# Mathematics

[ undergraduate program | graduate program | faculty ]

*All courses, faculty listings, and curricular and degree requirements described herein are subject to change or deletion without notice. Updates may be found on the Academic Senate website: http://senate.ucsd.edu/catalog-copy/approved-updates/.*

## Courses

*For course descriptions not found in the *UC
San Diego General Catalog, 2015–16*, please contact the department
for more information.*

All prerequisites listed below may be replaced by an equivalent or higher-level course. The listings of quarters in which courses will be offered are only tentative. Please consult the Department of Mathematics to determine the actual course offerings each year.

## Lower Division

3C. Precalculus (4)

Functions and their graphs. Linear and polynomial functions, zeroes, inverse functions, exponential and logarithmic, trigonometric functions and their inverses. Emphasis on understanding algebraic, numerical and graphical approaches making use of graphing calculators. (No credit given if taken after Math 4C, 1A/10A, or 2A/20A.) Three or more years of high school mathematics or equivalent recommended. **Prerequisites:** Math Placement Exam qualifying score.

4C. Precalculus for Science and Engineering (4)

Review of polynomials. Graphing functions and relations: graphing rational functions, effects of linear changes of coordinates. Circular functions and right triangle trigonometry. Reinforcement of function concept: exponential, logarithmic, and trigonometric functions. Vectors. Conic sections. Polar coordinates. Three lectures, one recitation. (No credit given if taken after Math 1A/10A or 2A/20A. Two units of credit given if taken after Math 3C.) **Prerequisites:** Math Placement Exam qualifying score or Math 3C with a grade of C– or better.

10A. Calculus I (4)

Differential calculus of functions of one variable, with applications. Functions, graphs, continuity, limits, derivatives, tangent lines, optimization problems. (No credit given if taken after or concurrent with Math 20A.) **Prerequisites:** Math Placement Exam qualifying score, or AP Calculus AB score of 2, or SAT II Math Level 2 score of 600 or higher, or Math 3C, or Math 4C.

10B. Calculus II (4)

Integral calculus of functions of one variable, with applications. Antiderivatives, definite integrals, the Fundamental Theorem of Calculus, methods of integration, areas and volumes, separable differential equations. (No credit given if taken after or concurrent with Math 20B.) **Prerequisites:** AP Calculus AB score of 3, 4, or 5 (or equivalent AB subscore on BC exam), or Math 10A, or Math 20A.

10C. Calculus III (4)

Introduction to functions of more than one variable. Vector geometry, partial derivatives, velocity and acceleration vectors, optimization problems. (No credit given if taken after or concurrent with 20C.) **Prerequisites:** AP Calculus BC score of 3, 4, or 5, or Math 10B, or Math 20B.

11. Calculus-Based Introductory Probability and Statistics (4)

Events and probabilities, conditional probability, Bayes’ formula. Discrete random variables: mean, variance; binomial, Poisson distributions. Continuous random variables: densities, mean, variance; normal, uniform, exponential distributions, central limit theorem. Sample statistics, confidence intervals, hypothesis testing, regression. Applications. Intended for biology and social science majors. **Prerequisites:** AP Calculus BC score of 3, 4, or 5, or Math 10B or Math 20B, and concurrent enrollment in Math 11L.

11L. Elementary Probability and Statistics Laboratory (1)

Introduction to the use of software in probabilistic and statistical analysis. Emphasis on understanding connections between the theory of probability and statistics, numerical results of real data, and learning techniques of data analysis and interpretation useful for solving scientific problems. **Prerequisites:** AP Calculus BC score of 3, 4, or 5, or Math 10B with a grade of C– or better, or Math 20B with a grade of C– or better, and concurrent enrollment in Math 11.

15A. Introduction to Discrete Mathematics (4)

Basic discrete mathematical structure: sets, relations, functions, sequences, equivalence relations, partial orders, and number systems. Methods of reasoning and proofs: propositional logic, predicate logic, induction, recursion, and pigeonhole principle. Infinite sets and diagonalization. Basic counting techniques; permutation and combinations. Applications will be given to digital logic design, elementary number theory, design of programs, and proofs of program correctness. Credit not offered for both Math 15A and CSE 20. Equivalent to CSE 20. **Prerequisites:** CSE 8A or CSE 8B or CSE 11.

20A. Calculus for Science and Engineering (4)

Foundations of differential and integral calculus of one variable. Functions, graphs, continuity, limits, derivative, tangent line. Applications with algebraic, exponential, logarithmic, and trigonometric functions. Introduction to the integral. (Two credits given if taken after Math 1A/10A and no credit given if taken after Math 1B/10B or Math 1C/10C. Formerly numbered Math 2A.) **Prerequisites:** Math Placement Exam qualifying score, or AP Calculus AB score of 2 or 3 (or equivalent AB subscore on BC exam), or SAT II Math 2C score of 650 or higher, or Math 4C with a grade of C– or better, or Math 10A with a grade of C– or better.

20B. Calculus for Science and Engineering (4)

Integral calculus of one variable and its
applications, with exponential, logarithmic, hyperbolic, and trigonometric
functions. Methods of integration. Infinite series. Polar coordinates in
the plane and complex exponentials. (Two units of credits given if taken
after Math 1B/10B or Math 1C/10C.)** Prerequisites:** AP
Calculus AB score of 4 or 5, or AP Calculus BC score of 3, or Math 20A
with a grade of C– or better, or Math 10B with a grade of C– or better,
or Math 10C with a grade of C– or better.

20C. Calculus and Analytic Geometry for Science and Engineering (4)

Vector geometry, vector functions and their derivatives. Partial differentiation. Maxima and minima. Double integration. (Two units of credit given if taken after Math 10C. Credit not offered for both Math 20C and 31BH. Formerly numbered Math 21C.) **Prerequisites:** AP Calculus BC score of 4 or 5, or Math 20B with a grade of C– or better.

20D. Introduction to Differential Equations (4)

Ordinary differential equations: exact, separable,
and linear; constant coefficients, undetermined coefficients, variations
of parameters. Systems. Series solutions. Laplace transforms. Techniques
for engineering sciences. Computing symbolic and graphical solutions using
Matlab. (Formerly numbered Math 21D.) May be taken as repeat credit for
Math 21D.** Prerequisites:** Math 20C (or
Math 21C) or Math 31BH with a grade of C– or better.

20E. Vector Calculus (4)

Change of variable in multiple integrals, Jacobian, Line integrals, Green’s theorem. Vector fields, gradient fields, divergence, curl. Spherical/cylindrical coordinates. Taylor series in several variables. Surface integrals, Stoke’s theorem. Gauss’ theorem. Conservative fields. (Credit not offered for both Math 20E and 31CH.) **Prerequisites:** Math 20C (or Math 21C) or Math 31BH with a grade of C– or better.

20F. Linear Algebra (4)

Matrix algebra, Gaussian elimination, determinants. Linear and affine subspaces, bases of Euclidean spaces. Eigenvalues and eigenvectors, quadratic forms, orthogonal matrices, diagonalization of symmetric matrices. Applications. Computing symbolic and graphical solutions using Matlab. (Credit not offered for both Math 20F and 31AH.) **Prerequisites:** Math 20C (or Math 21C) with a grade of C– or better.

31AH. Honors Linear Algebra (4)

First quarter of three-quarter honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Topics include: real/complex number systems, vector spaces, linear transformations, bases and dimension, change of basis, eigenvalues, eigenvectors, diagonalization. (Credit not offered for both Math 31AH and 20F.) **Prerequisites:** AP Calculus BC score of 5 or consent of instructor.

31BH. Honors Multivariable Calculus (4)

Second quarter of three-quarter honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Topics include: derivative in several variables, Jacobian matrices, extrema and constrained extrema, integration in several variables. (Credit not offered for both Math 31BH and 20C.) **Prerequisites:** Math 31AH with a grade of B– or better, or consent of instructor.

