex bodies, convex functions, geometric inequalities, combinatorial geometry, and integral geometry. Offered in alternate years.—(II.) **115A. Number Theory (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 108. Divisibility and related topics, diophantine equations, selected topics from the theory of prime numbers.—I. (I.) **115B. Number Theory (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 108. Euler function, Moebius function, congruences, primitive roots, quadratic reciprocity law. Offered in alternate years.—II. **115C. Number Theory (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 108. Continued fractions, partitions. Offered in alternate years.—III. **116. Differential Geometry (4)** Lecture—3 hours; extensive problem solving. Prerequisite: courses 22A, 21D, or consent of instructor. Vector analysis, curves and surfaces in three dimensions. Offered in alternate years.—III. **118A. Partial Differential Equations: Elementary Methods (4)** Lecture—3 hours; extensive problem solving. Prerequisite: courses 22A, 22B, 21D. Derivation of partial differential equations; separation of variables; equilibrium solutions and Laplace’s equation; Fourier series; method of characteristics for the one-dimensional wave equation; solution of nonhomogeneous equations.—I. (I.) **118B. Partial Differential Equations: Eigenfunction Expansions (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 118A. Sturm-Liouville Theory; self-adjoint operators; mixed boundary conditions; partial differential equations in two and three dimensions; Eigenvalue problems in circular domains; nonhomogeneous problems and the method of eigenfunction expansions; Poisson’s Equations.—II. (II.) **118C. Partial Differential Equations: Green’s Functions and Transforms (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 118B. Green’s functions for one-dimensional problems and Poisson’s equation; Fourier transforms; Green’s Functions for time dependent problems; Laplace transform and solution of partial differential equations.—III. (III.) **119A-119B. Ordinary Differential Equations (4-4)** Lecture—3 hours; extensive problem solving. Prerequisite: courses 22A, 22B. Scalar and Planar Autonomous Systems, nonlinear systems and linearization. Phase plane analysis. Classification of singular points. Scalar and Planar maps. Bifurcations and the implicit function theorem. Notions of stability and Liapunov’s method. Periodic orbits and their bifurcations. Poincare Bendixon theory.—II-III. (II-III.) **121. Advanced Analysis for the Sciences (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 21D, 22A, 22B. Introduction to Fourier series and Fourier transforms. Theory and solutions of basic partial differential equations, such as Laplace, heat and wave equations. Green’s functions. Special functions of importance in physics and engineering.—I. (I.) **124. Mathematical Biology (4)** Lecture—3 hours; project—3 hours. Prerequisite: knowledge of a computer language or Matlab, course 22B or the equivalent. Methods of mathematical modeling of biological systems including difference equations, ordinary differential equations, stochastic and dynamic programming models. Computer simulation methods as applied to biological systems. Applications to population growth, cell biology, physiology, evolutionary ecology and protein clustering. Offered in alternate years.—III. **127A-127B-127C. Advanced Calculus (4-4-4)** Lecture/discussion—4 hours. Prerequisite: courses 21D, 22A, 108. Real number system, continuity, differentiation and integration on the real line; vector calculus and functions of several variables; theory of convergence.—I-II-III, II-III. (I-II-III, II-III.) **128A. Numerical Analysis (4)** Lecture—3 hours; term project. Prerequisite: course 21C; knowledge of a programming language such as Pascal, FORTRAN or BASIC. Error analysis, approximation, interpolation, numerical differentiation and integration.—I. (I.) **128B. Numerical Analysis in Solution of Equations (4)** Lecture—3 hours; term project. Prerequisite: courses 21C and 22A; knowledge of a programming language such as Pascal, FORTRAN or BASIC. Solution of nonlinear equations and nonlinear systems. Minimization of functions of several variables. Simultaneous linear equations. Eigenvalue problems.—II. (II.) **128C. Numerical Analysis in Differential Equations (4)** Lecture—3 hours; term project. Prerequisite: courses 22A, 22B, and a knowledge of a programming language such as Pascal, FORTRAN or BASIC. Difference equations, operators, numerical solution of ordinary and partial differential equations.—III. (III.) **131. Probability Theory (4)** Lecture—4 hours. Prerequisite: course 21C, 22A. Probability space, event, combinatorics; discrete, continuous distributions; random variables; joint, marginal, conditional densities; transformation; expectation; sums and moments; inequalities; laws of large numbers; central limit law; probability models via conditioning. Not open for credit to students who have completed Statistics 131A.—I, II, III. (I, II, III.) **132A-132B. Stochastic Processes (4-4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 131 or Statistics 131A. Markov chains, Poisson process, birth and death processes, renewal theory, queueing theory, Brownian motion, stationary processes. Course 132B is offered in alternate years.