Mathematics
Courses
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. Pre-Calculus (4) Functions and their
graphs. Linear and polynomial functions, zeroes, inverse functions,
exponential and logarithm, 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.) Prerequisite: two or more years of
high school mathematics or equivalent.
4C. Pre-Calculus 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.) Prerequisite: qualifying score on placement
examination. With a superior performance in Math. 3C, the placement
examination requirement may be waived.
10A. Calculus (4) Differentiation and
integration of algebraic functions. Fundamental theorem of calculus.
Applications. (No credit given if taken after Math. 2A/20A. Formerly
numbered Math. 1A.) Prerequisite: qualifying score on Math Placement
Exam, or completion of Math 3C with a grade of C or better.
10B. Calculus (4) Further applications
of the definite integral. Calculus of trigonometric, logarithmic, and
exponential functions. Complex numbers. (No credit given if taken after
Math. 2B/20B. Formerly numbered Math. 1B.) Prerequisite: Math. 1A
or 10A.
10C. Calculus (4) Vector geometry, velocity,
and acceleration vectors. (No credit given if taken after Math. 2C/20C.
Formerly numbered Math. 1C.) Prerequisite: Math. 1B or 10B.
11. Elementary Probability and Statistics (4) Events
and probabilities, combinatorics, conditional probability, Bayes formula.
Discrete random variables: mean, variance; binomial, multinomial, Poisson
distributions. Continuous random variables: densities, mean, variance;
normal, uniform, exponential distributions. Sample statistics, confidence
intervals, regression. Applications. Intended for biology and social
science majors. Prerequisites: Math. 10A-B or Math. 20A-B.
15A. Discrete Mathematics (4) Basic discrete
mathematical structures: sets, relations, functions, sequences, equivalence
relations, partial orders, number systems. Methods of reasoning and
proofs: propositional logic, predicate logic, induction, recursion,
pigeonhole principle. Infinite sets and diagonalization. Basic counting
techniques; permutations and combinations. Applications will be given
to digital logic design, elementary number theory, design of programs,
and proofs of program correctness. Equivalent to CSE 20. Credit not
offered for both Math. 15A and CSE 20. Prerequisites: CSE 8A or CSE
8B or CSE 11. CSE 8B or CSE 11 may be taken concurrently with Math.
15A/CSE 20.
15B. Mathematics for Algorithm and Systems (4) This
course introduces mathematical tools for the qualitative and quantitative
analysis of algorithms and computer systems. Topics to be covered include
basic enumeration and counting techniques; recurrence relations; graph
theory; asymptotic notation; elementary applied discrete probability.
Equivalent to CSE 21. Prerequisite: Math. 15A or CSE 20 or 160A;
CSE 12 is strongly recommended for CSE 21.
18. Computer Animated Statistics (4) Students
will acquire the basics of statistical analysis by working with computer-simulated
models rather than abstract mathematical language. Topics include hypothesis
testing, maximum likelihood estimation, sampling, chi-square tests and
construction of confidence intervals. Prerequisite: Math. 1B or 10B
or 20B.
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.) Prerequisite: qualifying score on the Math Placement
Exam or completion of Math. 4C with a grade of B or better.
20B. Calculus for Science and Engineering (4) Integral
calculus of one variable and its applications, with exponential, logarithmic,
hyperbolic, and trig-onometric functions. Methods of integration. Polar
coordinates in the plane. (Two units of credits given if taken after
Math. 1B/10B or Math. 1C/10C.) Prerequisite: Math. 20A or equivalent
/ Score of 4 or better on AB calculus AP test.
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. Formerly numbered Math. 21C. Prerequisite: Math. 20B
or equivalent or consent of instructor.
20D. Introduction to Differential Equations (4) Infinite
series. Ordinary differential equations: exact, separable, and linear;
constant coefficients, undetermined coefficients, variations of parameters.
Series solutions. Systems, Laplace transforms, technique for engineering
sciences. Computing symbolic and graphical solutions using Matlab. Formerly
numbered Math. 21D. May be taken as repeat credit for Math. 21D. Prerequisite:
Math. 20C or equivalent or consent of instructor.
20E. Vector Calculus (4) Change of variable
in multiple integrals, Jacobian Line integrals, Greens theorem.