31CH. Honors Vector Calculus (4)

Third quarter of honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Topics include: change of variables formula, integration of differential forms, exterior derivative, generalized Stoke’s theorem, conservative vector fields, potentials. (Credit not offered for both Math 31CH and 20E.) **Prerequisites:** Math 31BH with a grade of B– or better, or consent of instructor.

87. Freshman Seminar (1)

The Freshman Seminar Program is designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a small seminar setting. Freshman Seminars are offered in all campus departments and undergraduate colleges, and topics vary from quarter to quarter. Enrollment is limited to fifteen to twenty students, with preference given to entering freshman. **Prerequisites:** none.

95. Introduction to Teaching Math (2)

(Cross-listed with EDS 30.) Revisit students’ learning
difficulties in mathematics in more depth to prepare students
to make meaningful observations of how K–12 teachers deal with these difficulties.
Explore how instruction can use students’ knowledge to pose problems
that stimulate students’ intellectual curiosity. **Prerequisites:** none.

99R. Independent Study (1)

Independent study or research under direction of a member of the faculty. **Prerequisites:** Must be of first-year standing and a Regent’s Scholar.

## Upper Division

100A. Abstract Algebra I (4)

First course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include: groups, subgroups and factor groups, homomorphisms, rings, fields. (Students may not receive credit for both Math 100A and Math 103A.) **Prerequisites:** Math 31CH or Math 109 or consent of instructor.

100B. Abstract Algebra II (4)

Second course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include: rings (especially polynomial rings) and ideals, unique factorization, fields; linear algebra from perspective of linear transformations on vector spaces, including inner product spaces, determinants, diagonalization. (Students may not receive credit for both Math 100B and Math 103B.) **Prerequisites: **Math 100A or consent of instructor.

100C. Abstract Algebra III (4)

Third course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include: linear transformations, including Jordan canonical form and rational canonical form; Galois theory, including the insolvability of the quintic. **Prerequisites:** Math 100B or consent of instructor. (F,W,S)

102. Applied Linear Algebra (4)

Second course in linear algebra from a computational yet geometric point
of view. Elementary Hermitian matrices, Schur’s theorem, normal matrices,
and quadratic forms. Moore-Penrose generalized inverse and least square
problems. Vector and matrix norms. Characteristic and singular values.
Canonical forms. Determinants and multilinear algebra. **Prerequisites:** Math
20F. or Math 31AH. (W)

103A. Modern Algebra I (4)

First course in a two-quarter introduction to abstract algebra with some applications. Emphasis on group theory. Topics include: definitions and basic properties of groups, properties of isomorphisms, subgroups. (Students may not receive credit for both Math 100A and Math 103A.) **Prerequisites:** Math 31CH or Math 109 or consent of instructor.

103B. Modern Algebra II (4)

Second course in a two-quarter introduction to abstract algebra with some applications. Emphasis on rings and fields. Topics include: definitions and basic properties of rings, fields, and ideals, homomorphisms, irreducibility of polynomials. (Students may not receive credit for both Math 100B and Math 103B.) **Prerequisites:** Math 103A or Math 100A or consent of instructor.

104A. Number Theory I (4)

Elementary number theory with applications. Topics include
unique factorization, irrational numbers, residue systems,
congruences, primitive roots, reciprocity laws, quadratic forms,
arithmetic functions, partitions, Diophantine equations, distribution
of primes. Applications include fast Fourier transform, signal
processing, codes, cryptography. **Prerequisites:** Math
109 or Math 31CH, or consent of instructor.

104B. Number Theory II (4)

Topics in number theory such as finite fields, continued fractions,
Diophantine equations, character sums, zeta and theta functions,
prime number theorem, algebraic integers, quadratic and cyclotomic
fields, prime ideal theory, class number, quadratic forms,
units, Diophantine approximation, *p*-adic numbers,
elliptic curves. **Prerequisites:** Math 104A
or consent of instructor.

104C. Number Theory III (4)

Topics in algebraic and analytic number theory, with an advanced
treatment of material listed for Math 104B. **Prerequisites:** Math
104B or consent of instructor.

109. Mathematical Reasoning (4)

This course uses a variety of topics in mathematics to introduce the students to rigorous mathematical proof, emphasizing quantifiers, induction, negation, proof by contradiction, naive set theory, equivalence relations and epsilon-delta proofs. Required of all departmental majors. **Prerequisites:** Math 20F or Math 31AH or consent of instructor.

110A. Introduction to Partial Differential Equations (4)

Fourier series, orthogonal expansions, and eigenvalue problems.
Sturm-Liouville theory. Separation of variables for partial
differential equations of mathematical physics, including topics
on Bessel functions and Legendre polynomials. Formerly Math
110. (Students may not receive credit for Math 110A and
Math 110.) **Prerequisites:** Math 20D
and either 20F or Math 31AH, or consent of instructor. (F,S)

110B. Elements of Partial Differential Equations and Integral Equations (4)

Basic concepts and classification of partial differential
equations. First order equations, characteristics. Hamilton-Jacobi
theory, Laplace’s equation, wave equation, heat equation. Separation
of variables, eigenfunction expansions, existence and uniqueness
of solutions. (Formerly Math 132A. Students may not receive
credit for Math 110B and Math 132A.) **Prerequisites:** Math
110A or consent of instructor. (W)

111A. Mathematical Modeling I (4)

An introduction to mathematical modeling in the physical and
social sciences. Topics vary, but have included mathematical
models for epidemics, chemical reactions, political organizations,
magnets, economic mobility, and geographical distributions
of species. May be taken for credit two times when topics change. **Prerequisites:** Math
20D and either Math 20F or Math 31AH, and Math 109, or consent of
instructor.

111B. Mathematical Modeling II (4)

Continued study on mathematical modeling in the physical and social sciences, using advanced techniques that will expand upon the topics selected and further the mathematical theory presented in Math 111A. **Prerequisites:** Math 111A or consent of instructor.

120A. Elements of Complex Analysis (4)

Complex numbers and functions. Analytic functions, harmonic functions,
elementary conformal mappings. Complex integration. Power series.
Cauchy’s theorem. Cauchy’s formula. Residue theorem. **Prerequisites:**
Math 20E or Math 31CH, or consent of instructor. (F,W)

120B. Applied Complex Analysis (4)

Applications of the residue theorem. Conformal
mapping and applications to potential theory, flows, and temperature
distributions. Fourier transformations. Laplace transformations, and applications
to integral and differential equations. Selected topics such as Poisson’s
formula, Dirichlet’s problem, Neumann’s problem, or special functions. **Prerequisites:** Math
120A or consent of instructor. (W,S)

121A. Foundations of Teaching and Learning Mathematics I (4)

(Cross-listed with EDS 121A.) Develop teachers’ knowledge base (knowledge of mathematics content, pedagogy, and student learning) in the context of advanced mathematics. This course builds on the previous courses where these components of knowledge were addressed exclusively in the context of high-school mathematics. **Prerequisites:** EDS 30/Math 95, Calculus 10C or 20C.

121B. Foundations of Teaching and Learning Math II (4)

(Cross-listed with EDS 121B.) Examine how learning theories can consolidate observations about conceptual development with the individual student as well as the development of knowledge in the history of mathematics. Examine how teaching theories explain the effect of teaching approaches addressed in the previous courses. **Prerequisites:** EDS 121A/Math 121A.

