Courses in Mathematics

(Schedule of Classes designation: Math)

LOWER-DIVISION

1A-B Pre-Calculus. Lecture, three hours; discussion, one hour.

1A (0) F, Summer. Course may be offered online. Basic equations and inequalities, linear and quadratic functions, and systems of simultaneous equations. Four units of workload credit only.

1B (4) F, W, Summer. Course may be offered online. Preparation for calculus and other mathematics courses. Exponentials, logarithms, trigonometry, polynomials, and rational functions. Satisfies no requirements other than contribution to the 180 units required for graduation. Prerequisite: Mathematics 1A, or Mathematics 1B placement via the Calculus Placement exam (fee required), or a score of 450 or higher on the Mathematics section of the SAT Reasoning Test.

2A-B Single-Variable Calculus (4-4) F, W, S, Summer. Lecture, three hours; discussion, two hours. 2A: Introduction to derivatives, calculation of derivatives of algebraic and trigonometric functions; applications including curve sketching, related rates, and optimization. Exponential and logarithm functions. Prerequisite: Mathematics 2A placement via the Calculus Placement exam (fee required), or a grade of C or better in Mathematics 1B at UCI, or a score of 3 on the AP Calculus AB exam, or a score of 600 or higher on the Mathematics section of the SAT Reasoning Test. 2B: Definite integrals; the fundamental theorem of calculus. Applications of integration including finding areas and volumes. Techniques of integration. Infinite sequences and series. Parametric and polar equations. Prerequisite for Mathematics 2B: 2A. Majors in the Schools of Physical Sciences, Engineering, and Information and Computer Sciences have first consideration for enrollment. (2A, 2B: Vb)

2D-E Multivariable Calculus. Lecture, three hours; discussion, two hours.

2D (4) F, W, S, Summer. Differential and integral calculus of real-valued functions of several real variables, including applications. Polar coordinates. Prerequisites: Mathematics 2A-B. Mathematics 2D and H2D may not both be taken for credit. Majors in the Schools of Physical Sciences, Engineering, and Information and Computer Sciences have first consideration for enrollment. (Vb)

2E (4) F, W, S, Summer. The differential and integral calculus of vector-valued functions. Implicit and inverse function theorems. Line and surface integrals, divergence and curl, theorems of Green, Gauss, and Stokes. Prerequisite: 2D. Mathematics 2E and H2E may not both be taken for credit.

H2D-E Honors Multivariable Calculus (4-4) W, S. Lecture, three hours; discussion, two hours. Covers the same material as Mathematics 2D-E, but with a greater emphasis on the theoretical structure of the subject matter. Especially recommended for prospective Mathematics majors and others with a particular interest in mathematics. Satisfies the same requirements and prerequisites as 2D-E. Prerequisites for H2D: a grade of B (3.0) or better in Mathematics 2B or a score of 4 or 5 on the Advanced Placement Calculus BC examination; for H2E: a grade of C (2.0) or better in Mathematics H2D. Mathematics 2D-E and H2D-E may not both be taken for credit. (H2D: Vb)

2J Infinite Series and Basic Linear Algebra (4) F, W, S, Summer. Lecture, three hours; discussion, two hours. Systems of linear equations: matrix operations; determinants; eigenvalues, and eigenvectors. Infinite sequences and series. Complex numbers. Prerequisites: Mathematics 2A-B. Majors in the Schools of Physical Sciences and Engineering have first consideration for enrollment. (Vb)

3A Introduction to Linear Algebra (4) F, W, S. Lecture, three hours; discussion, two hours. Systems of linear equations, matrix operations, determinants, eigenvalues and eigenvectors, vector spaces, subspaces and dimension. Prerequisite: Mathematics 2B. Only one course from Mathematics 3A, Mathematics 6G, and ICS 6N may be taken for credit. Mathematics and School of Engineering majors have first consideration for enrollment.