—II. (II-III.) **141. Euclidean Geometry (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 108. An axiomatic and analytic examination of Euclidean geometry from an advanced point of view. In particular, a discussion of its relation to other geometries.—II. (II.) **145. Combinatorics (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 108. Combinatorial methods using basic graph theory counting methods, generating functions, and recurrence relations.—III. (III.) **147. Topology (4)** Lecture—3 hours; extensive problem solving. Prerequisite: courses 108, 127A. Basic notions of point-set and combinatorial topology.—III. (III.) **149A-149B. Discrete Mathematics (4-4)** Lecture/discussion—4 hours. Prerequisite: courses 22A and 108. Coding theory and counting theory and the algebraic concepts needed in their development.—II-III. (II-III.) **150A-150B-150C. Modern Algebra (4-4-4)** Lecture/discussion—4 hours. Prerequisite: course 108. Basic concepts of groups, rings, and fields. Emphasis on the techniques used in the proof of the ideas (Lemmas, Theorems, etc.) developing these concepts. Precise thinking, writing, and the ability to deal with abstraction.—I-II-III. (I-II-III.) **160. Mathematical Foundations of Database Theory, Design and Performance (4)** Lecture—3 hours; term project. Prerequisite: course 108 and familiarity with one high-level computer language. The relational model; relational algebra, relational calculus, normal forms, functional and multivalued dependencies. Separability. Cost benefit analysis of physical database design and reorganization. Performance via analytical modeling, simulation, and queueing theory. Block accesses; buffering; operating system contention; CPU intensive operations. Offered in alternate years.—(I.) **165. Mathematics and Computers (4)** Lecture—3 hours; project—3 hours. Prerequisite: Computer Science Engineering 30 or the equivalent, course 22B, 108. Computational mathematics and computer generated/verified proofs in algebra, analysis and geometry. Investigation of rigorous new mathematics developed in conjunction with modern computational questions and the role that computers play in mathematical conjecture and experimentation.—III. (III.) **167. Advanced Linear Algebra (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 22A. Introduction to linear algebra; linear equations, orthogonal projections, similarity transformations, quadratic forms, eigenvalues and eigenvectors. Applications to physics, engineering, economics, biology and statistics.—I, II, III. (I, II, III.) **168. Mathematical Programming (4)** Lecture—3 hours; extensive problem solving. Prerequisite: courses 21C, and 22A or 167; knowledge of a programming language. Linear programming, simplex method. Basic properties of unconstrained nonlinear problems, descent methods, conjugate direction method. Constrained minimization.—III. (III.) **180. Special Topics (3)** Lecture—3 hours. Prerequisite: course 22B or consent of instructor. Special topics from various fields of modern pure and applied mathematics. Some recent topics include Knot Theory, General Relativity, and Fuzzy Sets. May be repeated for credit when topic differs.—I, II, III. (I, II, III.) **185A. Complex Analysis with Applications (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 21D. Complex number systems, analyticity and the Cauchy-Riemann equations, elementary functions, complex integration, power and Laurent series expansions, residue theory.—II. (II.) **185B. Complex Analysis with Applications (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 185A or consent of instructor. Analytic functions, elementary functions and their mapping properties, applications of Cauchy’s integral theorem, conformal mapping and applications to heat flow and fluid mechanics. Offered in alternate years.—(III.) **189. Advanced Problem Solving (1)** Lecture—1 hour. Prerequisite: courses 21A-21B-21C-21D, 22A, 22B, or the equivalent. Introduction to advanced problem solving techniques outside the usual context of homework style problems. The problems only require a background in second year college mathematics. May be repeated once for credit. (P/NP grading only.)—I, II, III. **192. Internship in Applied Mathematics (1-3)** Internship; final report. Prerequisite: upper division standing; project approval by faculty sponsor prior to enrollment. Supervised work experience in applied mathematics. May be repeated for credit for a total of 10 units. (P/NP grading only.) **194. Undergraduate Thesis (3)** Prerequisite: consent of instructor. Independent research under supervision of a faculty member. Student will submit written report in thesis form. May be repeated with consent of Vice Chairperson. (P/NP grading only.)