Vector fields, gradient fields, divergence, curl. Spherical and cylindrical
coordinates. Taylor series in several variables. Surface integrals,
Stokes theorem. Gauss theorem and its applications. Conservative
fields. (Zero units given if Math. 2F previously. Formerly numbered
Math. 2F) Prerequisite: Math. 20C (or 21C) or equivalent, or consent
of instructor.
20F. Linear Algebra (4) Matrix algebra,
solution of systems of linear equations by Gaussian elimination, determinants.
Linear and affine subspaces, bases of Euclidean spaces. Eigenval-ues
and eigenvectors, quadratic forms, orthogonal matrices, diagonalization
of symmetric matrices. Applications. Computing symbolic and graphical
solutions using Matlab. (Zero units given if Math. 2EA previously. Formerly
numbered 2EA.) Prerequisite: Math. 20C (or 21C) or equivalent 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 15 to 20 students, with preference given to entering freshman. Prerequisite:
none.
Upper-Division
100A-B-C. Modern Algebra (4-4-4) An introduction
to the methods and basic structures of higher algebra: sets and mappings,
the integers, rational, real and complex numbers, groups, rings (especially
polynomial rings) and ideals, fields, real and complex vector spaces,
linear transformations, inner product spaces, matrices, triangular form,
diagonalization. Both 100 and 103 cannot be taken for credit. Three
lectures, one recitation. Prerequisites: Math. 20F, and Math. 109
or consent of instructor. (F,W,S)
102. Applied Linear Algebra (4) A second
course in linear algebra from a computational yet geometric point of
view. Elementary Hermitian matrices, Schurs 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. Three lectures,
one recitation. Prerequisite: Math. 20F. (W)
103A-B. Modern Applied Algebra (4-4) Abstract
algebra with applications to computation. Set algebra and graph theory.
Finite state machines. Boolean algebras and switching theory. Lattices.
Groups, rings and fields: applications to coding theory. Recurrent sequences.
Three lectures, one recitation. Both 100 and 103 cannot be taken for
credit. Prerequisites: Math. 20F and Math. 109 (may be taken concurrently).
(F,W)
104A-B-C. Number Theory (4-4-4) Topics
from number theory with applications and computing. Possible topics
are: congruences, reciprocity laws, quadratic forms, prime number theorem,
Riemann zeta function, Fermats conjecture, diophantine equations,
Gaussian sums, algebraic integers, unique factorization into prime ideals
in algebraic number fields, class number, units, splitting of prime
ideals in extensions, quadratic and cyclotomic fields, partitions. Possible
applications are Fast Fourier Transform, signal processing, coding,
cryptography. Three lectures. Prerequisite: consent of instructor.
107A-B. Computer Algebra (4) An introduction
to algebraic computation. Compu-tational aspects of groups, rings, fields,
etc. Data representation and algorithms for symbolic computation. Polynomials
and their arithmetic. The use of a computer algebra system as an experimental
tool in mathematics. Programming using algebra systems. Prerequisite:
prior or concurrent enrollment in the Math. 100 or 103 sequence.
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. Prerequisite:
Math. 20F.
110. 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. Prerequisites: Math. 20D (or 21D) and 20F, or consent
of instructor. (F,S)
120A. Elements of Complex Analysis (4) Complex
numbers and functions. Analytic functions, harmonic functions, elementary
conformal mappings. Complex integration. Power series. Cauchys
theorem. Cauchys formula. Residue theorem. Three lectures, one
recitation. Prerequisite or co-registration: Math. 20E, 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 Poissons formula. Dirichlet
problem. Neumanns problem, or special functions. Three lectures,
one recitation. Prerequisite: Math. 120A. (W,S)
130A. Ordinary Differential Equations (4) Linear
and nonlinear systems of differential equations. Stability theory, perturbation
theory. Applications and introduction to numerical solutions. Three
lectures. Prerequisites: Math. 20D/21D and 20F. (F)
130B. Ordinary Differential Equations (4) Existence
and uniqueness of solutions to differential equations. Local and global
theorems of continuity and differentiabillity. Three lectures. Prerequisites:
Math. 20D/21D and 20F, and Math. 130A. (W)
131. Variational Methods in Optimization (4) Maximum-minimum
problems. Normed vector spaces, functionals, Gateaux variations. Euler-Lagrange
multiplier theorem for an extremum with constraints. Calculus of variations
via the multiplier theorem. Applications may be taken from a variety
of areas such as the following: applied mechanics, elasticity, economics,
production planning and resource allocation, astronautics, rocket control,
physics, Fermats principle and Hamiltons principle, geometry,
geodesic curves, control theory, elementary bang-bang problems. Three
lectures, one recitation. Prerequisites: Math. 20D/21D and 20F or
consent of instructor. (S)
132A. Elements of Partial Differential Equations and Integral Equations
(4) Basic concepts and classification of
partial differential equations. First order equations, characteristics.
