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Mathematics
194 Baskin Engineering
(831) 459-2969
http://www.math.ucsc.edu
Program Description
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Changes to 2006-08 Catalog Highlighted
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Faculty
Lower-Division Courses
2.
College Algebra for Calculus.
F,W
Operations on real numbers, complex numbers, polynomials, and
rational expressions; exponents and radicals; solving linear and
quadratic equations and inequalities; functions, algebra of functions,
graphs; conic sections; mathematical models; sequences and series.
Prerequisite(s): placement exam score of 12 or higher. The Staff
3.
Precalculus.
F,W,S
Inverse functions and graphs; exponential and logorithmic
functions, their graphs, and use in mathematical models of the real
world; rates of change; trigonometry, trigonometric functions, and
their graphs; and geometric series. Students cannot receive credit for
both course 3 and Applied Mathematics and Statistics 3. Applied
Mathematics and Statistics 3 can substitute for course 3.
Prerequisite(s): course 2 or placement exam score of 20 or higher.
(General Education Code(s): Q.) The Staff
4.
Mathematics of Choice and Argument.
S
Techniques of analyzing and creating quantitative arguments.
Application of probability theory to questions in justice, medicine,
and economics. Analysis and avoidance of statistical bias.
Understanding the application and limitations of quantitative
techniques. Prerequisite(s): course 2, or placement exam score of 12 or
higher, or AP calculus exam score of 3 or higher. Enrollment limited to
30. (General Education Code(s): Q.) The Staff
11A.
Calculus with Applications.
F,W,S
A modern course stressing conceptual understanding, relevance, and
problem solving. The derivative of polynomial, exponential, and
trigonometric functions of a single variable is developed and applied
to a wide range of problems involving graphing, approximation, and
optimization. Students cannot receive credit for both this course and
course 19A or Applied Mathematics and Statistics 11A or Economics 11A.
Prerequisite(s): course 3 or Applied Mathematics and Statistics 3; or
placement exam score of 31 or higher; or AP Calculus AB exam score of 3
or higher. (General Education Code(s): IN, Q.) The Staff
11B.
Calculus with Applications.
F,W,S
Starting with the fundamental theorem of calculus and related
techniques, the integral of functions of a single variable is developed
and applied to problems in geometry, probability, physics, and
differential equations. Polynomial approximations, Taylor series, and
their applications conclude the course. Students cannot receive credit
for this course and course 19B, or Applied Mathematics and Statistics
11B, or Economics 11B. Prerequisite(s): course 11A or AP Calculus AB
exam score of 4 or 5, or BC exam score of 3 or higher, or IB
Mathematics Higher Level exam score of 5 or higher. (General Education
Code(s): IN, Q.) The Staff
19A.
Calculus for Science, Engineering, and Mathematics.
F,W,S
The limit of a function, calculating limits, continuity, tangents,
velocities, and other instantaneous rates of change. Derivatives, the
chain rule, implicit differentiation, higher derivatives. Exponential
functions, inverse functions, and their derivatives. The mean value
theorem, monotonic functions, concavity, and points of inflection.
Applied maximum and minimum problems. Students cannot receive credit
for both this course and course 11A or Applied Mathematics and
Statistics 11A or Economics 11A. Prerequisite(s): course 3 or Applied
Mathematics and Statistics 3 or placement exam score of 40 or higher or
AP Calculus AB exam score of 3 or higher. (General Education Code(s):
IN, Q.) The Staff
19B.
Calculus for Science, Engineering, and Mathematics.
F,W,S
The definite integral and the fundamental theorem of calculus.
Areas, volumes. Integration by parts, trigonometric substitution, and
partial fractions methods. Improper integrals. Sequences, series,
absolute convergence and convergence tests. Power series, Taylor and
Maclaurin series. Students cannot receive credit for both this course
and course 11B, Applied Math and Statistics 11B, or Economics 11B.
Prerequisite(s): course 19A or AP Calculus AB exam score of 4 or 5, or
BC exam score of 3 or higher, or IB Mathematics Higher Level exam score
of 5 of higher. (General Education Code(s): IN, Q.) The Staff
20A.
Honors Calculus.
F
Challenging course designed to approach single-variable calculus
from the perspective of modern mathematics. Emphasis is on the
evolution and historical development of core concepts underlying
calculus and analysis. Prerequisite(s): placement exam score of 46 or
higher; or AP Calculus AB exam score of 4 or 5; or BC exam of 3 or
higher; or IB Mathematics Higher Level exam score of 5 or higher.
Enrollment limited to 60. (General Education Code(s): IN, Q.) The Staff
20B.
Honors Calculus.
W
Challenging course designed to approach single-variable calculus
from the perspective of modern mathematics. Emphasis is on the
evolution and historical development of core concepts underlying
calculus and analysis. Prerequisite(s): course 20A. Enrollment limited
to 60. (General Education Code(s): IN, Q.) The Staff
21.
Linear Algebra.
F,W,S
Systems of linear equations, matrices, determinants. Introduction
to abstract vector spaces, linear transformation, inner products,
geometry of Euclidean space, and eigenvalues. One quarter of college
mathematics is recommended as preparation. (General Education Code(s):
Q.) The Staff
22.