130A. Ordinary Differential Equations I (4)

Linear and nonlinear systems of differential equations. Stability theory, perturbation theory. Applications and introduction to numerical solutions. Three lectures. **Prerequisites:** Math 20D and either Math 20F or Math 31AH or consent of instructor. (F)

130B. Ordinary Differential Equations II (4)

Existence and uniqueness of solutions to differential equations. Local and global theorems of continuity and differentiabillity. Three lectures. **Prerequisites:** Math 130A or consent of instructor. (W)

140A. Foundations of Real Analysis I (4)

First course in a rigorous three-quarter sequence on real analysis. Topics include: the real number system, basic topology, numerical sequences and series, continuity. (Students may not receive credit for both Math 140A and Math 142A.) **Prerequisites:** Math 31CH or Math 109, or consent of instructor.

140B. Foundations of Real Analysis II (4)

Second course in a rigorous three-quarter sequence on real analysis. Topics include: differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and special functions. (Students may not receive credit for both Math 140B and Math 142B.) **Prerequisites:** Math 140A or consent of instructor.

140C. Foundations of Real Analysis III (4)

Third course in a rigorous three-quarter sequence on real analysis. Topics include: differentiation of functions of several real variables, the implicit and inverse function theorems, the Lebesgue integral, infinite-dimensional normed spaces. **Prerequisites:** Math 140B or consent of instructor.

142A. Introduction to Analysis I (4)

First course in an introductory two-quarter
sequence on analysis. Topics include: the real number system,
numerical sequences and series, limits of functions, continuity.
(Students may not receive credit for both Math 140 and Math
142A.) **Prerequisites:** Math
31CH or Math 109, or consent of instructor.

142B. Introduction to Analysis II (4)

Second course in an introductory two-quarter sequence on analysis. Topics include: differentiation, the Rieman integral, sequences and series of functions, uniform convergence, Taylor and Fourier series, special functions. (Students may not receive credit for both Math 140B and Math 142B.) **Prerequisites:** Math 142A or Math 140A, or consent of instructor.

150A. Differential Geometry (4)

Differential geometry of curves and surfaces.
Gauss and mean curvatures, geodesics, parallel displacement, Gauss-Bonnet
theorem. Three lectures. **Prerequisites:** Math
20E with a grade of C– or better and Math 20F with a grade of C– or better,
or consent of instructor. (F)

150B. Calculus on Manifolds (4)

Calculus of functions of several variables,
inverse function theorem. Further topics may include exterior
differential forms, Stokes’ theorem, manifolds, Sard’s theorem, elements
of differential topology, singularities of maps, catastrophes, further
topics in differential geometry, topics in geometry of physics. **Prerequisites:** Math
150A or consent of instructor. (W)

152. Applicable Mathematics and Computing (4)

This course will give students experience
in applying theory to real world applications such as Internet and wireless
communication problems. The course will incorporate talks by experts from
industry and students will be helped to carry out independent projects.
Topics include graph visualization, labelling, and embeddings, random graphs
and randomized algorithms. May be taken 3 times for credit. **Prerequisites:** Math 20D
and either 20F or Math 31AH, or consent of instructor.

153. Geometry for Secondary Teachers (4)

Two- and three-dimensional Euclidean geometry
is developed from one set of axioms. Pedagogical issues will emerge from
the mathematics and be addressed using current research in teaching and
learning geometry. This course is designed for prospective secondary school
mathematics teachers. **Prerequisites:** Math
109 or Math 31CH, or consent of instructor.

154. Discrete Mathematics and Graph Theory (4)

Basic concepts in graph theory. Combinatorial
tools, structures in graphs (Hamiltonian cycles, perfect matching). Properties
of graphics and applications in basic algorithmic problems (planarity,
k-colorability, traveling salesman problem). **Prerequisites:** Math
109 or Math 31CH, or consent of instructor.

155A. Geometric Computer Graphics (4)

Bezier curves and control lines, de Casteljau construction for subdivision,
elevation of degree, control points of Hermite curves, barycentric coordinates,
rational curves. Programming knowledge recommended. (Students may not receive
credit for both Math 155A and CSE 167.) **Prerequisites:** Math
20F or Math 31AH, or consent of instructor. (F)

160A. Elementary Mathematical Logic I (4)

An introduction to recursion theory, set theory, proof theory, model theory. Turing machines. Undecidability of arithmetic and predicate logic. Proof by induction and definition by recursion. Cardinal and ordinal numbers. Completeness and compactness theorems for propositional and predicate calculi. **Prerequisites:** Math 100A, or Math 103A, or Math 140A, or consent of instructor.

160B. Elementary Mathematical Logic II (4)

A continuation of recursion theory, set theory, proof theory, model theory. Turing machines. Undecidability of arithmetic and predicate logic. Proof by induction and definition by recursion. Cardinal and ordinal numbers. Completeness and compactness theorems for propositional and predicate calculi. **Prerequisites:** Math 160A or consent of instructor.

163. History of Mathematics (4)

Topics will vary from year to year in areas of mathematics and their development. Topics may include the evolution of mathematics from the Babylonian period to the eighteenth century using original sources, a history of the foundations of mathematics and the development of modern mathematics. **Prerequisites:** Math 20B or consent of instructor. (S)

168A. Topics in Applied Mathematics—Computer Science (4)

Topics to be chosen in areas of applied mathematics
and mathematical aspects of computer science. May be repeated
once for credit with different topics. **Prerequisites:** Math
20F or Math 31AH, or consent of instructor. (W,S)

170A. Introduction to Numerical Analysis: Linear Algebra (4)

Analysis of numerical methods for linear algebraic systems and least squares problems. Orthogonalization methods. Ill conditioned problems. Eigenvalue and singular value computations. Three lectures, one recitation. Knowledge of programming recommended. **Prerequisites:** Math 20F. (F,S)

170B. Introduction to Numerical Analysis: Approximation and Nonlinear Equations (4)

Rounding and discretization errors. Calculation of roots of polynomials and nonlinear equations. Interpolation. Approximation of functions. Three lectures, one recitation. Knowledge of programming recommended. **Prerequisites:** Math 170A. (W)

170C. Introduction to Numerical Analysis: Ordinary Differential Equations (4)

Numerical differentiation and integration. Ordinary differential equations and their numerical solution. Basic existence and stability theory. Difference equations. Boundary value problems. Three lectures, one recitation. **Prerequisites:** Math 20D or 21D and Math 170B, or consent of instructor. (S)

171A. Introduction to Numerical Optimization: Linear Programming (4)

Linear optimization and applications. Linear programming, the simplex method, duality. Selected topics from integer programming, network flows, transportation problems, inventory problems, and other applications. Three lectures, one recitation. Knowledge of programming recommended. (Credit not allowed for both Math 171A and Econ 172A.) **Prerequisites:** Math 20F or consent of instructor.

171B. Introduction to Numerical Optimization: Nonlinear Programming (4)

Convergence of sequences in Rn, multivariate Taylor series. Bisection and related methods for nonlinear equations in one variable. Newton’s methods for nonlinear equations in one and many variables. Unconstrained optimization and Newton’s method. Equality-constrained optimization, Kuhn-Tucker theorem. Inequality-constrained optimization. Three lectures, one recitation. Knowledge of programming recommended. (Credit not allowed for both Math 171B and Econ 172B.) **Prerequisites:** Math 171A or consent of instructor.

174. Numerical Methods for Physical Modeling (4)

(Conjoined with Math 274.) Floating point
arithmetic, direct and iterative solution of linear equations,
iterative solution of nonlinear equations, optimization, approximation
theory, interpolation, quadrature, numerical methods for initial
and boundary value problems in ordinary differential equations.
(Students may not receive credit for both Math 174 and PHYS
105, AMES 153 or 154. Students may not receive credit for Math 174 if
Math 170A, B, or C has already been taken.) Graduate students will do an
extra assignment/exam. **Prerequisites:** Math
20D or Math 21D, and either Math 20F or Math 31AH, or consent
of instructor.