Hamilton-Jacobi theory, Laplaces equation, wave equation, heat
equation. Separation of variables, eigenfunction expansions, existence
and uniqueness of solutions. Three lectures. Prerequisite: Math.
110 or consent of instructor. (W)
132B. Elements of Partial Differential Equations and Integral Equations
(4) Relation between differential and integral
equations, some classical integral equations, Volterra integral equations,
integral equations of the second kind, degenerate kernels, Fredholm
alternative, Neumann-Liouville series, the resolvent kernel. Three lectures.
Prerequisite: Math. 132A. (S)
140A-B-C. Foundations of Analysis (4-4-4) Axioms,
the real number system, topology of the real line, metric spaces, continuous
functions, sequences of functions, differentiation, Riemann-Stieltjes
integration, partial differentiation, multiple integration, Jacobians.
Additional topics at the discretion of the instructor: power series,
Fourier series, successive approximations of other infinite processes.
Three lectures, one recitation. Prerequisites: Math. 20F and Math.
109 or consent of instructor. Credit cannot be obtained for both Math.
140A-B and 142A-B. (F,W,S)
142A-B. Advanced Calculus (4-4) The number
system. Functions, sequences, and limits. Continuity and differentiability.
The Riemann integral. Transcendental functions. Limits and continuity.
Infinite series. Sequences and series of functions. Uniform convergence.
Taylor series. Improper integrals. Gamma and Beta functions. Fourier
series. Three lectures. Prerequisites: Math. 20F and Math. 109 (concurrent
enrollment in Math. 109 allowed.) Credit cannot be obtained for both
Math. 140A-B and 142A-B.
150A. Differential Geometry (4) Differential
geometry of curves and surfaces. Gauss and mean curvatures, geodesics,
parallel displacement, Gauss-Bonnet theorem. Three lectures. Prerequisite:
Math. 20E or consent of instructor. (F)
150B. Calculus on Manifolds (4) Calculus
of functions of several variables, inverse function theorem. Further
topics, selected by instructor, such as exterior differential forms,
Stokes theorem, manifolds, Sards theorem, elements of differential
topology, singularities of maps, catastrophes, further topics in differential
geometry, topics in geometry of physics. Three lectures. Prerequisite:
Math. 150A. (W)
151. Topics in Geometry (4) A topic,
selected by the instructor, from Euclidean geometry, non-Euclidean geometry,
projective geometry, algebraic geometry, or other geometries. May be
repeated for credit with a different topic. Three lectures. Prerequisite:
consent of instructor. (S)
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 or 21D, and 20F 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. Prerequisite:
Math. 109.
155A. 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. Three lectures, one recitation, and approximately eight
laboratory hours per week. Prerequisites: Math. 20F and programming
experience. [Warning: There are duplicate credit restrictions on
this course. See section on Duplication of Credit.] (F)
155B. Topics in Computer Graphics (4) Spline
curves, spline interpolation, affine and affine cross ratios, polar
forms (blossoming), the Oslo algorithm for knot insertion, NURBS and
geometric continuity. Three lectures, one recitation, and approximately
eight laboratory hours per week. Prerequisite: Math. 155A or consent
of instructor. (W)
160A-B. Elementary Mathematical Logic (4-4) An
introduction to recursion theory, set theory, proof theory, and 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. Three lectures. Prerequisite: Math. 100A,
103A, 140A, or consent of instructor.