Introduction to Calculus of Several Variables.
F,W,S
Functions of several variables. Continuity and partial
derivatives. The chain rule, gradient and directional derivative.
Maxima and minima, including Lagrange multipliers. The double and
triple integral and change of variables. Surface area and volumes.
Applications from biology, chemistry, earth sciences, engineering, and
physics. Students cannot receive credit for this course and course 23A.
Prerequisite(s): course 11B or 19B or 20B. The Staff
23A.
Multivariable Calculus.
F,W,S
Vectors in n-dimensional Euclidean space. The inner and cross
products. The derivative of functions from n-dimensional to
m-dimensional Euclidean space is studied as a linear transformation
having matrix representation. Paths in 3-dimensions, arc length, vector
differential calculus. Taylor's theorem in several variables, extrema
of real-valued functions, constrained extrema and Lagrange multipliers,
the implicit function theorem, some applications. Students cannot
receive credit for this course and course 22. Prerequisite(s): course
19B or 20B. The Staff
23B.
Multivariable Calculus.
F,W,S
Double integral, changing the order of integration. Triple
integrals, maps of the plane, change of variables theorem, improper
double integrals. Path integrals, line integrals, parametrized
surfaces, area of a surface, surface integrals. Green's theorem, Stokes
theorem, conservative fields, Gauss' theorem. Applications to physics
and differential equations, differential forms. Prerequisite(s): course
23A. The Staff
24.
Ordinary Differential Equations.
S
First and second order ordinary differential equations, with
emphasis on the linear case. Methods of integrating factors,
undetermined coefficients, variation of parameters, power series,
numerical computation. Students cannot receive credit for this course
and Applied Mathematics and Statistics 27. Prerequisite(s): course 22
or 23A; course 21 is recommended as preparation. The Staff
30.
Mathematical Problem Solving.
F
Students learn techniques of problem solving such as induction,
contradiction, exhaustion, dissection, analogy, generalization,
specialization, and others in the context of solving problems drawn
from number theory, probability, combinatorics, graph theory, geometry,
and logic. Prerequisite(s): course 11A or 19A or 20A or Math Placement
Exam score of 40 or higher. B. Cooperstein
99.
Tutorial.
F,W,S
The Staff
Upper-Division Courses
100.
Introduction to Proof and Problem Solving.
F,W,S
Students learn the basic concepts and ideas necessary for
upper-division mathematics and techniques of mathematical proof in the
context of specific topics. Introduction to sets, relations, elementary
mathematical logic, proof by contradiction, mathematical induction, and
counting arguments. Prerequisite(s): courses 11A and 11B or 19A and 19B
or 20A and 20B. Enrollment limited to 40. The Staff
103.
Complex Analysis.
F,W
Complex numbers, analytic and harmonic functions, complex
integration, the Cauchy integral formula, Laurent series, singularities
and residues, conformal mappings. Prerequisite(s): course 23B; and
either course 100 or Computer Science 101. The Staff
105A.
Real Analysis.
F,S
The basic concepts of one-variable calculus are treated carefully
and rigorously. Set theory, the real number system, numerical sequences
and series, continuity, differentiation. Prerequisite(s): course 23B
and either course 100 or Computer Science 101. The Staff
105B.
Real Analysis.
*
Metric spaces, differentiation and integration of functions. The Riemann-Stieltjes integral. Sequences and series of functions.
Prerequisite(s): course 105A.
The Staff
105C.
Real Analysis.
*
The Stone-Weierstrass theorem, Fourier series, differentiation and integration of functions of several variables.
Prerequisite(s): course 105B.
The Staff
106A.
Systems of Ordinary Differential Equations.
F
Linear systems, exponentials of operators, existence and
uniqueness, stability of equilibria, periodic attractors, and
applications. Prerequisite(s): either Applied Mathematics and
Statistics 27 or preferably courses 21 and 24; and either course 100 or
Computer Science 101. The Staff
106B.
Partial Differential Equations.
W
Topics covered include first and second order linear partial
differential equations, the heat equation, the wave equation, Laplace's
equation, separation of variables, eigenvalue problems, Green's
functions, Fourier series. Prerequisite(s): either courses 21 and 24 or
Applied Mathematics and Statistics 27; and either course 100 or
Computer Science 101; course 106A is recommended as preparation. The Staff
110.
Introduction to Number Theory.
F
Prime numbers, unique factorization, congruences with applications
(e.g., to magic squares). Rational and irrational numbers. Continued
fractions. Introduction to Diophantine equations. No calculus required.
An introduction to some of the ideas and outstanding problems of modern
mathematics. Prerequisite(s): course 100 or Computer Science 101.
(General Education Code(s): Q.) The Staff
111A.
Algebra.
W,S
Group theory including the Sylow theorem, the structure of abelian
groups, permutation groups. Introduction to rings and fields including
polynomial rings, factorization, the classical geometric constructions,
and Galois theory. Prerequisite(s): course 21 or Applied Mathematics
and Statistics 27 and either course 100 or Computer Science 101. The Staff
111B.