—I, II, III. (I, II, III.) **197TC. Tutoring Mathematics in the Community (1-5)** Seminar—1-2 hours; laboratory—2-6 hours. Prerequisite: upper division standing and consent of instructor. Special projects in mathematical education developing techniques for mathematics instruction and tutoring on an individual or small group basis. May be repeated once for credit. (P/NP grading only.) **198. Directed Group Study (1-5)** Prerequisite: consent of instructor. (P/NP grading only.) **199. Special Study for Advanced Undergraduates (1-5)** (P/NP grading only.) #### Graduate Courses **201A-201B-201C. Analysis (4-4-4)** Lecture—3 hours; discussion—1 hour. Prerequisite: course 127C or 203C or consent of instructor. Topological, metric, normed spaces. Stone-Weierstrass theorem. Contraction mapping theorem. Banach spaces. Bounded linear maps. Lebesgue measure. Fubini and Radon-Nikodym theorems. Lp spaces. Distributions, Fourier transform. Linear operators on Hilbert spaces. Spectral theorem. Variational methods.—I-II-III. (I-II-III.) **202. Functional Analysis (4)** Lecture—3 hours; term paper. Prerequisite: course 201A-201B-201C. The theory of Fredholm operators. Examples of Fredholm operators (singular integral operators, elliptic operators in Sobolov spaces). Index theory for Fredholm operators. Unbounded self-adjoint operators. Schrodinger operators and other differential operators. The spectral theorem for these and for unitary operators. Offered in alternate years.—II. **203A-203B-203C. Modern Applied Analysis (4-4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: graduate standing or consent of instructor. Metric and normed spaces. Continuous functions. Topological, Hilbert, and Banach spaces. Fourier series. Spectrum of bounded linear operators. Compact and linear differential operators. Green’s functions. Distributions, Fourier transform. Measure theory. Lp and Sobolev spaces. Calculus of Variations.—I-II-III. (I-II-III.) **204. Applied Asymptotic Analysis (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: graduate standing or consent of instructor. Scaling and non-dimensionalization. Asymptotic expansions. Regular and singular perturbation methods. Applications to algebraic and ordinary and partial differential equations in the natural sciences and engineering. Offered in alternate years.—I. **205. Complex Analysis (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 185 or the equivalent or consent of instructor. Analytic continuation, Riemann mapping theorem, elliptic functions, modular forms, Riemann zeta function, Riemann surfaces. Offered in alternate years.—I. **206. Measure Theory (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 127C. Introduction to measure theory. The study of lengths, surface areas, and volumes in general spaces, as related to integration theory. Offered in alternate years.—III. **210A. Topics in Geometry (3)** Lecture—3 hours. Prerequisite: bachelor’s degree in mathematics or consent of instructor. Topics in advanced geometry related to curriculum at all levels. Required for M.A.T. degree program for prospective teachers. May be repeated for credit with prior consent of instructor.—I. (I.) **210AL. Topics in Geometry: Discussion (1)** Lecture/discussion—1 hour (to be arranged). Prerequisite: course 210A (concurrently); consent of instructor. Special topics related to course 210A which are of special interest to teachers and candidates for M.A.T. degree program. May be repeated for credit.—I. (I.) **210B. Topics in Algebra (3)** Lecture—3 hours. Prerequisite: bachelor’s degree in mathematics or consent of instructor. Topics in advanced algebra related to curriculum at all levels. Required for M.A.T. degree program for prospective teachers. May be repeated for credit with prior consent of instructor.—II. (II.) **210BL. Topics in Algebra: Discussion (1)** Lecture/discussion—1 hour (to be arranged). Prerequisite: course 210B (concurrently); consent of instructor. Special topics related to course 210B which are of special interest to teachers and candidates for M.A.T. degree program. May be repeated for credit.—II. (II.) **210C. Topics in Analysis (3)** Lecture—3 hours. Prerequisite: bachelor’s degree in mathematics or consent of instructor. Topics in advanced analysis related to curriculum at all levels. Required for M.A.T. degree program for prospective teachers. May be repeated for credit with prior consent of instructor.—III. (III.) **210CL. Topics in Analysis: Discussion (1)** Lecture/discussion—1 hour (to be arranged). Prerequisite: course 210C (concurrently); consent of instructor. Special topics related to course 210C which are of special interest to teachers and candidates for M.A.T. degree program. May be repeated for credit.—III. (III.) **215A-215B-215C. Topology (4-4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: graduate standing or consent of instructor. Fundamental group and covering space theory. Homology and cohomology. Manifolds and duality. CW complexes. Fixed point theorems. Offered in alternate years.—I-II-III. **218A-218B. Partial Differential Equations (4-4)** Lecture—3 hours; term paper or discussion. Prerequisite: courses 22A, 127C. Initial and boundary value problems for elliptic, parabolic and hyperbolic partial differential equations; existence, uniqueness and regularity for linear and nonlinear equations; maximum principles; weak solutions, Holder and Sobolev spaces, energy methods; Euler-Lagrange equations.—II-III. (II-III.) **219. Ordinary Differential Equations (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 22A, 22B, 127C or consent of instructor. Theory of ordinary differential equations. Dynamical systems. Geometric theory. Normal forms. Bifurcation theory. Chaotic systems. Offered in alternate years.