161. Mathematical Computing (2 or 4) Programming
in higher level mathematical language such as Mathematica: Lists, Functions,
Expressions, Recursion, Iteration, graphics, packages. Application to
diverse areas of mathematics such as differential equations, dynamical
systems, fractals, chaos, probability, financial models. Prerequisite:
Math. 20A-B, 21C-D, 20E-F or equivalent.
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.
Prerequisite: Math. 20B or consent of instructor. (S)
165. Introduction to Set Theory (4) Sets,
relations, and functions. Partial, linear, and well-orders. The axiom
of choice, proof by induction and definition by recursion. Cardinal
and ordinal numbers and their arithmetic. Prerequisite: Math. 100A
or 140A or 103, or consent of instructor.
166. Intro to the Theory of Computation (4) Introduction
to formal languages; regular languages; regular expressions, finite
automata, minimization, closure properties, decision algorithms, and
non-regular languages; context-free languages, context-free grammars,
push-down automata, parsing theory, closure properties, and noncontext-free
languages; computable languages; turing machines, recursive functions,
Churchs thesis, undecidability and the halting problem. Equivalent
to CSE 105. Prerequisites: CSE 8B or 9B or 10 or 65 or 62B AND CSE
20 or 160A or Math. 15A or 109 or 100A or 103A.
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. Three lectures, one recitation. Prerequisite: consent of
instructor. (W,S)
170A. Numerical 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. Prerequisites:
Math. 20F and knowledge of programming. (F,S)
170B. Numerical Analysis (4) Rounding
and discretization errors. Calculation of roots of polynomials and nonlinear
equations. Interpolation. Approximation of functions. Three lectures,
one recitation. Prerequisites: Math. 20F and knowledge of programming.
(W)
170C. Numerical 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.
Prerequisite: Math. 170B or consent of instructor. (S)
171A-B. Mathematical ProgrammingNumerical Optimization (4-4) Mathematical
optimization and applications. Linear programming, the simplex method,
duality. Nonlinear programming, Kuhn-Tucker theorem. Selected topics
from integer programming, network flows, transportation problems, inventory
problems, and other applications. Three lectures, one recitation. Prerequisites:
Math. 20F and knowledge of programming.
172. Numerical Partial Differential Equations (4) Finite
difference methods for the numerical solution of hyperbolic and parabolic
partial differential equations; finite difference and finite element
methods for elliptic partial differential equations. Three lectures.
Prerequisites: Math. 170A or Math. 110 and programming experience.
(S)
173. Mathematical SoftwareScientific Programming (4) Development
of high quality mathematical software for the computer solution of mathematical
problems. Three lectures, one recitation. Prerequisites: Math. 170A
or Math. 174 and knowledge of FORTRAN. (W)
174. Numerical Methods in Science and Engineering (4) Floating
point arithmetic, linear equations, interpolation, integration, differential
equations, nonlinear equations, optimization, least squares. Students
may not receive credit for both Math. 174 and Physics 105 or MAE 153
or 154. Students may not receive credit for Math. 174 if Math. 170 A,B,
or C has already been taken. Prerequisites: Math. 20D (21D) and Math.
20F.
176. Advanced Data Structures (4) Descriptive
and analytical presentation of data structures and algorithms. Lists,
tables, priority queues, disjoint subsets, and dictionaries data types.
Data structuring techniques include linked lists, arrays, hashing, and
trees. Performance evaluation involving worst case, average and expected
case, and amortized analysis. Crecit not offered for both Math. 176
and CSE 100. Equivalent to CSE 100. Prerequisites: CSE 12, CSE 21,
or Math. 15B, and CSE 30, or consent of instructor.
180A. Introduction to Probability (4) Probability
spaces, random variables, independence, conditional probability, distribution,
expectation, joint distributions, central-limit theorem. Three lectures.
Prerequisites: Math. 20D/21D. [Warning: There are duplicate credit
restrictions on this course. See section on Duplication of Credit.]
(F)
180B. Introduction to Probability (4) Random
vectors, multivariate densities, covariance matrix, multivariate normal
distribution. Random walk, Poisson process. Other topics if time permits.
Three lectures. Prerequisites: Math. 180A and Math. 20E. (W)
180C. Introduction to Probability (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. Prerequisite: Math. 180B.