Algebra.
S
Group theory including the Sylow theorem, the structure of abelian
groups, permutation groups. Introduction to rings and fields including
polynomial rings, factorization, the classical geometric constructions,
and Galois theory. Prerequisite(s): course 111A. The Staff
112.
Mathematical Probability Theory.
*
Introductory probability course for mathematicians, designed as a
prerequisite for advanced probability courses at the graduate level.
Moving from elementary topics of probability spaces and random
variables, independent identical trials, the law of large numbers, the
Demoivre-Laplace central limit theorem, also includes basic Martingale
theory, finite Markov chains, percolations, and branching processes.
Prerequisite(s): courses 21 and 23B. The Staff
113.
Discrete Mathematics.
*
Basic course in theorems and applications of discrete mathematics.
Sequences and series, matrix operations, recursion relations, discrete
probability, algorithms, finite state machines, boolean functions,
trees, elementary number theory, generating functions, graph theory.
Particular emphasis on combinatorics. Applications dealing with
searching and sorting, cryptography, coding, quantum mechanics, and
Markov processes. Prerequisite(s): courses 19A-B, 21, or equivalent. The Staff
114.
Introduction to Financial Mathematics.
*
Financial derivatives: contracts and options. Hedging and risk
managment. Arbitrage, interest rate, and discounted value. Geometric
random walk and Brownian motion as models of risky assets. Ito's
formula. Initial boundary value problems for the heat and related
partial differential equations. Self-financing replicating portfolio;
Black-Scholes pricing of European options. Dividends. Implied
volatility. American options as free boundary problems.
Prerequisite(s): course 23B or Applied Mathematics and Statistics 27.
Corequisite(s): course 112 or Applied Mathematics and Statistics 131 or
Computer Engineering 107. The Staff
115.
Graph Theory.
W
Graph theory, trees, vertex and edge colorings, Hamilton cycles,
Eulerian circuits, decompositions into isomorphic subgraphs, extremal
problems, cages, Ramsey theory, Cayley's spanning tree formula, planar
graphs, Euler's formula, crossing numbers, thickness, splitting
numbers, magic graphs, graceful trees, rotations, and genus of graphs.
Prerequisite(s): course 21 or Applied Mathematics and Statistics 27 and
either course 100 or Computer Science 101. The Staff
117.
Advanced Linear Algebra.
W
Review of abstract vector spaces. Dual spaces, bilinear forms, and
the associated geometry. Normal forms of linear mappings. Introduction
to tensor products and exterior algebra. Prerequisite(s): course 21 or
Applied Mathematics and Statistics 27 and either course 100 or Computer
Science 101. The Staff
118.
Advanced Number Theory.
W
Topics include divisibility and congruences, arithmetical
functions, quadratic residues and quadratic reciprocity, quadratic
forms and representations of numbers as sums of squares, Diophantine
approximation and transcendence theory, quadratic fields. Additional
topics as time permits. Prerequisite(s): course 110 or 111A. The Staff
120.
Coding Theory.
*
An introduction to mathematical theory of coding. Construction and
properties of various codes, such as cyclic, quadratic residue, linear,
Hamming, and Golay codes; weight enumerators; connections with modern
algebra and combinatorics. Prerequisite(s): course 21. The Staff
121A.
Differential Geometry.
W
Topics include Euclidean space, tangent vectors, directional
derivatives, curves and differential forms in space, mappings. Curves,
the Frenet formulas, covariant derivatives, frame fields, the
structural equations. The classification of space curves up to rigid
motions. Vector fields and differentiable forms on surfaces; the shape
operator. Gaussian and mean curvature. The theorem Egregium; global
classification of surfaces in three space by curvature.
Prerequisite(s): courses 21 and 23B and either course 100 or Computer
Science 101. Course 105A strongly recommended. The Staff
121B.
Differential Geometry and Topology.
S
Examples of surfaces of constant Gauss curvature, surfaces of
revolutions, minimal surfaces. Abstract manifolds; integration theory;
Riemannian manifolds. Total curvature and geodesics; the Euler
characteristic, the theorem of Gauss-Bonnet. Length-minimizing
properties of geodesics, complete surfaces, curvature and conjugate
points covering surfaces. Surfaces of constant curvature; the theorems
of Bonnet and Hadamard. Prerequisite(s): course 121A. The Staff
124.
Introduction to Topology.
F
Topics include introduction to point set topology (topological
spaces, continuous maps, connectedness, compactness), homotopy
relation, definition and calculation of fundamental groups and homology
groups, Euler characteristic, classification of orientable and
nonorientable surfaces, degree of maps, and Lefschetz fixed-point
theorem. Prerequisite(s): course 100; course 111A recommended. The Staff
126.
Mathematical Control Theory.
*
Control theory concerns steering and stabilizing systems by means
of tunable parameters. Examples are flight controllers, CD players, and
biological or robotic locomotion. Studies the mathematical foundations,
tools, and basic theorems of linear and nonlinear deterministic
control. Prerequisite(s): courses 23B and 24 or Applied Mathematics and
Statistics 27, and either course 100 or Computer Science 101. The Staff
128A.