175. Numerical Methods for Partial Differential Equations (4)

(Conjoined with Math 275.) Mathematical background for working with partial differential equations. Survey of finite difference, finite element, and other numerical methods for the solution of elliptic, parabolic, and hyperbolic partial differential equations. (Formerly Math 172. Students may not receive credit for Math 175/275 and Math 172.) Graduate students do an extra paper, project, or presentation, per instructor. **Prerequisites:** Math 174 or Math 274, or consent of instructor.

179. Projects in Computational and Applied Mathematics (4)

(Conjoined with Math 279.) Mathematical models of physical systems arising in science and engineering, good models and well-posedness, numerical and other approximation techniques, solution algorithms for linear and nonlinear approximation problems, scientific visualizations, scientific software design and engineering, project-oriented. Graduate students will do an extra paper, project, or presentation per instructor. **Prerequisites:** Math 174 or Math 274 or consent of instructor.

180A. Introduction to Probability (4)

Probability spaces, random variables, independence, conditional probability,
distribution, expectation, variance, joint distributions, central
limit theorem. (Two units of credit offered for Math 180A
if Econ 120A previously, no credit offered if Econ 120A concurrently.) Prior or concurrent enrollment in Math 109 is highly recommended. **Prerequisites:** Math
20C or Math 31BH, or consent of instructor. (F)

180B. Introduction to Stochastic Processes I (4)

Random vectors, multivariate densities, covariance matrix, multivariate
normal distribution. Random walk, Poisson process. Other topics if time
permits. **Prerequisites:** Math 20D and either Math 20F or Math 31AH, and Math 109, and Math 180A, or consent of instructor.
(W)

180C. Introduction to Stochastic Processes II (4)

Markov chains in discrete and continuous time, random walk, recurrent events. If time permits, topics chosen from stationary normal processes, branching processes, queuing theory. Three lectures. **Prerequisites:** Math
180B or consent of instructor. (S)

181A. Introduction to Mathematical Statistics I (4)

Multivariate distribution, functions of random variables, distributions
related to normal. Parameter estimation, method of moments, maximum likelihood.
Estimator accuracy and confidence intervals. Students completing Econ 120A instead of Math 180A must obtain consent of instructor to enroll. Prior or concurrent enrollment in Math 109 is highly recommended. **Prerequisites:** Math 180A, and Math 20F or Math 31AH, or consent of instructor. (W)

181B. Introduction to Mathematical Statistics II (4)

Hypothesis testing. Linear models, regression, and analysis of variance. Goodness of fit tests. Nonparametric statistics. Two units of credit offered for Math 181B if Econ 120B previously; no credit offered if Econ 120B concurrently. Prior enrollment in Math 109 is highly recommended. **Prerequisites:** Math
181A or consent of instructor. (S)

181C. Mathematical Statistics—Nonparametric Statistics (4)

Topics covered may include the following: classical rank test, rank correlations,
permutation tests, distribution free testing, efficiency, confidence
intervals, nonparametric regression and density estimation,
resampling techniques (bootstrap, jackknife, etc.) and cross validations. Prior enrollment in Math 109 is highly recommended. **Prerequisites:** Math
181B or consent of instructor.

181E. Mathematical Statistics—Time Series (4)

Analysis of trends and seasonal effects, autoregressive and moving averages
models, forecasting, informal introduction to spectral analysis. **Prerequisites:** Math
181B or consent of instructor.

183. Statistical Methods (4)

Introduction to probability. Discrete and continuous random variables–binomial, Poisson and Gaussian distributions. Central limit theorem. Data analysis and inferential statistics: graphical techniques, confidence intervals, hypothesis tests, curve fitting. (Credit not offered for Math 183 if Econ 120A, ECE 109, Math 180A, Math 181A, or Math 186 previously or concurrently taken.) **Prerequisites:** Math 20C (21C) with a grade of C– or better, or consent of instructor. (F,S)

184A. Combinatorics (4)

Introduction to the theory and applications of combinatorics. Enumeration of combinatorial structures. Ranking and unranking. Graph theory with applications and algorithms. Recursive algorithms. Inclusion-exclusion. Generating functions. Polya theory. **Prerequisites:** Math 109 with a grade of C– or better, or consent of instructor. (W,S)

185. Introduction to Computational Statistics (4)

Statistical analysis of data by means of package programs. Regression, analysis of variance, discriminant analysis, principal components, Monte Carlo simulation, and graphical methods. Emphasis will be on understanding the connections between statistical theory, numerical results, and analysis of real data. Recommended preparation: exposure to computer programming (such as CSE 5A, CSE 7, or ECE 15) highly recommended. **Prerequisites:** Math 11, or Math 181A, or Math 183, or Math 186, or MAE 108, or ECE 109, or Econ 120A, and either Math 20F or Math 31AH or consent of instructor.

186. Probability Statistics for Bioinformatics (4)

This course will cover discrete and random variables, data analysis and inferential statistics, likelihood estimators and scoring matrices with applications to biological problems. Introduction to Binomial, Poisson, and Gaussian distributions, central limit theorem, applications to sequence and functional analysis of genomes and genetic epidemiology. (Credit not offered for Math 186 if Econ 120A, ECE 109, Math 180A, Math 181A, or Math 183 previously or concurrently.) **Prerequisites:** Math 20C (21C) with a grade of C– or better, or consent of instructor.

187. Introduction to Cryptography (4)

An introduction to the basic concepts and techniques of modern cryptography. Classical cryptanalysis. Probabilistic models of plaintext. Monalphabetic and polyalphabetic substitution. The one-time system. Caesar-Vigenere-Playfair-Hill substitutions. The Enigma. Modern-day developments. The Data Encryption Standard. Public key systems. Security aspects of computer networks. Data protection. Electronic mail. Three lectures, one recitation. **Prerequisites:** programming experience. (S)

189. Exploratory Data Analysis and Inference (4)

An introduction to various quantitative methods and statistical techniques for analyzing data—in particular big data. Quick review of probability continuing to topics of how to process, analyze, and visualize data using statistical language R. Further topics include basic inference, sampling, hypothesis testing, bootstrap methods, and regression and diagnostics. Offers conceptual explanation of techniques, along with opportunities to examine, implement, and practice them in real and simulated data. **Prerequisites:** Math 20F or Math 31AH, and Math 180A. Students who have not completed listed prerequisites may enroll with consent of instructor.

190. Introduction to Topology (4)

Topological spaces, subspaces, products, sums
and quotient spaces. Compactness, connectedness, separation
axioms. **Prerequisites:** Math 31CH or Math 140A. Students who have not completed prerequisites may enroll with consent of instructor.

191. Topics in Topology (4)

Topics to be chosen by the instructor from the fields of differential algebraic, geometric, and general topology. Three lectures. **Prerequisites:** Math 190 or consent of instructor. (S)

192. Senior Seminar in Mathematics (1)

The Senior Seminar Program is designed to allow senior undergraduates to meet with faculty members in a small group setting to explore an intellectual topic in mathematics at the upper-division level. Topics will vary from quarter to quarter. Senior Seminars may be taken for credit up to four times, with a change in topic, and permission of the department. Enrollment is limited to twenty students, with preference given to seniors. **Prerequisites:** department stamp and/or consent of instructor.

193A. Actuarial Mathematics I (4)

Probabilistic Foundations of Insurance. Short-term
risk models. Survival distributions and life tables. Introduction
to life insurance. **Prerequisites:** Math
180A or Math 183, or consent of instructor.

193B. Actuarial Mathematics II (4)

Life Insurance and Annuities. Analysis of premiums and premium reserves. Introduction to multiple life functions and decrement models as time permits. **Prerequisites:** Math 193A or consent of instructor.