—I. **221A. Mathematical Fluid Dynamics (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 118B or consent of instructor. Kinematics and dynamics of fluids. The Euler and Navier-Stokes equations. Vorticity dynamics. Irrotational flow. Low Reynolds number flows and the Stokes equations. High Reynolds number flows and boundary layers. Compressible fluids. Shock waves. Offered in alternate years.—I. **221B. Mathematical Fluid Dynamics (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 118B or consent of instructor. Kinematics and dynamics of fluids. The Euler and Navier-Stokes equations. Vorticity dynamics. Irrotational flow. Low Reynolds number flows and the Stokes equations. High Reynolds number flows and boundary layers. Compressible fluids. Shock waves. Offered in alternate years. **222. Introduction to Biofluid Dynamics (3)** Lecture—3 hours. Prerequisite: Population Biology 231/Ecology 231 and Neurobiology, Physiology and Behavior 245 or consent of instructor. The basic principles of fluid dynamics are introduced in the first half of the course by describing various phenomena studies from a biofluids perspective. The equations of fluid motion associated with these phenomena are derived and studied in the second half.—III. (III.) **227. Mathematical Biology (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: graduate standing or consent of instructor. Nonlinear ordinary and partial differential equations and stochastic processes of cell and molecular biology. Scaling, qualitative, and numerical analysis of mathematical models. Applications to nerve impulse, chemotaxis, muscle contraction, and morphogenesis. Offered in alternate years.—I. **228A-228B-228C. Numerical Solution of Differential Equations (4-4-4)** Lecture—3 hours; term paper or discussion. Prerequisite: course 128C. Numerical solutions of initial-value, eigenvalue and boundary-value problems for ordinary differential equations. Numerical solution of parabolic and hyperbolic partial differential equations. Offered in alternate years.—I-II-III. **229A-229B. Numerical Methods in Linear Algebra (4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 22A and 167 or the equivalent or consent of instructor. Computational methods for the solution of linear algebraic equations and matrix eigenvalue problems. Analysis of direct and iterative methods. Special methods for sparse matrices. Offered in alternate years—II-III. **235A-235B-235C. Probability Theory (4-4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 127C and 131 or Statistics 131A or consent of instructor. Measure-theoretic foundations, abstract integration, independence, laws of large numbers, characteristic functions, central limit theorems. Weak convergence in metric spaces, Brownian motion, invariance principle. Conditional expectation. Topics selected from martingales, Markov chains, ergodic theory. (Same course as Statistics 235A-235B-235C.)—I-II-III. **236A-236B. Stochastic Dynamics and Applications (4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 201C or course/Statistics 235B; course/Statistics 235A-235B-235C recommended. Stochastic processes, Brownian motion, Stochastic integration, martingales, stochastic differential equations. Diffusions, connections with partial differential equations, mathematical finance. Offered in alternate years.—I-II. **240A-240B-240C. Differential Geometry (4-4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 116 or consent of instructor. Manifolds. Differentiable structures. Vector fields and tangent spaces. Bundles, tensors, forms, Grassman algebras. DeRham cohomology. Riemannian geometry. Connections, curvature, geodesics, submanifolds. Curves and surfaces. Positive and negative curvature; Morse Theory; homogeneous spaces; Hodge theory; applications. Offered in alternate years.—I-II-III. **245. Enumerative Combinatorics (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 145, 150 or the equivalent, or consent of instructor. Introduction to modern combinatorics and its applications. Emphasis on enumerative aspects of combinatorial theory. Offered in alternate years.—I. **246. Algebraic Combinatorics (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 245 or consent of instructor. Algebraic and geometric aspects of combinatorics. The use of structures such as groups, polytopes, rings, and simplicial complexes to solve combinatorial problems. Offered in alternate years—II. **250A-250B-250C. Algebra (4-4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: graduate standing in mathematics or consent of instructor. Group and rings. Sylow theorems, abelian groups, Jordan-Holder theorem. Rings, unique factorization. Algebras, and modules. Fields and vector spaces over fields. Field extensions. Commutative rings. Representation theory and its applications.—I-II-III. (I-II.) **258A. Numerical Optimization (4)** Lecture—3 hours; term paper or discussion. Prerequisite: courses 127A, 167. Numerical methods for infinite dimensional optimization problems. Newton and Quasi-Newton methods, linear and sequential quadratic programming, barrier methods; large-scale optimization; theory of approximations; infinite and semi-infinite programming; applications to optimal control, stochastic optimization and distributed systems.