3D Elementary Differential Equations (4) F, W, S, Summer. Lecture, three hours; discussion, two hours. Linear differential equations, variation of parameters, constant coefficient cookbook, systems of equations, Laplace transforms, series solutions. Prerequisites: Mathematics 2D; 2J or 3A. School of Physical Sciences and School of Engineering majors have first consideration for enrollment.

4 Mathematics for Economists (4) F, S. Lecture, three hours; discussion, two hours. Topics in linear algebra and multivariable differential calculus suitable for economic applications. Prerequisite: Mathematics 2B. No credit for Mathematics 4 if taken after both Mathematics 2D and either 2J or 3A. Economics, Business Economics, and Quantitative Economics majors have first consideration for enrollment. (Vb)

6G Linear Algebra (4) F, W, S, Summer. Lecture, three hours; discussion, two hours. Linear equations, vector spaces and subspaces, linear functions and matrices, linear codes, determinants, scalar products. Prerequisite: high school mathematics through trigonometry. Only one course from Mathematics 3A, Mathematics 6G, and ICS 6N may be taken for credit. NOTE: Mathematics majors must take 3A. (Vb)

8 Explorations in Functions and Modeling (4) S. Lecture, three hours; discussion, one hour. Explorations of applications and connections in topics in algebra, geometry, calculus, and statistics for future secondary math educators. Emphasis on nonstandard modeling problems. Corequisite: Mathematics 2A.

13 Introduction to Abstract Mathematics (4) F, W, S. Lecture, three hours; discussion, two hours. Introduction to formal definition and rigorous proof writing in mathematics. Topics include basic logic, set theory, equivalence relations, and various proof techniques such as direct, induction, contradiction, contrapositive, and exhaustion. Prerequisite: Mathematics 2A or Mathematics 6D/ICS 6D. Mathematics majors have first consideration for enrollment.

77A, B, C, D Topics in Mathematics and Computation in the Digital Age. Lecture, three hours; laboratory, two hours. Corequisite: Mathematics 2J or 6G, or consent of instructor. Prerequisites: Mathematics 2A-B; ICS 21/CSE21 or Informatics 41 or consent of instructor. First- and second-year students only.

77A Introduction to Signal Processing (4). Signals in MATLAB; blurring, mixing, filtering; elements of linear algebra, statistics, optimization; blind matrix inversion; de-correlation method, stochastic gradient descent method, applications to sounds and images. Same as ICS 77A. (II, Va)

77B Introduction to Collaborative Filtering (4). Basic concepts of collaborative filtering; clustering; matrix factorization and principal components analysis; regression; classification; naive Bayes classifier, decision trees, Perceptron (neural networks). Same as ICS 77B. (II, Va)

77C Introduction to Image Processing (4). Image de-noising, de-blurring, low pass filtering; image segmention and classification; sparse representation; visualization. Same as ICS 77C. (II, Va)

77D Introduction to Game Simulation and Analysis (4). Combinatorial Game Theory—game classification, tree graphs, strategy analysis, Sprague Grundy functions, Bouton's Theorem; Zero-Sum and General-Sum Game Theory—payoff matrices, Minimax Theorem, Nash equilibrium; machine learning—search algorithms. Same as ICS 77D. (II, Va)

UPPER-DIVISION

NOTE: Some of the upper-division courses listed below have one or two hours of discussion weekly in addition to the lectures. Not all courses are offered every year.

105A-B Numerical Analysis (4-4) F, W. Lecture, three hours. Introduction to the theory and practice of numerical computation. 105A: Floating point arithmetic, roundoff; solving transcendental equations; quadrature; linear systems, eigenvalues, power method. Corequisite: Mathematics 105LA if offered. Prerequisites: Mathematics 2J or 3A; some acquaintance with computer programming. Mathematics 105A and Engineering MAE185 may not both be taken for credit. 105B: Lagrange interpolation, finite differences, splines, Padé approximations; Gaussian quadrature; Fourier series and transforms. Corequisite: Mathematics 105LB if offered. Prerequisite: Mathematics 105A.

105LA-LB Numerical Analysis Laboratory (1-1) F, W. Laboratory, two hours. Provides practical experience to complement the theory developed in Mathematics 105A-B. Corequisite: concurrent enrollment in Mathematics 105A-B.