Classical Geometry: Euclidean and Non-Euclidean.
F
Rigorous foundations for Euclidean and non-Euclidean geometries.
History of attempts to prove the parallel postulate and of the
simultaneous discovery by Gauss, J. Bolyai, and Lobachevsky of
hyperbolic geometry. Consistency proved by Euclidean models.
Classification of rigid motions in both geometries. Prerequisite(s):
either course 100 or Computer Science 101. The Staff
128B.
Classical Geometry: Projective.
*
Theorems of Desargue, Pascal, and Pappus; projectivities;
homogeneous and affine coordinates; conics; relation to perspective
drawing and some history. Prerequisite(s): course 21. The Staff
130.
Celestial Mechanics.
*
Solves the two-body (or Kepler) problem, then moves onto the
N-body problem where there are many open problems. Includes central
force laws; orbital elements; conservation of linear momentum, energy,
and angular momentum; the Lagrange-Jacobi formula; Sundman's theorem
for total collision; virial theorem; the three-body problem; Jacobi
coordinates; solutions of Euler and of Lagrange; and restricted
three-body problem. Prerequisite(s): courses 19A-B and course 23A or
Physics 5A or 6A; courses 21 and 24 strongly recommended. Enrollment
limited to 35. The Staff
134.
Cryptography.
*
Introduces different methods in cryptography (shift cipher, affine
cipher, Vigenere cipher, Hill cipher, RSA cipher, ElGamal cipher,
knapsack cipher). The necessary material from number theory and
probability theory is developed in the course. Common methods to attack
ciphers discussed. Prerequisite(s): course 100; course 110 recommended
as preparation. R. Boltje
141.
Introduction to Nonlinear Mathematics.
*
Modeling problems involving nonlinear differential equations.
Applications to chemical reactions, electrical circuits, shock waves,
ecosystems, microeconomics, stochastic processes. Exact solutions,
intuitive and pictorial methods of analysis. Prerequisite(s): courses
21 and 24 or Applied Mathematics and Statistics 27; course 100 or
Computer Science 101; 106A recommended. The Staff
145.
Introductory Chaos Theory.
S
The Lorenz and Rossler attractors, measures of chaos, attractor
reconstruction, applications from the sciences. Students cannot receive
credit for this course and Applied Mathematics and Statistics 146.
Prerequisite(s): course 22 or 23A; course 21; course 100 or Computer
Science 101. Concurrent enrollment in course 145L is required. The Staff
145L.
Introductory Chaos Laboratory
(1 credit).
S
Laboratory sequence illustrating topics covered in course 145. One three-hour session per week in microcomputer laboratory.
Concurrent enrollment in course 145 is required.
The Staff
148.
Numerical Analysis.
*
The theory of constructive methods in mathematical analysis and
its application with scientific computation. Some typical topics are
difference equations, linear algebra, iteration, Bernoulli's method,
quotient difference algorithm, the interpolating polynomial, numerical
differentiation and integration, numerical solution of differential
equations, finite Fourier series. Prerequisite(s): course 22 or 23A;
course 21 and 24 or Applied Mathematics and Statistics 27; course 100
or Computer Science 101. Concurrent enrollment in course 148L is
required. The Staff
148L.
Numerical Analysis Laboratory
(1 credit).
*
Laboratory sequence illustrating topics covered in course 148. One
three-hour session per week in microcomputer laboratory. Concurrent
enrollment in course 148 is required. The Staff
160.
Mathematical Logic I.
*
Propositional and predicate calculus. Resolution, completeness,
compactness, and Löwenheim-Skolem theorem. Recursive functions, Gödel
incompleteness theorem. Undecidable theories. Hilbert's 10th problem.
Prerequisite(s): course 100 or Computer Science 101. The Staff
161.
Mathematical Logic II.
S
Continuation of course 160: arithmetization of syntax, Tarski's
theorem on the undefinability of truth, Gödel's first incompleteness
theorem, naive set theory and its limitations (Russell's paradox),
cardinal numbers, cardinal arithmetic, Axiom of Choice, finite,
countable and uncountable sets, and Continuum Hypothesis.
Prerequisite(s): course 160. Enrollment limited to 45. The Staff
181.
History of Mathematics.
W
A survey from a historical point of view of various developments in mathematics. Specific topics and periods to vary yearly.
The Staff
188.
Supervised Teaching.
F,W,S
Supervised tutoring in self-paced courses. May not be repeated for credit. Students submit petition to sponsoring agency.
The Staff
194.
Senior Seminar.
W,S
Designed to expose the student to topics not normally covered in
the standard courses. The format varies from year to year. In recent
years each student has written a paper and presented a lecture on it to
the class. Prerequisite(s): course 103 or 105A or 111A. Enrollment
priority given to seniors. The Staff
195.
Senior Thesis.
F,W,S
Students research a mathematical topic under the guidance of a
faculty sponsor and write a senior thesis demonstrating knowledge of
the material. Students submit petition to sponsoring agency. May be
repeated for credit. The Staff
199.
Tutorial.
F,W,S
Students submit petition to sponsoring agency.
May be repeated for credit.