194. The Mathematics of Finance (4)

Introduction to the mathematics of financial models. Basic probabilistic models and associated mathematical machinery will be discussed, with emphasis on discrete time models. Concepts covered will include conditional expectation, martingales, optimal stopping, arbitrage pricing, hedging, European and American options. **Prerequisites: **Math
20D, and either Math 20F or Math 31AH, and Math 180A, or consent of instructor.

195. Introduction to Teaching in Mathematics (4)

Students will be responsible for and teach a class section of a lower-division mathematics course. They will also attend a weekly meeting on teaching methods. (Does not count towards a minor or major.) Five lectures, one recitation. **Prerequisites:** consent of instructor. (F,W,S)

196. Student Colloquium (1)

A variety of topics and current research results in mathematics will be presented by guest lecturers and students under faculty direction. May be taken for P/NP grade only. **Prerequisites:** upper-division status.

197. Mathematics Internship (2 or 4)

An enrichment program which provides work
experience with public/private sector employers. Subject to
the availability of positions, students will work in a local
company under the supervision of a faculty member and site
supervisor. Units may not be applied towards major graduation
requirements. **Prerequisites:** completion
of ninety units, two upper-division mathematics courses, an overall
2.5 UC San Diego GPA, consent of mathematics faculty coordinator,
and submission of written contract. Department stamp required.

199. Independent Study for Undergraduates (2 or 4)

Independent reading in advanced mathematics by individual students. Three periods. (P/NP grades only.) **Prerequisites:** permission of department. (F,W,S)

199H. Honors Thesis Research for Undergraduates (2–4)

Honors thesis research for seniors participating in the Honors Program. Research is conducted under the supervision of a mathematics faculty member. **Prerequisites:** admission to the Honors Program in mathematics, department stamp.

## Graduate

200A-B-C. Algebra (4-4-4)

Group actions, factor groups, polynomial rings, linear algebra, rational and Jordan canonical forms, unitary and Hermitian matrices, Sylow theorems, finitely generated abelian groups, unique factorization, Galois theory, solvability by radicals, Hilbert Basis Theorem, Hilbert Nullstellensatz, Jacobson radical, semisimple Artinian rings. **Prerequisites:** consent of instructor.

201A. Basic Topics in Algebra I (4)

Recommended for all students specializing in algebra. Basic topics include categorical algebra, commutative algebra, group representations, homological algebra, nonassociative algebra, ring theory. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites: **Math 200C. Students who have not taken Math 200C may enroll with consent of instructor.

201B. Basic Topics in Algebra II (4)

Continued development of a basic topic in algebra. Topics include categorical algebra, commutative algebra, group representations, homological algebra, nonassociative algebra, ring theory. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 201A. Students who have not taken Math 201A may enroll with consent of instructor.

202A. Applied Algebra I (4)

Introduction to algebra from a computational perspective. Groups, rings, linear algebra, rational and Jordan forms, unitary and Hermitian matrices, matrix decompositions, perturbation of eigenvalues, group representations, symmetric functions, fast Fourier transform, commutative algebra, Grobner basis, finite fields. **Prerequisites:** graduate standing or consent of instructor.

202B. Applied Algebra II (4)

Second course in algebra from a computational perspective. Groups, rings, linear algebra, rational and Jordan forms, unitary and Hermitian matrices, matrix decompositions, perturbation of eigenvalues, group representations, symmetric functions, fast Fourier transform, commutative algebra, Grobner basis, finite fields. **Prerequisites:** Math 202A or consent of instructor.

202C. Applied Algebra III (4)

Third course in algebra from a computational perspective. Groups, rings, linear algebra, rational and Jordan forms, unitary and Hermitian matrices, matrix decompositions, perturbation of eigenvalues, group representations, symmetric functions, fast Fourier transform, commutative algebra, Grobner basis, finite fields. **Prerequisites:** Math 202B or consent of instructor.

203A-B-C. Algebraic Geometry (4-4-4)

Places, Hilbert Nullstellensatz, varieties, product of varieties: correspondences, normal varieties. Divisors and linear systems; Riemann-Roch theorem; resolution of singularities of curves. Grothendieck schemes; cohomology, Hilbert schemes; Picard schemes. **Prerequisites:** Math 200A-B-C. (F,W,S)

204A. Number Theory I (4)

First course in graduate-level number theory. Local fields: valuations and metrics on fields; discrete valuation rings and Dedekind domains; completions; ramification theory; main statements of local class field theory. **Prerequisites:** Math 200C. Students who have not taken Math 200C may enroll with consent of instructor.

204B. Number Theory II (4)

Second course in graduate-level number theory. Global fields: arithmetic properties and relation to local fields; ideal class groups; groups of units; ramification theory; adèles and idèles; main statements of global class field theory. **Prerequisites:** Math 204A. Students who have not taken Math 204A may enroll with consent of instructor.

204C. Number Theory III (4)

Third course in graduate-level number theory. Zeta and L-functions; Dedekind zeta functions; Artin L-functions; the class-number formula and generalizations; density theorems. **Prerequisites:** Math 204B. Students who have not taken Math 204B may enroll with consent of instructor.

205. Topics in Number Theory (4)

Topics in algebraic and analytic number theory, such as: L-functions, sieve methods, modular forms, class field theory, p-adic L-functions and Iwasawa theory, elliptic curves and higher dimensional abelian varieties, Galois representations and the Langlands program, p-adic cohomology theories, Berkovich spaces, etc. May be taken for credit nine times. **Prerequisites:** graduate standing.

207A. Topics in Algebra (4)

Introduction to varied topics in algebra. In recent years, topics have included number theory, commutative algebra, noncommutative rings, homological algebra, and Lie groups. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites: **graduate standing. Nongraduate students may enroll with consent of instructor.

207B. Further Topics in Algebra (4)

Introduction to varied topics in algebra. In recent years, topics have included number theory, commutative algebra, noncommutative rings, homological algebra, and Lie groups. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 207A. Students who have not completed Math 207A may enroll with consent of instructor.

209. Seminar in Number Theory (1)

Various topics in number theory. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

210A. Mathematical Methods in Physics and Engineering (4)

Complex variables with applications. Analytic functions, Cauchy’s theorem, Taylor and Laurent series, residue theorem and contour integration techniques, analytic continuation, argument principle, conformal mapping, potential theory, asymptotic expansions, method of steepest descent. **Prerequisites:** Math 20DEF,140A/142A or consent of instructor.

210B. Mathematical Methods in Physics and Engineering (4)

Linear algebra and functional analysis. Vector spaces, orthonormal bases, linear operators and matrices, eigenvalues and diagonalization, least squares approximation, infinite-dimensional spaces, completeness, integral equations, spectral theory, Green’s functions, distributions, Fourier transform. **Prerequisites:** Math 210A or consent of instructor. (W)

210C. Mathematical Methods in Physics and Engineering (4)

Calculus of variations: Euler-Lagrange equations, Noether’s theorem. Fourier analysis of functions and distributions in several variables. Partial differential equations: Laplace, wave, and heat equations; fundamental solutions (Green’s functions); well-posed problems. **Prerequisites:** Math 210B or consent of instructor. (S)

217. Topics in Applied Mathematics (4)

In recent years, topics have included applied complex analysis, special functions, and asymptotic methods. May be repeated for credit with consent of adviser as topics vary. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

220A-B-C. Complex Analysis (4-4-4)

Complex numbers and functions. Cauchy theorem and its applications, calculus of residues, expansions of analytic functions, analytic continuation, conformal mapping and Riemann mapping theorem, harmonic functions. Dirichlet principle, Riemann surfaces. **Prerequisites:** Math 140A-B or consent of instructor. (F,W,S)

221A. Topics in Several Complex Variables (4)

Introduction to varied topics in several complex variables. In recent years, topics have included formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** Math 200A and 220C. Students who have not completed Math 200A and 220C may enroll with consent of instructor.