107 Numerical Differential Equations (4) S. Lecture, three hours. Theory and applications of numerical methods to initial and boundary-value problems for ordinary and partial differential equations. Corequisite: concurrent enrollment in Mathematics 107L if offered. Prerequisites: Mathematics 3D and 105A-B.

107L Numerical Differential Equations Laboratory (1) S. Laboratory, two hours. Provides practical experience to complement the theory developed in Mathematics 107. Corequisite: concurrent enrollment in Mathematics 107.

112A-B-C Introduction to Partial Differential Equations and Applications (4-4-4). Lecture, three hours. Introduction to ordinary and partial differential equations and their applications in engineering and science. Basic methods for classical PDEs (potential, heat, and wave equations). 112A: Classification of PDEs, separation of variables and series expansions, special functions, eigenvalue problems. 112B: Green functions and integral representations, method of characteristics. 112C: Galerkin method and other discretization techniques. Prerequisites for 112A: Mathematics 2D, 3D; for 112B: 2E and 112A.

113A-B-C Mathematical Modeling in Biology (4-4-4). Lecture, three hours; discussion, two hours. 113A: Discrete mathematical and statistical models; difference equations, population dynamics, Markov chains, and statistical models in biology. 113B: Linear algebra; differential equations models; dynamical systems; stability; hysteresis; phase plane analysis; applications to cell biology, viral dynamics, and infectious diseases. 113C: Partial differential equations models in biology such as one dimensional blood flow, morphogen gradients, and tumor growth; stochastic models in cancer and epidemiology. Prerequisite for 113A: Mathematics 2B; for 113B: Mathematics 113A; for 113C: Mathematics 113B.

114A Applied Complex Analysis (4) F. Lecture, three hours. Introduction to complex functions and their applications to engineering and science. Complex numbers, elementary functions; analytic functions; complex integration; power series; residue theory; conformal maps; applications. Prerequisites: Mathematics 2D and either 2J or 3A. Mathematics 2E and 3D recommended. Mathematics 114A may not be taken for credit after Mathematics 147.

115 Mathematical Modeling (4). Lecture, three hours. Mathematical modeling and analysis of phenomena that arise in engineering physical sciences, biology, economics, or social sciences. Corequisite or prerequisite: Mathematics 112A or Engineering MAE140. Prerequisites: Mathematics 2D; 3A or 6G; 3D.

117 Dynamical Systems (4). Lecture, three hours; discussion, two hours. Introduction to the modern theory of dynamical systems including contraction mapping principle, fractals and chaos, conservative systems, Kepler problem, billiard models, expanding maps, Smale's horseshoe, topological entropy. Prerequisites: Mathematics 3D and 140A. Mathematics 117 and 118B may not both be taken for credit.

118A-B-C The Theory of Differential Equations (4-4-4). Lecture, three hours; discussion, one hour. 118A: Existence and uniqueness of solutions, continuous dependence of solutions on initial conditions and parameters, Lyapunov and asymptotic stability, Floquet theory, nonlinear systems, and bifurcations. 118B: Dynamical systems. 118C: Boundary value problems in ordinary differential equations. Prerequisites for 118A: Mathematics 3D and 140A. Mathematics 118B and 117 may not both be taken for credit. Mathematics 118C and 119 may not both be taken for credit.

119 Boundary Value Problems (4). Lecture, three hours; discussion, two hours. Introduction to boundary value problems including Green's function representations, maximum principle, variational formulations, Sturm-Liouville problems, eigenfunction expansions, existence and uniqueness for nonlinear problems, method of shooting, finite difference methods. Prerequisites: Mathematics 3D and 140A; Mathematics 118A recommended. Mathematics 119 and 118C may not both be taken for credit.

120A Introduction to Abstract Algebra: Groups (4) F, W. Lecture, three hours; discussion, two hours. Axioms for group theory; permutation groups, matrix groups. Isomorphisms, homomorphisms, quotient groups. Advanced topics as time permits. Special emphasis on doing proofs. Prerequisites: Mathematics 3A or 6G; 13. Mathematics majors have first consideration for enrollment.