The Staff
Graduate Courses
200.
Algebra I.
F
Subgroups, cosets, normal subgroups, homomorphisms, isomorphisms,
quotient groups, free groups, generators and relations, group actions
on a set. Sylow theorems, semidirect products, simple groups, nilpotent
groups, and solvable groups. Ring theory: Chinese remainder theorem,
prime ideals, localization. Euclidean domains, PIDs, UFDs, polynomial
rings. Prerequisite(s): courses 111A and 117 are recommended as
preparation. May be repeated for credit. The Staff
201.
Algebra II.
W
Vector spaces, linear transformations, eigenvalues and
eigenvectors, Jordan canonical forms, bilinear forms, quadratic forms,
real symmetric forms and real symmetric matrices, orthogonal
transformations and orthogonal matrices, Euclidean space, Hermitian
forms and Hermitian matrices, Hermitian spaces, unitary transformations
and unitary matrices, skew-symmetric forms, tensor products of vector
spaces, tensor algebras, symmetric algebras, exterior algebras,
Clifford algebras and spin groups. Course 200 is recommended as
preparation. The Staff
202.
Algebra III.
S
Submodules, quotient modules, module homomorphisms, generators of
modules, direct sums, free modules, torsion modules, modules over PIDs,
and applications to rational and Jordan canonical forms. Field theory:
field extensions, algebraic and transcendental extensions, splitting
fields, algebraic closures, separable and normal extensions, the Galois
theory, finite fields, Galois theory of polynomials. Course 201 is
recommended as preparation. The Staff
203.
Algebra IV.
F
Topics include tensor product of modules over rings, projective
modules and injective modules, Jacobson radical, Weederburns' theorem,
category theory, Noetherian rings, Artinian rings, affine varieties,
projective varieties, Hilbert's Nullstellensatz, prime spectrum,
Zariski topology, discrete valuation rings, and Dedekind domains.
Prerequisite(s): courses 200, 201, and 202. The Staff
204.
Analysis I.
F
Completeness and compactness for real line, sequences and infinite
series of functions, Fourier series, calculus on Euclidean space and
implicit function theorem, metric spaces and contracting mapping
theorem, Arzela-Ascoli theorem, basics of general topological spaces,
Baire category theorem, Urysohn lemma, and Tychnoff theorem. Course
105A and course 105B or equivalent are recommended as preparation. The Staff
205.
Analysis II.
W
Lebesgue measure theory, abstract measure theory, measurable
functions, integration, space of absolutely integrable functions,
dominated convergence theorem, convergence in measure, Riesz
representation theorem, product measure and Fubini theorem, Lp spaces, derivative of a measure and Radon-Nikodym theorem, fundamental theorem of calculus.
Prerequisite(s): course 204.
The Staff
206.
Analysis III.
S
Banach space, Hahn-Banach theorem, uniform boundedness theorem,
open mapping theorem and closed graph theorem, weak and weak* topology
and Banach-Alaoglu theorem, Hilbert space, self-adjoint operators,
compact operators, spectral theory, Fredholm operators, space of
distributions and Fourier transform, Sobolev spaces. Courses 204 and
205 recommended as preparation. The Staff
207.
Complex Analysis.
W
Holomorphic and harmonic functions, the Cauchy integral theorem,
the maximum principle and its consequences, conformal mapping, analytic
continuation. The Riemann mapping theorem. Course 103 is recommended as
preparation. The Staff
208.
Manifolds I.
F
Definition of manifolds, tangent bundle, inverse and implicit
function theorems, transversality, Sard's theorem and the Whitney
embedding theorem, vector fields, flows, and Lie bracket, Frobenius's
theorem. Course 204 recommended for preparation. The Staff
209.
Manifolds II.
W
Tensor algebra. Differential forms and associated formalism of
pullback, wedge product, exterior derivative, Stokes theorem,
integration. Cartan's formula for Lie derivative. Cohomology via
differential forms. Poincaré lemma and the Mayer-Vietoris sequence.
Theorems of deRham and Hodge. Prerequisite(s): course 208. Course 201
is recommended as preparation. The Staff
210.
Manifolds III.
S
The fundamental group, covering space theory and Van Kampen's
theorem (with a discussion of free and amalgamated products of groups),
CW complexes, higher homotopy groups, cellular and singular cohomology,
the Eilenberg-Steenrod axioms, computational tools including
Mayer-Vietoris, cup products, Poincaré duality, and the Lefschetz fixed
point theorem, homotopy exact sequence of a fibration and the Hurewicz
isomorphism theorem, remarks on characteristic classes. Courses 208 and
209 recommended as preparation. The Staff
211.
Algebraic Topology.
F
Continuation of course 210. Topics include theory of
characteristic classes of vector bundles, cobordism theory, and
homotopy theory. Courses 200 and 210 recommended as preparation. The Staff
212.
Differential Geometry.
S
Principle bundles, associated bundles and vector bundles,
connections and curvature on principle and vector bundles. More
advanced topics: introduction to cohomology, the Chern-Weil
construction and characteristic classes, the Gauss-Bonnet theorem or
Hodge theory, eigenvalue estimates for Beltrami Laplacian, comparison
theorems in Riemannian geometry. Prerequisite(s): course 208. The Staff
213A.