221B. Further Topics in Several Complex Variables (4)

Continued development of a topic in several complex variables. Topics include: formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 221A. Students who have not completed Math 221A may enroll with consent of instructor.

231A-B-C. Partial Differential Equations (4-4-4)

Existence and uniqueness theorems. Cauchy-Kowalewski theorem, first order systems. Hamilton-Jacobi theory, initial value problems for hyperbolic and parabolic systems, boundary value problems for elliptic systems. Green’s function, eigenvalue problems, perturbation theory. **Prerequisites:** Math 210A-B or 240A-B-C or consent of instructor.

237A. Topics in Differential Equations (4)

Introduction to varied topics in differential equations. In recent years, topics have included Riemannian geometry, Ricci flow, and geometric evolution. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

237B. Further Topics in Differential Equations (4)

Continued development of a topic in differential equations. Topics include: Riemannian geometry, Ricci flow, and geometric evolution. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 237A. Students who have not completed Math 237A may enroll with consent of instructor.

240A-B-C. Real Analysis (4-4-4)

Lebesgue integral and Lebesgue measure, Fubini theorems, functions of bounded variations, Stieltjes integral, derivatives and indefinite integrals, the spaces L and C, equi-continuous families, continuous linear functionals general measures and integrations. **Prerequisites:** Math 140A-B-C. (F,W,S)

241A-B. Functional Analysis (4-4)

Metric spaces and contraction mapping theorem;
closed graph theorem; uniform boundedness principle; Hahn-Banach
theorem; representation of continuous linear functionals; conjugate
space, weak topologies; extreme points; Krein-Milman theorem; fixed-point
theorems; Riesz convexity theorem; Banach algebras. **Prerequisites:** Math
240A-B-C or consent of instructor.

242. Topics in Fourier Analysis (4)

In recent years, topics have included Fourier analysis in Euclidean spaces, groups, and symmetric spaces. May be repeated for credit with consent of adviser as topics vary. **Prerequisites:** Math 240C, students who have not completed Math 240C may enroll with consent of instructor.

243. Seminar in Operator Algebras (1)

Various topics in operator algebras. May be taken for credit nine times. **Prerequisites:** graduate standing or consent of instructor. (S/U grades only.)

245A. Convex Analysis and Optimization I (4)

Convex sets and functions, convex and affine hulls, relative
interior, closure, and continuity, recession and existence
of optimal solutions, saddle point and min-max theory, subgradients
and subdifferentials. **Prerequisites:** Math
20F and Math 142A, or graduate standing, or consent of instructor.

245B. Convex Analysis and Optimization II (4)

Optimality conditions, strong duality and the primal function,
conjugate functions, Fenchel duality theorems, dual derivatives
and subgradients, subgradient methods, cutting plane methods. **Prerequisites:** Math
245A or consent of instructor.

245C. Convex Analysis and Optimization III (4)

Convex optimization problems, linear matrix inequalities,
second-order cone programming, semidefinite programming, sum
of squares of polynomials, positive polynomials, distance geometry. **Prerequisites:** Math
245B or consent of instructor.

247A. Topics in Real Analysis (4)

Introduction to varied topics in real analysis. In recent years, topics have included Fourier analysis, distribution theory, martingale theory, operator theory. May be taken for credit six times with consent of adviser. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

247B. Further Topics in Real Analysis (4)

Continued development of a topic in real analysis. Topics include: Fourier analysis, distribution theory, martingale theory, operator theory. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 247A. Students who have not completed Math 247A may enroll with consent of instructor.

248. Seminar in Real Analysis (1)

Various topics in real analysis. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

250A-B-C. Differential Geometry (4-4-4)

Differential manifolds, Sard theorem, tensor bundles, Lie derivatives, DeRham theorem, connections, geodesics, Riemannian metrics, curvature tensor and sectional curvature, completeness, characteristic classes. Differential manifolds immersed in Euclidean space. **Prerequisites:** consent of instructor. (F,W,S)

251A-B-C. Lie Groups (4-4-4)

Lie groups, Lie algebras, exponential map, subgroup subalgebra correspondence, adjoint group, universal enveloping algebra. Structure theory of semi-simple Lie groups, global decompositions, Weyl group. Geometry and analysis on symmetric spaces. **Prerequisites:** Math 200 and 250 or consent of instructor. (F,W,S)

256. Seminar in Lie Groups and Lie Algebras (1)

Various topics in Lie groups and Lie algebras, including structure theory, representation theory, and applications. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.) (F,W,S)

257A. Topics in Differential Geometry (4)

Introduction to varied topics in differential geometry. In recent years, topics have included Morse theory and general relativity. May be taken for credit six times with consent of adviser. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

257B. Further Topics in Differential Geometry (4)

Continued development of a topic in differential geometry. Topics include Morse theory and general relativity. May be taken for credit three times with consent of adviser. **Prerequisites:** Math 257A. Students who have not completed Math 257A may enroll with consent of instructor.

258. Seminar in Differential Geometry (1)

Various topics in differential geometry. May be taken for credit nine times. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

259A-B-C. Geometrical Physics (4-4-4)

Manifolds, differential forms, homology, deRham’s theorem. Riemannian geometry, harmonic forms. Lie groups and algebras, connections in bundles, homotopy sequence of a bundle, Chern classes. Applications selected from Hamiltonian and continuum mechanics, electromagnetism, thermodynamics, special and general relativity, Yang-Mills fields. **Prerequisites:** graduate standing in mathematics, physics, or engineering, or consent of instructor.

260A. Mathematical Logic I (4)

Propositional calculus and first-order logic. Theorem proving, Model theory, soundness, completeness, and compactness, Herbrand’s theorem, Skolem-Lowenheim theorems, Craig interpolation. **Prerequisites:** graduate standing or consent of instructor.

260B. Mathematical Logic II (4)

Theory of computation and recursive function theory, Church’s thesis, computability and undecidability. Feasible computability and complexity. Peano arithmetic and the incompleteness theorems, nonstandard models. **Prerequisites:** Math 260A or consent of instructor.

261A. Probabilistic Combinatorics and Algorithms (4)

Introduction to the probabilistic method. Combinatorial applications of the linearity of expectation, second moment method, Markov, Chebyschev, and Azuma inequalities, and the local limit lemma. Introduction to the theory of random graphs. **Prerequisites:** graduate standing or consent of instructor.

261B. Probabilistic Combinatorics and Algorithms II (4)

Introduction to probabilistic algorithms. Game theoretic techniques. Applications of the probabilistic method to algorithm analysis. Markov Chains and Random walks. Applications to approximation algorithms, distributed algorithms, online and parallel algorithms. Math 261A must be taken before Math 261B. **Prerequisites:** Math 261A.

261C. Probabilistic Combinatorics and Algorithms III (4)

Advanced topics in the probabilistic combinatorics and probabilistics algorithms. Random graphs. Spectral Methods. Network algorithms and optimization. Statistical learning. Math 261B must be taken before Math 261C. **Prerequisites:** Math 261B.

262A. Topics in Combinatorial Mathematics (4)

Introduction to varied topics in combinatorial mathematics. In recent years topics have included problems of enumeration, existence, construction, and optimization with regard to finite sets. Recommended preparation: some familiarity with computer programming desirable but not required. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

262B. Further Topics in Combinatorial Mathematics (4)

Continued development of a topic in combinatorial mathematics. Topics include: problems of enumeration, existence, construction, and optimization with regard to finite sets. Recommended preparation: some familiarity with computer programming desirable but not required. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 262A. Students who have not completed Math 262A may enroll with consent of instructor.