120B Introduction to Abstract Algebra: Rings and Fields (4) W, S. Lecture, three hours; discussion, two hours. Basic properties of rings; ideals, quotient rings; polynomial and matrix rings. Elements of field theory. Prerequisite: Mathematics 120A. Mathematics majors have first consideration for enrollment.

120C Introduction to Abstract Algebra: Galois Theory (4) S. Lecture, three hours. Galois Theory: proof of the impossibility of certain ruler-and-compass constructions (squaring the circle, trisecting angles); nonexistence of analogues to the "quadratic formula" for polynomial equations of degree 5 or higher. Prerequisite: Mathematics 120B. Mathematics majors have first consideration for enrollment.

121A-B Linear Algebra (4-4). Lecture, three hours; discussion, two hours. Introduction to modern abstract linear algebra. Special emphasis on students doing proofs. 121A: Vector spaces, linear independence, bases, dimension. Linear transformations and their matrix representations. Theory of determinants. 121B: Canonical forms; inner products; similarity of matrices. Prerequisites for 121A: Mathematics 3A or 6G; 13. Mathematics majors have first consideration for enrollment.

130B-C Probability and Stochastic Processes (4-4). Lecture, three hours. Introductory course emphasizing applications. 130B: Conditional probability and conditional expectations; Markov chains. 130C: Exponential distribution and Poisson process; Brownian motion; additional topics, such as option pricing, as time permits. Prerequisites: for 130B: Mathematics 2A-B, and either 130A, 131A, 132A, Statistics 120A, or Mathematics 67 and either 6G or 3A; for 130C: Mathematics 130B.

131A-B-C Introduction to Probability and Statistics (4-4-4). Lecture, three hours; discussion, one to two hours. Introductory course covering basic principles of probability and statistical inference. 131A: Axiomatic definition of probability, random variables, probability distributions, expectation. 131B: Point estimation, interval estimating, and testing hypotheses, Bayesian approaches to inference. 131C: Linear regression, analysis of variance, model checking. Prerequisites for 131A-B: Mathematics 2A-B; 2D and 2J or 4; for 131C: Mathematics 131A-B; 3A or 6G. Same as Statistics 120A-B-C.

133A-B Statistical Methods with Applications to Finance (4-4) W, S. Lecture three hours; discussion, one hour. Introduction to Monte Carlo (MC) methods. 133A: Overview of probability, statistics, financial concepts; linear and logistic regressions; time series models; Brownian motion; MC simulations. 133B: Elliptic and parabolic partial differential equations; MC methods; vanilla and exotic derivatives; Greeks, portfolio management, and value-at-risk. Prerequisites for 133A: Mathematics 67 and either 2D or 4, or Mathematics 131A, or Statistics 120A; for 133B: Mathematics 133A.

140A-B Elementary Analysis (4-4). Lecture, three hours; discussion, two hours. Introduction to real analysis including convergence of sequences, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs. Prerequisites for 140A: Mathematics 2D; 2J or 3A; 13; for 140B: Mathematics 140A. Mathematics majors have first consideration for enrollment.

140C Analysis in Several Variables (4). Lecture, three hours; discussion, two hours. Rigorous treatment of multivariable differential calculus. Jacobians, Inverse and Implicit Function theorems. Prerequisite: Mathematics 140B.

141 Introduction to Topology (4) S. Lecture, three hours. The elements of naive set theory and the basic properties of metric spaces. Introduction to topological properties. Prerequisite: Mathematics 140A.

146 Fourier Analysis (4) S. Lecture, three hours. Rigorous introduction to the theory of Fourier series and orthogonal expansions. Fourier transform. Prerequisites: Mathematics 3D and 140A-B. Mathematics 112A recommended.