Partial Differential Equations I.
F
First of the two PDE series covering basically Part I in Evans' book,
Partial Differential Equations, which includes transport
equations, Laplace equations, heat equations, wave equations,
characteristics of nonlinear first order PDE, Hamilton-Jacobi
equations, equations of conservation laws, some methods to solve
equations with close forms, and Cauchy-Kovalevskaya theorem. Courses
106A and 106B are recommended as preparation. The Staff
213B.
Partial Differential Equations II.
W
Second of the PDE series covering basically most of Part II in
Evans' book and some topics in nonlinear PDE including Sobolev space,
Sobolev inequalities, existence, regularity and a priori estimates of
solutions to second order elliptic PDE, parabolic equations, hyperbolic
equations and systems of conservation laws, and calculus of variations
and its applications to PDE. Courses 106A, 106B, and 213A are
recommended as preparation. The Staff
214.
Theory of Finite Groups.
*
Nilpotent groups, solvable groups, Hall subgroups, the Frattini
subgroup, the Fitting subgroup, Schur-Zassenhaus theorem, fusion in
p-subgroups, the transfer map, Frobenius theorem on normal
p-complements. Courses 200 and 201 are recommended as preparation. The Staff
215.
Operator Theory.
S
Operators on Banach and Hilbert spaces. The spectral theorem.
Compact and Fredholm operators. Other special classes of operators.
Courses 204, 205, 206, and 207 are recommended as preparation. The Staff
216.
Advanced Analysis.
*
Topics include maximal function, the Lebesgue set, the
Marcinkiewicz interpolation theorem, singular integrals,
Calderon-Zygmund theorem, Hardy Littlewood-Sobolev theorem,
pseudodifferential operators, compensated compactness, concentration
compactness, and applications to PDE. Course 204 is recommended as
preparation. The Staff
217.
Advanced Elliptic Partial Differential Equations.
*
Topics include elliptic equations, existence of weak solutions,
Lax-Milgram theorem, interior and boundary regularity, maximum
principles, Harnack inequality, eigenvalues for symmetric and
non-symmetric elliptic operators, calculus of variations (first
variation: Euler-Lagrange equations, second variation: existence of
minimizers). Other topics covered as time permits. Courses 204 and 205
are recommended as preparation. The Staff
218.
Advanced Parabolic and Hyperbolic Partial Differential Equations.
*
Topics include linear evolution equations, second order parabolic
equations, maximum principles, second order hyperbolic equations,
propagation of singularities, hyperbolic systems of first order,
semigroup theory, systems of conservation laws, Riemann problem, simple
waves, rarefaction waves, shock waves, Riemann invariants, and entropy
criteria. Other topics covered as time permits. The Staff
219.
Nonlinear Functional Analysis.
*
Topological methods in nonlinear partial differential equations,
including degree theory, bifurcation theory, and monotonicity. Topics
also include variational methods in the solution of nonlinear partial
differential equations. Enrollment restricted to graduate students. The Staff
220A.
Representation Theory I.
*
Lie groups and Lie algebras, and their finite dimensional representations.
Prerequisite(s): courses 200 and 204.
The Staff
220B.
Representation Theory II.
*
Lie groups and Lie algebras, and their finite dimensional representations.
Prerequisite(s): course 220A.
The Staff
222A.
Algebraic Number Theory.
F
Topics include algebraic integers, completions, different and
discriminant, cyclotomic fields, parallelotopes, the ideal function,
ideles and adeles, elementary properties of zeta functions and
L-series, local class field theory, global class field theory. Courses
200, 201, and 202 are recommended as preparation. The Staff
222B.
Algebraic Number Theory.
*
Topics include geometric methods in number theory, finiteness
theorems, analogues of Riemann-Roch for algebraic fields (after A.
Weil), inverse Galois problem (Belyi theorem) and consequences. The Staff
223A.
Algebraic Geometry I.
*
Topics include examples of algebraic varieties, elements of
commutative algebra, local properties of algebraic varieties, line
bundles and sheaf cohomology, theory of algebraic curves. Weekly
problem solving. Courses 200, 201, 202, and 208 are recommended as
preparation. The Staff
223B.
Algebraic Geometry II.
*
A continuation of course 223A. Topics include theory of schemes
and sheaf cohomology, formulation of Riemann-Roch theorem, birational
maps, theory of surfaces. Weekly problem solving. Course 223A is
recommended as preparation. The Staff
225A.
Lie Algebras.
*
Basic concepts of Lie algebras. Engel's theorem, Lie's theorem,
Weyl's theorem are proved. Root space decomposition for semi-simple
algebras, root systems and the classification theorem for semi-simple
algebras over the complex numbers. Isomorphism and conjugacy theorems.
Course 202 is recommended as preparation. The Staff
225B.
Infinite Dimensional Lie Algebra.
*
Finite dimensional semi-simple Lie algebras: PBW theorem,
generators and relations, highest weight representations, Weyl
character formula. Infinite dimensional Lie algebras: Heisenberg
algebras, Virasoro algebras, loop algebras, affine Kac-Moody algebras,
vertex operator representations. The Staff
226A.