(S)
181A. Introduction to Mathematical Statistics (4) Random
samples, linear regression, least squares, testing hypotheses, and estimation.
Neyman-Pearson lemma, likelihood ratios. Three lectures, one recitation.
Prerequisites: Math. 180A and 20F. [Warning: There are duplicate
credit restrictions on this course. See section on Duplication of Credit.]
(W)
181B. Introduction to Mathematical Statistics (4) Goodness
of fit, special small sample distribution and use, nonparametric methods.
Kolmogorov-Smirnov statistics, sequential analysis. Three lectures.
Prerequisite: Math. 181A. (S)
181C. Mathematical Statistics (4) Nonparametric
Statistics. 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. Prerequisites: Math. 181A, 181B previously or concurrently.
181D. Mathematical Statistics (4) Sampling
Theory. Basic notions of estimation: bias, variance, and sampling errors.
Sampling from finite populations: simple random, stratified, cluster,
sampling with unequal probabilities. Ratio and regression estimaters,
multistage sampling. Prerequisites: Math. 181A, 181B previously or
concurrently.
181E. Mathematical Statistics (4) Time
Series. Analysis of trends and seasonal effects, autoregressive and
moving averages models, forecasting, informal introduction to spectral
analysis. Prerequisites: Math. 181A, 181B previously or concurrently.
182. Introduction to Combinatorics (4) Combinatorial
methods and their computer implementation. Permutations and combinations,
generating functions, partitions, principle of inclusion and exclusion.
Polyas theory of counting. Halls theorem, assignment problem,
backtrack technique, error-correcting codes, combinatorial optimization
problems. Three lectures, one recitation. Prerequisites: Math. 20F
and programming experience. (W)
183. Statistical Methods (4) Introduction
to probability. Discrete and continuous random variablesbinomial,
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, Math. 180A, or Math. 181A previously or concurrently.) Prerequisite:
Math. 20C (21C). (F,S)
184A. Mathematical Foundations of Computer Science (4) Enumeration
of combinatorial structures. Ranking and unranking. Graph theory with
applications and algorithms. Recursive algorithms. Circuit design. Inclusion-exclusion.
Generating functions. Polya theory. Three lectures, one recitation.
Prerequisite: Math. 15B or CSE 21 or Math. 109 or consent of instructor.
(W,S)
186. Probability Statistics for Bioinformatics (4) This
course will cover an introduction to probability and statistics, the
use of discrete and random variables, different types of distributions,
data analysis and inferential statistics, likelihood estimators and
scoring matrices with applications to biological problems. Introduction
to probability, Binomial, Poisson, and Gaussian distributions, central
limit theorem, applications to sequence and functional analysis of genomes
and genetic epidemiology. Prerequisite: Math. 20A, Math. 20B, Math.
20C (21C).
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. Prerequisite:
programming experience. (S)
188. Design and Analysis of Algorithms (4) Introduction
to the design and analysis of efficient algorithms. Basic techniques
for analyzing the time requirements of algorithms. Algorithms for sorting,
searching, and pattern matching, algorithms for graphs and networks.
NP-complete problems. Equivalent to CSE 101. Prerequisites: CSE 100
or Math. 176A for Math. 188; CSE 12, CSE 21, and CSE 100 for CSE 101.
190. Introduction to Topology (4) Topological
spaces, subspaces, products, sums and quotient spaces. Compactness,
connectedness, separation axioms. Selected further topics such as fundamental
group, classification of surfaces, Morse theory, topological groups.
May be repeated for credit once when topics vary, with consent of instructor.
Three lectures. Prerequisite: Math. 109 or consent of instructor.
(W)
191. Topics in Topology (4) Topics to
be chosen by the instructor from the fields of differential algebraic,
geometric, and general topology. Three lectures. Prerequisite: Math.
190 or consent of instructor. (S)
193A. Actuarial Mathematics (4) Probabilistic
Foundations of Insurance. Short-term risk models. Survival distributions
and life tables. Introduc-tion to life insurance. Prerequisite: Math.
180A or 183, or consent of instructor.
193B. Actuarial Mathematics (4) Life
Insurance and Annuities. Analysis of premiums and premium reserves.