147 Complex Analysis (4) W. Rigorous treatment of basic complex analysis: complex numbers, analytic functions, Cauchy integral theory and its consequences (Morera's Theorem, Argument Principle, Fundamental Theorem of Algebra, Maximum Modulus Principle, Liouville's Theorem), power series, residue calculus harmonic functions, conformal mapping. Students are expected to do proofs. Corequisite: Mathematics 140B. Prerequisite: Mathematics 140A. Mathematics 114A may not be taken for credit after 147.

150 Introduction to Mathematical Logic (4) F. Lecture, three hours. First-order logic through the Completeness Theorem for predicate logic. Prerequisites: Mathematics 13 or ICS 6B and 6D. Only one course from Mathematics 150, Philosophy 105B, and Logic and Philosophy of Science 105B may be taken for credit.

151 Set Theory (4) W. Lecture, three hours. Axiomatic development; infinite sets; cardinal and ordinal numbers. Prerequisite: Mathematics 150. Only one course from Mathematics 151, Philosophy 105A, and Logic and Philosophy of Science 105A may be taken for credit.

152 Computability (4) S. Lecture, three hours. Computable functions; undecidability; Gödel's Incompleteness Theorem. Prerequisite: Mathematics 150. Only one course from Mathematics 152, Philosophy 105C, and Logic and Philosophy of Science 105C may be taken for credit.

161 Modern Geometry (4). Lecture, three hours. Euclidean geometry; Hilbert's axioms; absolute geometry; hyperbolic geometry; the Poincare models; geometric transformations. Prerequisites: Mathematics 13 or ICS 6B and 6D. Mathematics majors have first consideration for enrollment.

162A-B Introduction to Differential Geometry (4-4) F, W. Lecture, three hours. Applications of advanced calculus and linear algebra to the geometry of curves and surfaces in space. Prerequisites: Mathematics 2E, 3A, and 3D. Not offered every year.

171A-B Mathematical Methods in Operations Research. Lecture, three hours. Offered summer only.

171A Linear Programming (4). Simplex algorithm, duality, optimization in networks. Prerequisite: Mathematics 3A or 6G.

171B Nonlinear Programming (4). Conditions for optimality, quadratic and convex programming, search methods, geometric programming. Prerequisites: Mathematics 2D and either 3A or 6G.

173A-B Introduction to Cryptology (4-4). Lecture, three hours. Introduction to some of the mathematics used in the making and breaking of codes, with applications to classical ciphers and public key systems. The mathematics which is covered includes topics from number theory, probability, and abstract algebra. Prerequisites for 173A: Mathematics 2B and 3A or 6G; Mathematics 13 or ICS 6B and 6D; for 173B: Mathematics 173A.

174A-B Modern Graph Theory I, II. Lecture, three hours. An introduction to fundamental concepts of graph theory by developing abilities to produce examples, following and devising simple proofs, and current applications of graph theory.

174A Modern Graph Theory I (4). Topics include graph types, matching in graphs; Menger's Theorem; Kuratowski's Theorem. Prerequisites: Mathematics 2B, 3A or 6G, and Mathematics 13 or ICS 6B and 6D.

174B Modern Graph Theory II (4). Topics include coloring maps, plane graphs, vertices, and edges; Hadwiger's Conjecture; Hamilton Cycles; Ramsey Theory. Prerequisite: Mathematics 174A.

175 Combinatorics (4). Lecture, three hours; discussion, two hours. Introduction to combinatorics including basic counting principles, permutations, combinations, binomial coeffecients, inclusion-exclusion, derangements, ordinary and exponential generating functions, recurrence relations, Catalan numbers, Stirling numbers, and partition numbers. Prerequisites: Mathematics 2B and 13.

176 Mathematics of Finance (4). Lecture, three hours; discussion, one hour. Introduces the mathematics of finance with an emphasis on financial derivatives. After a review of certain tools from probability, statistics, and elementary differential and partial differential equations, concepts such as hedging, arbitrage, Puts, Calls, and the design of portfolios, the derivation and solution of the Blac-Scholes and other equations are discussed. Prerequisite: Mathematics 2J or 3A. Same as Economics 135. Mathematics and Economics majors have first consideration for enrollment.