Infinite Dimensional Lie Algebras and Quantum Field Theory I.
*
Introduction to the infinite-dimensional Lie algebras that arise
in modern mathematics and mathematical physics: Heisenberg and Virasoro
algebras, representations of the Heisenberg algebra, Verma modules over
the Virasoro algebra, Kac determinant formula, and unitary and discrete
series representations. Enrollment restricted to graduate students. The Staff
226B.
Infinite Dimensional Lie Algebras and Quantum Field Theory II.
*
Continuation of I: Kac-Moody and affine Lie algebras and their
representations, integrable modules, representations via vertex
operators, modular invariance of characters, and introduction to vertex
operator algebras. Enrollment restricted to graduate students. The Staff
227.
Lie Groups.
W
Lie groups and algebras, the exponential map, the adjoint action,
Lie's three theorems, Lie subgroups, the maximal torus theorem, the
Weyl group, some topology of Lie groups, some representation theory:
Shur's lemma, Peter-Weyl theorem, roots, weights, classification of Lie
groups, the classical groups. Prerequisite(s): courses 203 and 208. The Staff
228.
Lie Incidence Geometries.
*
Linear incidence geometry is introduced. Linear and classical
groups are reviewed, and geometries associated with projective and
polar spaces are introduced. Characterizations are obtained. The Staff
229.
Kac-Moody Algebras.
*
Theory of Kac-Moody algebras and their representations. Weil-Kac
character formula. Emphasis on representations of affine superalgebras
by vertex operators. Connections to combinatorics, PDE, the monster
group. The Virasoro algebra. The Staff
232.
Morse Theory.
*
Classical Morse Theory. The fundamental theorems relating to
critical points to the topology of a manifold are treated in detail.
The Bolt Periodicity Theorem. A specialized course offered every few
years. The Staff
233.
Random Matrix Theory.
*
Classical matrix ensembles; Wigner semi-circle law; method of
moments. Gaussian ensembles. Method of orthogonal polynomials; Gaudin
lemma. Distribution functions for spacings and largest eigenvalue.
Asymptotics and Riemann-Hilbert problem. Painleve theory and
Tracy-Widom distribution. Selberg's Integral. Matrix ensembles related
to classical groups; symmetric functions theory. Averages of
characteristic polynomials. Fundamentals of free probability theory.
Overview of connections with physics, combinatorics, and number theory.
Prerequisite(s): courses 103 and 204; course 117 recommended as
preparation. Enrollment restricted to graduate students. The Staff
234.
Riemann Surfaces.
*
Riemann surfaces, conformal maps, harmonic forms, holomorphic forms, the theorem of Reimann-Roch, the theory of moduli.
The Staff
235.
Dynamical Systems Theory.
*
An introduction to the qualitative theory of systems of ordinary
differential equations. Structural stability, critical elements, stable
manifolds, generic properties, bifurcations of generic arcs.
Prerequisite(s): courses 106A and 203. The Staff
236.
Probability Theory.
*
Probability theory taught at the graduate level. Topics covered
are weak convergence of probability measures, law of large numbers,
central limit theorems, infinitely divisible distributions, dependent
random variables, conditional expectation and conditioned probability,
Markov chains, basic martingale theory, and ergodic theorems. The Staff
237.
Stochastic Calculus.
*
Introduces Ito's stochastic calculus. Topics covered include
Brownian motion, stochastic integration, exit times, elliptic and
parabolic partial differential equations, stochastic differential
equations, one dimensional diffusions and functional integration. The Staff
238.
Elliptic Functions and Modular Forms.
*
The course, aimed at second-year graduate students, will cover the
basic facts about elliptic functions and modular forms. The goal is to
provide the student with foundations suitable for further work in
advanced number theory, in conformal field theory, and in the theory of
Riemann surfaces. Successful completion of graduate sequence (courses
200-202) and either 207 or 103 are recommended as preparation. The Staff
239.
Homological Algebra.
*
Homology and cohomology theories have proven to be powerful tools
in many fields (topology, geometry, number theory, algebra).
Independent of the field, these theories use the common language of
homological algebra. The aim of this course is to acquaint the
participants with basic concepts of category theory and homological
algebra, as follows: chain complexes, homology, homotopy, several
(co)homology theories (topological spaces, manifolds, groups, algebras,
Lie groups), projective and injective resolutions, derived functors
(Ext and Tor). Depending on time, spectral sequences or derived
categories may also be treated. Courses 200 and 202 strongly
recommended. The Staff
240A.
Representations of Finite Groups I.
*
Introduces ordinary representation theory of finite groups (over
the complex numbers). Main topics are characters, orthogonality
relations, character tables, induction and restriction, Frobenius
reciprocity, Mackey's formula, Clifford theory, Schur indicator, Schur
index, Artin's and Braver's induction theorems. Recommended: successful
completion of courses 200-202. The Staff
240B.
Representations of Finite Groups II.