Introduction to multiple life functions and decrement models as time
permits. Prerequisite: Math. 193A.
193C. Actuarial Mathematics (4) Topics
to be selected from pension plans, collective risk models, advanced
topics in insurance. Prerequisite: Math. 193B.
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 (21D), Math. 20F,
and Math. 180A or 183.
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. Prerequisite: consent of instructor. (F,W,S)
196. Student Colloquium (1-2) A variety
of topics and current research results in mathematics will be presented
by guest lecturers and students under faculty direction. Prerequisites:
upper-division status or consent of instructor (for one unit) and consent
of instructor (for two units).
197. Mathematics Internship (2 or 4) An
enrichment program which provides work experience with public/private
sector employers. Subject to the availablility 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 90 units, 2 upper-division
mathematics courses, an overall 2.5 UCSD G.P.A., 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.) Prerequisite: 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. Prerequisite: consent of instructor.
201A. Basic Topics in Algebra (4) Recommended
for all students specializing in algebra. Basic topics include categorical
algebra, commutative algebra, group representations, homological algebra,
nonassociative algebra, ring theory. Prerequisites: Math. 200A-B-C
or consent of instructor. (F,W,S)
202B-C. Applied Algebra (4-4) Algebra
from a computational perspective using Maple, Mathematica and Matlab.
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. Prerequisite:
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)
204. Topics in Number Theory (4) Topics
in analytic number theory, such as zeta functions and L-functions and
the distribution of prime numbers, zeros of zeta functions and Siegels
theorem, transcendence theory, modular forms, finite and infinite symmetric
spaces. Prerequisite: consent of instructor.
205. Topics in Algebraic Number Theory (4) Topics
in algebraic number theory, such as cyclotomic and Kummer extensions,
class number, units, splitting of primes in extensions, zeta functions
of number fields and the Brauer-Siegel Theorem, class field theory,
elliptic curves and curves of higher genus, complex multiplication.
Prerequisite: consent of instructor.
207A-B-C. Topics in Algebra (4-4-4) In
recent years, topics have included number theory, commutative algebra,
noncommutative rings, homological algebra, and Lie groups. May be repeated
for credit with consent of adviser. Prerequisite: consent of instructor.
208. Seminar in Algebra (1-4) Prerequisite:
consent of instructor. (S/U grades permitted.)
209. Seminar in Number Theory (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
210A. Mathematical Methods in Physics and Engineering (4) Complex
variables with applications. Analytic functions, Cauchys 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, Greens functions, distributions, Fourier transform.
Prerequisite: Math. 210A or consent of instructor. (W)
210C. Mathematical Methods in Physics and Engineering (4) Calculus
of variations: Euler-Lagrange equations, Noethers theorem. Fourier
analysis of functions and distributions in several variables. Partial
differential equations: Laplace, wave, and heat equations; fundamental
solutions (Greens functions); well-posed problems. Prerequisite:
Math. 210B or consent of instructor. (S)
211. Fourier Analysis on Finite Groups (4) Applied
group representations. Emphasis on the integers, mod n, finite matrix
groups. Applications may include: the fast Fourier tranform, digital
signal processing, finite difference equations, spectral graph theory,
error-correcting codes, vibrating systems, finite wavelet tranforms.
Prerequisite: none.
212A. Introduction to the Mathematics of Systems and Control (4) Linear
and nonlinear systems, and their input-output behavior, linear continuous
time and discrete-time systems, reachability and controllability for linear
systems, feedback and stabilization, eigenvalue placement, nonlinear controllability,
feedback linearization, disturbance rejection, nonlinear stabilization,
Lyapunov and control-Lyapunov functions, linearization principle for stability.
Prerequisites: Math. 102 or equivalent, Math. 120A or equivalent, Math.
142A or equivalent.
212B. Introduction to the Mathematics of Systems and Control (4) Observability
notions, linearization principle for observability. Realization theory
for linear systems, observers and dynamic feedback, detectability, external
stability for linear systems, frequency-domain considerations, dynamic
programming, quadratic cost, state estimation and Kalman filtering, nonlinear
stabilizing optimal controls, calculus of variations, and the Maximum
Principle. Prerequisite: Math. 212A.