—I. (I.) **258B. Variational Analysis (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 127A and 167 or consent of instructor. Foundations of optimization theory. The design of solution procedures for optimization problems. Modeling issues, and stability analysis. Offered in alternate years.—II. (II.) **261A-261B. Lie Groups and Their Representations (4-4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 215A, 240A, 250A-250B or the equivalent or consent of instructor. Lie groups and Lie algebras. Classification of semi-simple Lie groups. Classical and compact Lie groups. Representations of Lie groups and Lie algebras. Root systems, weights, Weil character formula. Kac-Moody and Virasoro algebras. Applications. Offered in alternate years.—II-III. **265. Mathematical Quantum Mechanics (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 201 or consent of instructor. Mathematical foundations of quantum mechanics: the Hilbert space and Operator Algebra formulations; the Schrödinger and Heisenberg equations, symmetry in quantum mechanics, basics of spectral theory and perturbation theory. Applications to atoms and molecules. The Dirac equation. Offered in alternate years.—I. **266. Mathematical Statistical Mechanics and Quantum Field Theory (4)** Lecture—3 hours; term paper or discussion—1 hour. Prerequisite: course 265 or consent of instructor. Mathematical principles of statistical mechanics and quantum field theory. Topics include classical and quantum lattice systems, variational principles, spontaneous symmetry breaking and phase transitions, second quantization and Fock space, and fundamentals of quantum field theory. Offered in alternate years.—II. **271. Applied and Computational Harmonic Analysis (4)** Lecture—3 hours; extensive problem solving. Prerequisite: course 121, 127C, 128A, 128B, 167, 201C, or the equivalent, or consent of instructor. Introduction to mathematical basic building blocks (wavelets, local Fourier basis, and their relatives) useful for diverse fields (signal and image processing, numerical analysis, and statistics). Emphasis on the connection between the continuum and the discrete worlds. Offered in alternate years.—II. **280. Topics in Pure and Applied Mathematics (3)** Lecture—3 hours. Prerequisite: graduate standing. Special topics in various fields of pure and applied mathematics. Topics selected based on the mutual interests of students and faculty. May be repeated for credit when topic differs.—I, II, III. (I, II, III.) **290. Seminar (1-6)** Seminar—1-6 hours. Advanced study in various fields of mathematics, including analysis, applied mathematics, discrete mathematics, geometry, mathematical biology, mathematical physics, optimization, partial differential equations, probability, and topology. May be repeated for credit. (S/U grading only.)—I, II, III. **298. Group Study (1-5)** **299. Individual Study (1-12)** (S/U grading only.) **299D. Dissertation Research (1-12)** (S/U grading only.) #### Professional Courses **301A-301B-301C. Mathematics Teaching Practicum (3-3-3)** Fieldwork—5 hours; discussion—1 hour. Prerequisite: course 302A-302B-302C and 303A-303B-303C concurrently or consent of instructor. Specialist training in mathematics teaching. Teaching, training, and cross observing classes taught using large group Socratic techniques, small group guided inquiry experiences, and/or other approaches to teaching at various grade levels. Required for advanced degrees in mathematics education. May be repeated once for credit.—I-II-III. **302A-302B-302C. Curriculum Development in Mathematics (1-1-1)** Lecture/discussion—1 hour. Prerequisite: course 303A-303B-303C concurrently or consent of instructor. Mathematics curriculum development for all grade levels. Required for advanced degrees in mathematics education. May be repeated once for credit.—I-II-III. **303A-303B-303C. Mathematics Pedagogy (1-1-1)** Lecture/discussion—1 hour. Prerequisite: course 302A-302B-302C or 210L concurrently or consent of instructor. An investigation of the interplay of mathematical pedagogy and mathematical content, including a historical survey of past and present methods in view of some of the influences that shaped their development. May be repeated once for credit.—I-II-III. **390. Methods of Teaching Mathematics (3)** Lecture—1 hour; discussion—1 hour; laboratory—2 hours. Prerequisite: graduate standing. Practical experience in methods and problems of the teaching of mathematics at the university level. Includes discussion of lecturing techniques, analysis of tests and supporting material, preparation and grading of examinations, and related topics. Required of departmental teaching assistants. May be repeated for credit. (S/U grading only.)—I. (I.) **399. Individual Study (2-4)** Independent study—2-3 hours; discussion—1 hour. Individual study of some aspect of mathematics education or a focused work on a curriculum design project under supervision of a faculty member in mathematics. May be repeated for credit. (S/U grading only.)—I, II, III. (I, II, III.) |