*
Introduces modular representation theory of finite groups (over a
field of positive characteristic). Main topics are Grothendieck groups,
Brauer characters, Brauer character table, projective covers,
Brauer-Cartan triangle, relative projectivity, vertices, sources, Green
correspondence, Green's indecomposability theorem. Recommended
completion of courses 200-202 and 240A. The Staff
246.
Representations of Algebras.
*
Material includes associative algebras and their modules;
projective and injective modules; projective covers; injective hulls;
Krull-Schmidt Theorem; Cartan matrix; semisimple algebras and modules;
radical, simple algebras; symmetric algebras; quivers and their
representations; Morita Theory; and basic algebras. Prerequisite(s):
courses 200, 201, and 202. The Staff
248.
Symplectic Geometry.
S
Basic definitions. Darboux theorem. Basic examples: cotangent
bundles, Kahler manifolds and co-adjoint orbits. Normal form theorems.
Hamiltonian group actions, momentum maps. Reduction by symmetry groups.
Atiyah-Guillemin-Sternberg convexity. Introduction to Floer homological
methods. Relations with other geometries including contact, Poisson,
and Kahler. Prerequisite(s): course 204; course 280 is recommended as
preparation. The Staff
249A.
Mechanics I.
*
Covers symplectic geometry and classical Hamiltonian dynamics.
Some of the key subjects are the Darboux theorem, Poisson brackets,
Hamiltonian and Langrangian systems, the Legendre transformation,
variational principles, Hamilton-Jacobi theory, godesic equations, and
an introduction to Poisson geometry. Course 208 is recommended as
preparation. The Staff
249B.
Mechanics II.
*
Hamiltonian dynamics with symmetry. Key topics center around the
momentum map and the theory of reduction in both the symplectic and
Poisson context. Applications are taken from geometry, rigid body
dynamics, and continuum mechanics. Course 249A is recommended as
preparation. The Staff
249C.
Mechanics III.
*
Introduces students to active research topics tailored according
to the interests of the students. Possible subjects are complete
integrability and Kac-Moody Lie algebras; Smale's topological program
and bifurcation theory; KAM theory, stability and chaos; relativity;
quantization. Course 249B is recommended as preparation. Offered in
alternate academic years. The Staff
252.
Fluid Mechanics.
*
First covers a basic introduction to fluid dynamics equations and
then focuses on different aspects of the solutions to the Navier-Stokes
equations. Prerequisite(s): courses 106A and 106B are recommended.
Enrollment restricted to graduate students. The Staff
254.
Geometric Analysis.
*
Introduction to some basics in geometric analysis through the
discussions of two fundamental problems in geometry: the resolution of
the Yamabe problem and the study of harmonic maps. The analytic aspects
of these problems include Sobolev spaces, best constants in Sobolev
inequalities, and regularity and a priori estimates of systems of
elliptic PDE. Prerequisite(s): courses 208, 212, and 213 are
recommended as preparation. The Staff
256.
Algebraic Curves.
*
Introduction to compact Riemann surfaces and algebraic geometry
via an in-depth study of complex algebraic curves. Courses 200, 201,
202, 203, 204, and 207 are recommended as preparation. Enrollment
restricted to graduate mathematics and physics students. The Staff
260.
Combinatorics.
*
Combinatorial mathematics, including summation methods, binomial
coefficients, combinatorial sequences (Fibonacci, Stirling, Eulerian,
harmonic, Bernoulli numbers), generating functions and their uses,
Bernoulli processes and other topics in discrete probability. Oriented
toward problem solving applications. Applications to statistical
physics and computer science. The Staff
280.
Topics in Analysis.
*
The Staff
281.
Topics in Algebra.
F
The Staff
282.
Topics in Geometry.
*
The Staff
283.
Topics in Combinatorial Theory.
*
The Staff
284.
Topics in Dynamics.
*
The Staff
285.
Topics in Partial Differential Equations.
*
Topics such as derivation of the Navier-Stokes equations. Examples
of flows including water waves, vortex motion, and boundary layers.
Introductory functional analysis of the Navier-Stokes equation. The Staff
286.
Topics in Number Theory.
*
Topics in number theory, selected by instructor. Possibilities
include modular and automorphic forms, elliptic curves, algebraic
number theory, local fields, the trace formula. May also cover related
areas of arithmetic algebraic geometry, harmonic analysis, and
representation theory. Courses 200, 201, 202, and 205 are recommended
as preparation. The Staff
287.
Topics in Topology.
*
Topics in topology, selected by the instructor. Possibilities
include generalized (co)homology theory including K-theory, group
actions on manifolds, equivariant and orbifold cohomology theory. May
be repeated for credit. The Staff
292.
Seminar
(no credit).
F,W,S
A weekly seminar attended by faculty, graduate students, and
upper-division undergraduate students. All graduate students are
expected to attend. The Staff
296.
Special Student Seminar.
F,W,S
Students and staff studying in an area where there is no specific course offering at that time.
The Staff
297.
Independent Study.
F,W,S
Either study related to a course being taken or a totally independent study.
The Staff
298.
Master's Thesis Research.
F,W,S
The Staff
299.
Thesis Research.
F,W,S
The Staff
* Not offered in 2007-08
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