264A-B-C. Combinatorics (4-4-4)

Topics from partially ordered sets, Mobius functions, simplicial complexes and shell ability. Enumeration, formal power series and formal languages, generating functions, partitions. Lagrange inversion, exponential structures, combinatorial species. Finite operator methods, q-analogues, Polya theory, Ramsey theory. Representation theory of the symmetric group, symmetric functions and operations with Schur functions. (F,W,S)

267A. Topics in Mathematical Logic (4)

Introduction to varied topics in mathematical logic. Topics chosen from recursion theory, model theory, and set theory. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** graduate standing or consent of instructor. Nongraduate students may enroll with consent of instructor.

267B. Further Topics in Mathematical Logic (4)

Continued development of a topic in mathematical logic. Topics chosen from recursion theory, model theory, and set theory. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 267A or consent of instructor. Students who have not completed Math 267A may enroll with consent of instructor.

268. Seminar in Logic (1)

Various topics in logic. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

269. Seminar in Combinatorics (1)

Various topics in combinatorics. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

270A. Numerical Linear Algebra (4)

Error analysis of the numerical solution of linear equations and least squares problems for the full rank and rank deficient cases. Error analysis of numerical methods for eigenvalue problems and singular value problems. Iterative methods for large sparse systems of linear equations. **Prerequisites:** graduate standing or consent of instructor.

270B. Numerical Approximation and Nonlinear Equations (4)

Iterative methods for nonlinear systems of equations, Newton’s method. Unconstrained and constrained optimization. The Weierstrass theorem, best uniform approximation, least-squares approximation, orthogonal polynomials. Polynomial interpolation, piecewise polynomial interpolation, piecewise uniform approximation. Numerical differentiation: divided differences, degree of precision. Numerical quadrature: interpolature quadrature, Richardson extrapolation, Romberg Integration, Gaussian quadrature, singular integrals, adaptive quadrature. **Prerequisites:** Math 270A or consent of instructor.

270C. Numerical Ordinary Differential Equations (4)

Initial value problems (IVP) and boundary value problems (BVP) in ordinary differential equations. Linear methods for IVP: one and multistep methods, local truncation error, stability, convergence, global error accumulation. Runge-Kutta (RK) Methods for IVP: RK methods, predictor-corrector methods, stiff systems, error indicators, adaptive time-stepping. Finite difference, finite volume, collocation, spectral, and finite element methods for BVP; a priori and a posteriori error analysis, stability, convergence, adaptivity. **Prerequisites:** Math 270B or consent of instructor.

271A-B-C. Numerical Optimization (4-4-4)

Formulation and analysis of algorithms for constrained optimization. Optimality conditions; linear and quadratic programming; interior methods; penalty and barrier function methods; sequential quadratic programming methods. **Prerequisites:** consent of instructor. (F,W,S)

272A. Numerical Partial Differential Equations I (4)

Survey of discretization techniques for elliptic partial differential equations, including finite difference, finite element and finite volume methods. Lax-Milgram Theorem and LBB stability. A priori error estimates. Mixed methods. Convection-diffusion equations. Systems of elliptic PDEs. **Prerequisites:** graduate standing or consent of instructor.

272B. Numerical Partial Differential Equations II (4)

Survey of solution techniques for partial differential equations. Basic iterative methods. Preconditioned conjugate gradients. Multigrid methods. Hierarchical basis methods. Domain decomposition. Nonlinear PDEs. Sparse direct methods. **Prerequisites:** Math 272A or consent of instructor.

272C. Numerical Partial Differential Equations III (4)

Time dependent (parabolic and hyperbolic) PDEs. Method of lines. Stiff systems of ODEs. Space-time finite element methods. Adaptive meshing algorithms. A posteriori error estimates. **Prerequisites:** Math 272B or consent of instructor.

273A. Advanced Techniques in Computational Mathematics I (4)

Models of physical systems, calculus of variations, principle of least action. Discretization techniques for variational problems, geometric integrators, advanced techniques in numerical discretization. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. **Prerequisites:** graduate standing or consent of instructor.

273B. Advanced Techniques in Computational Mathematics II (4)

Nonlinear functional analysis for numerical treatment of nonlinear PDE. Numerical continuation methods, pseudo-arclength continuation, gradient flow techniques, and other advanced techniques in computational nonlinear PDE. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. **Prerequisites:** Math 273A or consent of instructor.

273C. Advanced Techniques in Computational Mathematics III (4)

Adaptive numerical methods for capturing all scales in one model, multiscale and multiphysics modeling frameworks, and other advanced techniques in computational multiscale/multiphysics modeling. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. **Prerequisites:** Math 273B or consent of instructor.

274. Numerical Methods for Physical Modeling (4)

(Conjoined with Math 174.) Floating point
arithmetic, direct and iterative solution of linear equations,
iterative solution of nonlinear equations, optimization, approximation
theory, interpolation, quadrature, numerical methods for initial
and boundary value problems in ordinary differential equations.
Students may not receive credit for both Math 174 and PHYS 105, AMES 153
or 154. (Students may not receive credit for Math 174 if Math 170A, B,
or C has already been taken.) Graduate students will complete an additional
assignment/exam. **Prerequisites:** Math
20D or 21D, and either Math 20F or Math 31AH, or consent of
instructor.

275. Numerical Methods for Partial Differential Equations (4)

(Conjoined with Math 175.) Mathematical background
for working with partial differential equations. Survey of finite difference,
finite element, and other numerical methods for the solution of elliptic,
parabolic, and hyperbolic partial differential equations. (Formerly Math
172; students may not receive credit for Math 175/275 and Math 172.) Graduate
students will do an extra paper, project, or presentation, per instructor. **Prerequisites:** Math
174 or Math 274 or consent of instructor.

276. Numerical Analysis in Multiscale Biology (4)

(Cross-listed with BENG 276/CHEM 276.) Introduces
mathematical tools to simulate biological processes at multiple scales.
Numerical methods for ordinary and partial differential equations (deterministic
and stochastic), and methods for parallel computing and visualization.
Hands-on use of computers emphasized, students will apply numerical methods
in individual projects. **Prerequisites:** consent
of instructor.

277A. Topics in Computational and Applied Mathematics (4)

Introduction to varied topics in computational and applied mathematics. In recent years, topics have included: applied functional analysis and approximation theory; numerical treatment of nonlinear partial differential equations; and geometric numerical integration for differential equations. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

277B. Further Topics in Computational and Applied Mathematics (4)

Continued development of a topic in computational and applied mathematics. Topics have included: applied functional analysis and approximation theory; numerical treatment of nonlinear partial differential equations; and geometric numerical integration for differential equations. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 277A. Students who have not completed Math 277A may enroll with consent of instructor.

278A. Seminar in Computational and Applied Mathematics (1)

Various topics in computational and applied mathematics. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor. (S/U grade only.)

278B. Seminar in Mathematical Physics/PDE (1)

Various topics in mathematical physics and partial differential equations. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

279. Projects in Computational and Applied Mathematics (4)

(Conjoined with Math 179.) Mathematical models of physical
systems arising in science and engineering, good models and
well-posedness, numerical and other approximation techniques,
solution algorithms for linear and nonlinear approximation
problems, scientific visualizations, scientific software design
and engineering, project-oriented. Graduate students will do
an extra paper, project, or presentation per instructor. **Prerequisites:** Math
174, or Math 274, or consent of instructor.

280A-B-C. Probability Theory (4-4-4)

Probability measures; Borel fields; conditional probabilities, sums of independent random variables; limit theorems; zero-one laws; stochastic processes. **Prerequisites:** advanced calculus and consent of instructor. (F,W,S)

281A. Mathematical Statistics (4)

Statistical models, sufficiency, efficiency, optimal estimation, least squares and maximum likelihood, large sample theory. **Prerequisites:** advanced
calculus and basic probability theory or consent of instructor.