180A-B Number Theory (4-4). Lecture, three hours; discussion, one hour. Introduction to number theory and applications. 180A: Divisibility, prime numbers, factorization. Arithmetic functions. Congruences. Quadratic residue. Diophantine equations. Introduction to cryptography. Formerly Mathematics 180. 180B: Analytic number theory, character sums, finite fields, discrete logarithm, computational complexity. Introduction to coding theory. Other topics as time permits. Prerequisites for 180A: Mathematics 2J or 3A, and 13; or consent of instructor; for 180B: Mathematics 180A. Mathematics majors have first consideration for enrollment.

184 History of Mathematics (4). Lecture, three hours. Topics vary from year to year. Some possible topics: mathematics in ancient times; the development of modern analysis; the evolution of geometric ideas. Students are assigned individual topics for term papers. Prerequisites: Mathematics 3D, 120A, 140A. Not offered every year. Mathematics majors have first consideration for enrollment.

184L History of Mathematics Lesson Lab (1). Laboratory, one hour. Aspiring math teachers research, design, present, and peer review middle school or high school math lessons that draw from history of mathematics topics. Corequisite: Mathematics 184. Prerequisite: Physical Sciences 5.

189 Special Topics in Mathematics (4). Lecture, three hours. Offered from time to time, but not on a regular basis. Content and prerequisites vary with the instructor. May be repeated for credit as topics vary.

191 Mathematical Modeling Seminar (2). Seminar, 1.5 hours. Developing, testing, and presenting mathematical models for real world problems. Students will prepare for and participate in the Mathematical Contest in Modeling (MCM) in late February. Separate contest registration fee required. Prerequisite: Mathematics 3D. May be taken for credit twice.

192 Studies in the Learning and Teaching of Secondary Mathematics (2) W, S. Lecture, two hours; fieldwork, two hours. Enrollment limited to upper-division Mathematics majors participating in the Mathematics for Education specialization, or students in related majors. Admission requires approval of Department Tutor Supervisor. Focus is on historic and current mathematical concepts related to student learning and effective math pedagogy, with fieldwork in grades 6-14. For students not in the specialization in Mathematics for Education, this course satisfies no requirements other than contribution to the 180 units required for graduation. Pass/Not Pass only. Prerequisites: Mathematics 2D; 2J; 3D; 13 or 120A or 140A. May be taken twice for credit.

193 SMPP Capstone (2) W, S. Lecture, two hours; fieldwork, two hours. Capstone course for the Mathematics Subject-Matter Preparation (SMP) program. Engages students in reviewing and conducting current research on significant issues related to the teaching and learning of mathematics in the secondary classroom. Corequisite: Mathematics 192 recommended. May be taken twice for credit.

194 Problem-Solving Seminar (2). Develops ability in analytical thinking and problem solving, using problems of the type found in the Mathematics Olympiad and the Putnam Mathematical Competition. Students taking the course in fall will prepare for and take the Putnam examination in December. Pass/Not Pass only. NOTE: satisfies no requirement other than contribution to the 180 units required for graduation. Recommended for prospective teachers. May be taken twice for credit.

H195A-B Honors Seminar (4-4) W, S. Topics vary from year to year. Provides an integrative experience, including problem-solving and oral and written presentations. Required for the Honors Program in Mathematics and open to others with consent of instructor.

199A-B-C Special Studies in Mathematics (4-4-4) F, W, S. Supervised reading. For outstanding undergraduate mathematics majors in supervised but independent reading or research of mathematical topics. Prerequisite: consent of Department. NOTE: Cannot normally be used to satisfy departmental requirements.

GRADUATE

NOTE: Undergraduates who are not in the Honors Program and wish to take 205A or 206A should first get the consent of the instructor.

205A-B-C Introduction to Graduate Analysis (5-5-5) F, W, S. Lecture, four hours. Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces. Prerequisites for 205A: Mathematics 2E, 3A, 13; or equivalent or consent of instructor; for 205B: Mathematics 205A; for 205C: Mathematics 205B.