212C. Introduction to the Mathematics of Systems and Control (4) Topics
of current interest on systems theory, control, and estimation to be chosen
by instructor. Prerequisite: Math. 212B.
216. Topics in Pure Mathematics (4) This
course brings together graduate students, postdocs, and faculty to examine
a current research topic of broad interest. Previously covered topics
include: noncommutative geometry, Loop groups, geometric quantization.
Prerequisite: consent of instructor.
217A. 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. Prerequisite: 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-B. Topics in Several Complex Variables (4-4) Formal
and convergent power series, Weierstrass preparation theorem; Cartan-Ruckert
theorem, analytic sets; mapping theorems; domains of holomorphy; proper
holomorphic mappings; complex manifolds; modifications. Prerequisites:
Math. 200A and 220A-B-C or consent of instructor.
227A-B. Topics in Complex Analysis (4-4) In
recent years, topics have included conformal mapping, Riemann surfaces,
value distribution theory, external length. May be repeated for credit
with consent of adviser. Prerequisite: consent of instructor.
229. Computing Technology for Mathematics (2) Preparation
for making effective use of computer technology in research and teaching
of mathematics. UNIX basics, document preparation using TEX, Internet
resources, HTML, computer technology in teaching. Prerequisite: graduate
status in mathematics.
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. Greens
function, eigenvalue problems, perturbation theory. Prerequisites:
Math. 210A-B or 240A-B-C or consent of instructor.
233. Singular Perturbation Theory for Differential Equations (4) Multivariable
techniques, matching techniques and averaging techniques, including
various approaches to proofs of asymptotic correctness, for singular
perturbation problems including initial value problems with nonuniformities
at infinity, initial value problems with initial nonuniformities, two
point boundary value problems, and problems for partial differential
equations. Applications taken from celestial mechanics, oscillation
problems, fluid dynamics, elasticity, and applied mechanics. Prerequisites:
Math. 130A-B or 132A-B or consent of instructor. (S/U grades permitted.)
(S)
237A-B-C. Topics in Differential Equations (4-4-4) May
be repeated for credit with consent of adviser. Prerequisite: 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.
247A-B-C. Topics in Real Analysis (4-4) In
recent years, topics have included Fourier analysis, distribution theory,
martingale theory, operator theory. May be repeated for credit with
consent of adviser. Prerequisite: consent of instructor.
248. Seminar in Real Analysis (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
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. Prerequisite: 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 (2 to 4) Various
topics in Lie groups and Lie algebras, including structure theory, representation
theory, and applications. Prerequisite: consent of instructor.
(F,W,S)
257A. Topics in Differential Geometry (4) In
recent years, topics have included Morse theory and general relativity.
May be repeated for credit with consent of adviser. Prerequisite:
consent of instructor.
259A-B-C. Geometrical Physics (4-4-4) Manifolds,
differential forms, homology, deRhams 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. Prerequisite: graduate standing
in mathematics, physics, or engineering, or consent of instructor.
260A-B. Mathematical Logic (4-4) Propositional
calculus and quantification theory. Completeness theorem, theory of
equality, compactness theorem, Skolem-Lowenheim theorems. Vaughts
test: Craigs lemma. Elementary number theory and recursive function
theory. Undecidability of true arithmetic and of Peanos axioms.
Churchs thesis; set theory; Zermelo-Frankel axiomatic formulation.
Cardinal and ordinal numbers. The axiom of choice and the generalized
continuum hypothesis. Incompleteness and undecidability of set theory.
Relative consistency proofs. Prerequisites: Math. 100A-B-C or consent
of instructor.
261A-B. Combinatorial Algorithms (4-4) Lexicographic
order, backtracking, ranking algorithms, isomorph rejection, sorting,
orderly algorithms, network flows and related topics, constructive Polya
theory, inclusion-exclusion and seiving methods, Mobius inversion, generating
functions, algorithmic graph theory, trees, recursion, depth firstsearch
and applications, matroids. Prerequisites: CSE 160A-B or Math.184A-B
or consent of instructor. (F,W,S)
262A-B. Topics in Combinatorial Mathematics (4-4) Development
of a topic in combinatorial mathematics starting from basic principles.
Problems of enumeration, existence, construction, and optimization with
regard to finite sets. Some familiarity with computer programming desirable
but not required. Prerequisites: Math. 100A-B-C.