281B. Mathematical Statistics (4)

Hypothesis testing and confidence intervals,
one-sample and two-sample problems. Bayes theory, statistical decision
theory, linear models and regression. **Prerequisites:** advanced
calculus and basic probability theory or consent of instructor.

281C. Mathematical Statistics (4)

Nonparametrics: tests, regression, density estimation, bootstrap and jackknife. Introduction to statistical computing using S plus. **Prerequisites:** advanced
calculus and basic probability theory or consent of instructor.

282A. Applied Statistics I (4)

General theory of linear models with applications to regression analysis. Ordinary and generalized least squares estimators and their properties. Hypothesis testing, including analysis of variance, and confidence intervals. Completion of courses in linear algebra and basic statistics are recommended prior to enrollment. **Prerequisites:** graduate standing or consent of instructor. (S/U grades permitted.)

282B. Applied Statistics II (4)

Diagnostics, outlier detection, robust regression. Variable selection, ridge regression, the lasso. Generalized linear models, including logistic regression. Data analysis using the statistical software R. Students who have not taken Math 282A may enroll with consent of instructor. **Prerequisites: **Math 282A or consent of instructor. (S/U grades permitted.)

283. Statistical Methods in Bioinformatics (4)

This course will cover material related to the analysis of modern genomic data; sequence analysis, gene expression/functional genomics analysis, and gene mapping/applied population genetics. The course will focus on statistical modeling and inference issues and not on database mining techniques. **Prerequisites:** one year of calculus, one statistics course or consent of instructor.

284. Survival Analysis (4)

Survival analysis is an important tool in many areas of applications including biomedicine, economics, engineering. It deals with the analysis of time to events data with censoring. This course discusses the concepts and theories associated with survival data and censoring, comparing survival distributions, proportional hazards regression, nonparametric tests, competing risk models, and frailty models. The emphasis is on semiparametric inference, and material is drawn from recent literature. **Prerequisites:** Math 282A or consent of instructor.

285. Stochastic Processes (4)

Elements of stochastic processes, Markov chains, hidden Markov models, martingales, Brownian motion, Gaussian processes. **Prerequisites:** Math 180A or equivalent, or consent of instructor.

286. Stochastic Differential Equations (4)

Review of continuous martingale theory. Stochastic integration for continuous semimartingales. Existence and uniqueness theory for stochastic differential equations. Strong Markov property. Selected applications. **Prerequisites:** Math 280A-B or consent of instructor.

287A. Time Series Analysis (4)

Discussion of finite parameter schemes in the Gaussian and non-Gaussian context. Estimation for finite parameter schemes. Stationary processes and their spectral representation. Spectral estimation. Students who have not taken Math 282A may enroll with consent of instructor. **Prerequisites:** Math 282A or consent of instructor.

287B. Multivariate Analysis (4)

Bivariate and more general multivariate normal distribution. Study of tests based on Hotelling’s T2. Principal components, canonical correlations, and factor analysis will be discussed as well as some competing nonparametric methods, such as cluster analysis. Students who have not taken Math 282A may enroll with consent of instructor. **Prerequisites:** Math 282A or consent of instructor.

287C. Advanced Time Series Analysis (4)

Nonparametric function (spectrum, density,
regression) estimation from time series data. Nonlinear time series models
(threshold AR, ARCH, GARCH, etc.). Nonparametric forms of ARMA and GARCH.
Multivariate time series. Students who have not taken Math 287A may enroll with consent of instructor. **Prerequisites:** Math 287A or consent of instructor.

287D. Statistical Learning (4)

Topics include regression methods: (penalized) linear regression
and kernel smoothing; classification methods: logistic regression
and support vector machines; model selection; and mathematical
tools and concepts useful for theoretical results such as VC
dimension, concentration of measure, and empirical processes. Students who have not taken Math 282A may enroll with consent of instructor. **Prerequisites:** Math
282A or consent of instructor.

288. Seminar in Probability and Statistics (1)

Various topics in probability and statistics. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

289A. Topics in Probability and Statistics (4)

Introduction to varied topics in probability and statistics. In recent years, topics have included Markov processes, martingale theory, stochastic processes, stationary and Gaussian processes, ergodic theory. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

289B. Further Topics in Probability and Statistics (4)

Continued development of a topic in probability and statistics. Topics include: Markov processes, martingale theory, stochastic processes, stationary and Gaussian processes, ergodic theory. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 289A. Students who have not completed Math 289A may enroll with consent of instructor.

289C. Exploratory Data Analysis and Inference (4)

An introduction to various quantitative methods and statistical techniques for analyzing data—in particular big data. Quick review of probability continuing to topics of how to process, analyze, and visualize data using statistical language R. Further topics include basic inference, sampling, hypothesis testing, bootstrap methods, and regression and diagnostics. Offers conceptual explanation of techniques, along with opportunities to examine, implement, and practice them in real and simulated data. Recommended preparation: familiarity with linear algebra and mathematical statistics highly recommended. **Prerequisites:** graduate standing.

290A-B-C. Topology (4-4-4)

Point set topology, including separation axioms, compactness, connectedness. Algebraic topology, including the fundamental group, covering spaces, homology and cohomology. Homotopy or applications to manifolds as time permits. **Prerequisites:** Math 100A-B-C and Math 140A-B-C. (F,W,S)

291A. Topics in Topology (4)

Introduction to varied topics in topology. In recent years topics have included: generalized cohomology theory, spectral sequences, K-theory, homotophy theory. May be taken for credit six times with consent of adviser as topics vary. **Prerequisites:** graduate standing. Nongraduate students may enroll with consent of instructor.

291B. Further Topics in Topology (4)

Continued development of a topic in topology. Topics include generalized cohomology theory, spectral sequences, K-theory, homotophy theory. May be taken for credit three times with consent of adviser as topics vary. **Prerequisites:** Math 291A. Students who have not completed Math 291A may enroll with consent of instructor.

292. Seminar in Topology (1)

Various topics in topology. May be taken for credit nine times. **Prerequisites:** graduate standing or consent of instructor. (S/U grade only.)

294. The Mathematics of Finance (4)

Introduction to the mathematics of financial models. Hedging, pricing by arbitrage. Discrete and continuous stochastic models. Martingales. Brownian motion, stochastic calculus. Black-Scholes model, adaptations to dividend paying equities, currencies and coupon-paying bonds, interest rate market, foreign exchange models. **Prerequisites:** Math 180A (or equivalent probability course) or consent of instructor.

295. Special Topics in Mathematics (1 to 4)

A variety of topics and current research results in mathematics will be presented by staff members and students under faculty direction.

296. Graduate Student Colloquium (1)

A variety of advanced topics and current research in mathematics will be presented by department faculty. (S/U grades only.) May be taken for credit six times. **Prerequisites:** graduate standing.

297. Mathematics Graduate Research Internship (2–4)

An enrichment program that provides work experience with public/private sector employers and researchers. Under supervision of a faculty adviser, students provide mathematical consultation services. **Prerequisites:** consent of instructor.

299. Reading and Research (1 to 12)

Independent study and research for the doctoral dissertation. One to three credits will be given for independent study (reading) and one to nine for research. **Prerequisites:** consent of instructor. (S/U grades permitted.)

## Teaching of Mathematics

500. Apprentice Teaching (1 to 4)

Supervised teaching as part of the mathematics instructional program on campus (or, in special cases such as the CTF program, off campus). **Prerequisites:** consent of adviser. (S/U grades only.)

501. Seminar in Teaching Development (1)

A seminar designed for graduate students serving as teaching
assistants in mathematics. Includes discussion of teaching
theories, techniques, and materials with a focus on career
development. **Prerequisites:** graduate standing
or consent of instructor. (S/U grades only.)