206A-B-C Introduction to Graduate Algebra (5-5-5). Lecture, four hours. Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials and Galois groups. Prerequisite: Mathematics 3A or equivalent or consent of instructor.

210A-B-C Real Analysis (4-4-4) F, W, S. Lecture, three hours. Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces. Prerequisites: Mathematics 140A-B-C or consent of instructor.

211A-B-C Topics in Analysis (4-4-4) F, W, S. Lecture, three hours. A continuation of Mathematics 210A-B-C; topics selected by instructor.

218A-B-C Introduction to Manifolds and Geometry (4-4-4) F, W, S. Lecture, three hours. General topology and fundamental groups, covering space; Stokes theorem on manifolds, selected topics on abstract manifold theory. Prerequisites: Mathematics 205A-B-C or consent of instructor.

220A-B-C Analytic Function Theory (4-4-4) F, W, S. Lecture, three hours. Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping. Prerequisites: Mathematics 140A-B-C or equivalent or consent of instructor.

225A-B-C Introduction to Numerical Analysis and Scientific Computing (4-4-4). Lecture, three hours. Introduction to fundamentals of numerical analysis from an advanced viewpoint. 225A: Error analysis, approximation of functions, nonlinear equations. 225B-C: Numerical linear algebra, numerical solutions of differential equations; stability. Prerequisites: Mathematics 3D; 105A-B or 140A-B; 121A; and Mathematics 112A or Engineering MAE140.

226A-B-C Computational Differential Equations (4-4-4). Lecture, three hours. Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods; mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively. Prerequisites: basic differential equations, such as in Mathematics 3D and either Mathematics 112A or Engineering MAE140; plus either abstract analysis (e.g., Mathematics 140A-B) or numerical analysis (Mathematics 105A-B or equivalent).

227A-B-C Mathematical and Computational Biology (4-4-4). Analytical and numerical methods for dynamical systems, temporal-spatial dynamics, steady state, stability, stochasticity. Application to life sciences: genetics, tissue growth and patterning, cancers, ion channels gating, signaling networks, morphogen gradients. 227A: Analytical methods. Prerequisites: Mathematics 2A-B and 3A, or equivalent, and background in basic discrete probability, or consent of instructor; Mathematics 3D recommended. 227B: Numerical simulations. Prerequisite: Mathematics 227A. 227C: Probabilistic methods. Prerequisite: Mathematics 227A or consent of instructor. Same as Computer Science 285.

230A-B-C Algebra (4-4-4) F, W, S. Lecture, three hours. Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules. Prerequisites: Mathematics 120A and 121A-B or equivalent, or consent of instructor.

232A-B-C Algebraic Number Theory (4-4-4) F, W, S. Lecture, three hours. Prime number theorem, quadratic reciprocity, Gauss sums, diophantine equations, zeta functions over finite fields. Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Galois extensions, Chebatarev density theorem. Representations of finite groups, L-functions, Hecke L-functions. Introduction to class field theory. Prerequisites: Mathematics 230A-B-C or consent of instructor.

233A-B-C Algebraic Geometry (4-4-4). Lecture, three hours. Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jocobian classification of curves and surfaces.

234B-C Topics in Algebra (4-4). Lecture, three hours. Group theory, homological algebra, and other selected topics. Prerequisites: Mathematics 230A-B-C or consent of instructor.

235B-C Mathematics of Cryptography (4-4). Lecture, three hours. 235B: Continuation of 235A. 235C: Topics to be determined by instructor. Prerequisites: Mathematics 230A-B-C or consent of instructor.

239B-C Analytic Methods in Arithmetic Geometry (4-4) W, S. Lecture, three hours. Riemann zeta function. Dirichlet L-functions, prime number theorem, zeta functions over finite fields, sieve methods, zeta functions of algebraic curves, algebraic coding theory. L-functions over number fields, L-functions of modular forms. Eisenstein series. Prerequisites: Mathematics 220A-B-C and 230A-B-C, or consent of instructor.

240A-B-C Differential Geometry (4-4-4). Lecture, three hours. Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology. Prerequisite: consent of instructor.