263. History of Mathematics (4) Mathematics
in the nineteenth century from the original sources. Foundations of
analysis and commutative algebra. For algebra the authors studied will
be Lagrange, Ruffini, Gauss, Abel, Galois, Dirichlet, Kummer, Kronecker,
Dedekind, Weber, M. Noether, Hilbert, Steinitz, Artin, E. Noether. For
analysis they will be Cauchy, Fourier, Bolzano, Dirichlet, Riemann,
Weierstrass, Heine, Cantor, Peano, Hilbert. Prerequisites: Math.
100A-B, Math. 140A-B.(S)
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-B-C. Topics in Mathematical Logic (4-4-4) Topics
chosen from recursion theory, model theory, and set theory. May be repeated
with consent of adviser. Prerequisite: consent of instructor.
(S/U grades permitted.)
268. Seminar in Logic (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
269. Seminar in Combinatorics (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
270A-B-C. Numerical Mathematics (4-4-4) 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.
Error analysis of numerical quadrature and of the numerical solution
of ordinary differential equations. Prerequisites: Math. 20F and
knowledge of programming.
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.
Prerequisite: consent of instructor. (F,W,S)
272A-B-C. Numerical Partial Differential Equations (4-4-4) The
numerical solution of elliptic, parabolic, and hyperbolic partial differential
equations; discretization and solution techniques. Prerequisite:
consent of instructor. (F,W,S)
273A-B-C. Scientific Computation (4-4-4) Continuum
mechanics models of physical and biological systems, finite element
methods and approximation theory, complexity of iterative methods for
linear and nonlinear equations, continuation methods, adaptive methods,
parallel computing, and scientific visualization. Project-oriented;
theoretical and software development projects designed around problems
of current interest in science and engineering. Prerequisite: experience
with Matlab and C, some background in numerical analysis, or consent
of instructor. (F,W,S)
277A. Topics in Numerical Mathematics (4) Topics
vary from year to year. May be repeated for credit with consent of adviser.
Prerequisite: consent of instructor.
278. Seminar in Numerical Mathematics (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
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 probablilty 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 probablilty 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 probablilty theory or consent of instructor.
282A-B. Applied Statistics (4-4) Sequence
in applied statistics. First quarter: general theory of linear models
with applications to regression analysis. Second quarter: analysis of
variance and covariance and experimental design. Third quarter: further
topics to be selected by instructor. Emphasis throughout is on the analysis
of actual data. Prerequisite: Math. 181B or equivalent 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.
285A-B. Stochastic Processes (4-4) Elements
of stochastic processes, Markov chains, hidden Markov models, Poisson
point processes, renewal processes martingales, Brownian motion, Gaussian
processes, Kalman filter. Other topics to be selected by instructor
depending on interest of class. Prerequisites: Math. 180A ( or equivalent
basic probablilty course) 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. Prerequisite:
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. Prerequisite: Math.
181B or equivalent or consent of instructor.
287B. Multivariate Analysis (4)
288. Seminar in Probability and Statistics (1 to 4) Prerequisite:
consent of instructor. (S/U grades permitted.)
289A-B. Topics in Probability and Statistics (4-4) In
recent years, topics have included Markov processes, martingale theory,
stochastic processes, stationary and Gaussian processes, ergodic theory.
May be repeated for credit with consent of adviser.
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-B-C. Topics in Topology (4) In recent
years, topics have included generalized cohomology theory, spectral
sequences, K-theory, homotophy theory. Prerequisites: consent of
instructor.
292. Seminar in Topology (1-4) Various
topics in topology. Prerequisites: consent of instructor.
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. Prerequisite: 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. Student Colloquium (1 to 2) A variety
of topics and current research in mathematics will be presented by guest
lecturers and students under faculty direction. Prerequisites: for
one unitupper-division status or consent of instructor (may only
be taken P/NP), or graduate status (may only be taken S/U); for two
unitsconsent of instructor, standard grading option allowed.
297. Mathematics Graduate Research Internship (2-4) An
enrichment program which 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.
Prerequisite: 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). Prerequisite:
consent of adviser. (S/U grades only.)
Mathematics Courses
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