245A-B-C Topics in Differential Geometry (4-4-4). Lecture, three hours. Continuation of Mathematics 240A-B-C. Topics to be determined by the instructor. Prerequisites: Mathematics 240A-B-C or consent of instructor. May be repeated for credit as topics vary.

250A-B-C Algebraic Topology (4-4-4). Lecture, three hours. Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group. Prerequisite: Mathematics 230A or consent of instructor.

260A-B-C Functional Analysis (4-4-4). Lecture, three hours. Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces. Prerequisites: Mathematics 210A-B-C and 220A-B-C or consent of instructor.

270A-B-C Probability (4-4-4). Lecture, three hours. Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes. Prerequisites: Mathematics 130A-B-C and 210A-B-C or consent of instructor.

271A-B-C Stochastic Processes (4-4-4). Lecture, three hours. Processes with independent increments, Wiener and Gaussian processes, function space integrals, stationary processes, Markov processes. Prerequisites: Mathematics 210A-B-C or consent of instructor. Mathematics 271A-B-C and Statistics 270 may not both be taken for credit.

272A-B-C Probability Models (4-4-4) F, W, S. Lecture, three hours. Spin systems, Ising models, contact process, exclusion process, percolation, increasing events, critical probabilities, sub- and super-critical phases, scaling theory, oriented percolation, concentration of measure, Gaussian fields, Borell's inequality, chaining, entropy. Prerequisites: Mathematics 271A-B-C or equivalent.

274 Topics in Probability (4-4-4). Lecture, three hours. Selected topics, such as theory of stochastic processes, martingale theory, stochastic integrals, stochastic differential equations. Prerequisites: Mathematics 270A-B-C or consent of instructor. May be repeated for credit as topics vary.

277A-B-C Topics in Mathematical Physics (4-4-4). Lecture, three hours. Topics to be determined by the instructor. Prerequisite: consent of instructor. May be repeated for credit as topics vary.

280A-B-C Mathematical Logic (4-4-4). Lecture, three hours. Basic set theory; models, compactness, and completeness; basic model theory; Incompleteness and Gödel's Theorems; basic recursion theory; constructible sets. Prerequisite: consent of instructor.

281A-B-C Set Theory (4-4-4). Lecture, three hours. Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure. Prerequisites: Mathematics 280A-B-C or consent of instructor.

282A-B-C Model Theory (4-4-4). Lecture, three hours. Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra. Prerequisites: Mathematics 280A-B-C.

285A-B-C Topics in Mathematical Logic (4-4-4). Lecture, three hours. Continuation of Mathematics 280A-B-C. Topics to be conducted by the instructor. Prerequisite: Mathematics 280A-B-C or consent of instructor. May be repeated for credit as topics vary.

290A-B-C Methods in Applied Mathematics (4-4-4). Lecture, three hours. Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

291C Topics in Applied and Computational Mathematics (4) S. Lecture, three hours. Topics to be determined by instructor. Prerequisite: consent of instructor. May be repeated for credit as topics vary.

295A-B-C Partial Differential Equations (4-4-4). Lecture, three hours. Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods. Prerequisites: Mathematics 112A-B-C, 210A-B-C or equivalent, or consent of instructor.

296 Topics in Partial Differential Equations (4). Lecture, three hours. Continuation of Mathematics 295A-B-C. Topics to be determined by the instructor. Prerequisites: Mathematics 295A-B-C or consent of instructor. May be repeated for credit as topics vary.

297 Mathematics Colloquium (1). Weekly colloquia on topics of current interest in mathematics. Satisfactory/Unsatisfactory Only. May be repeated for credit.

298A-B-C Seminar (1 to 3) F, W, S. Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable. Prerequisite: consent of the Department. May be repeated for credit.

299A-B-C Supervised Reading and Research (1 to 12) F, W, S, Summer. May be repeated for credit.

399 University Teaching (1 to 4) F, W, S. Limited to Teaching Assistants. Does not satisfy any requirements for the Master's degree. Satisfactory/Unsatisfactory Only. May be repeated for credit.