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Stanford Bulletin, Archive 2008-09
Stanford Bulletin, Archive 2008-09
Undergraduate courses in Mathematics
The department offers two sequences of introductory courses in single variable calculus. MATH 41,42 present single variable calculus; differential calculus is covered in the first quarter, integral calculus in the second. MATH 19,20,21 cover the material in 41,42 in three quarters instead of two. There are two options for studying multivariable mathematics. MATH 51,52,53 cover differential and integral calculus in several variables, linear algebra, and ordinary differential equations. These topics are taught in an integrated fashion and emphasize application. MATH 51 covers differential calculus in several variables and introduces matrix theory and linear algebra, 52 covers integral calculus in several variables and vector analysis, 53 studies further topics in linear algebra and applies them to the study of ordinary differential equations. This sequence is recommended for incoming freshmen with 10 units of advanced placement credit. MATH 51H,52H,53H cover the same material as 51,52,53, but with more emphasis on theory and rigor. The introductory course in linear algebra is 103 or 113. There are no formal prerequisites for these courses, but appropriate mathematical maturity is expected. Much of the material in 103 is covered in the sequence 51,52,53.
MATH 15. Overview of Mathematics
Broad survey of mathematics; its nature and role in society. GER:DB-Math
3 units, not given this year
MATH 19. Calculus
The content of MATH 19, 20, 21 is the same as the sequence MATH 41, 42 described below, but covered in three quarters, rather than two. GER:DB-Math
3 units, Aut (Butscher, A), Win (Staff), Sum (Staff)
MATH 20. Calculus
Continuation of 19. Prerequisite: 19. GER:DB-Math
3 units, Win (Butscher, A), Spr (Staff)
MATH 21. Calculus
Continuation of 20. Prerequisite: 20. GER:DB-Math
4 units, Spr (Butscher, A)
MATH 41. Calculus
Introduction to differential and integral calculus of functions of one variable. Topics: review of elementary functions including exponentials and logarithms, rates of change, and the derivative. Introduction to the definite integral and integration. Prerequisites: algebra, trigonometry. GER:DB-Math
5 units, Aut (Lucianovic, M)
MATH 41A. Calculus ACE
Students attend MATH 41 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html. GER:DB-Math
6 units, Aut (Lucianovic, M)
MATH 42. Calculus
Continuation of 41. Methods of symbolic and numerical integration, applications of the definite integral, introduction to differential equations. Infinite series. Prerequisite: 41 or equivalent. GER:DB-Math
5 units, Aut (Staff), Win (Lucianovic, M)
MATH 42A. Calculus ACE
Students attend MATH 41 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html. GER:DB-Math
6 units, Aut (Staff), Win (Lucianovic, M)
MATH 51. Linear Algebra and Differential Calculus of Several Variables
Geometry and algebra of vectors, systems of linear equations, matrices, vector valued functions and functions of several variables, partial derivatives, gradients, chain rule in several variables, vector fields, optimization. Prerequisite: 21, 42, or a score of 4 on the BC Advanced Placement exam or 5 on the AB Advanced Placement exam, or consent of instructor. GER:DB-Math
5 units, Aut (Staff), Win (Staff), Spr (Staff), Sum (Staff)
MATH 51A. Linear Algebra and Differential Calculus of Several Variables, ACE
Students attend MATH 51 lectures with different recitation sessions: four hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html. GER:DB-Math
6 units, Aut (Staff), Win (Staff), Spr (Staff)
MATH 51H. Honors Multivariable Mathematics
For prospective Mathematics majors in the honors program and students from other areas of science or engineering who have a strong mathematics background. Three quarter sequence covers the material of 51, 52, 53, and additional advanced calculus and ordinary and partial differential equations. Unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know one-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on BC Advanced Placement exam, or consent of instructor. GER:DB-Math
5 units, Aut (Simon, L)
MATH 51M. Introduction to MATLAB for Multivariable Mathematics
Corequisite: MATH 51.
1 unit, Aut (Staff)
MATH 52. Integral Calculus of Several Variables
Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. Prerequisite: 51. GER:DB-Math
5 units, Aut (Tzou, L), Win (Kerckhoff, S), Spr (Brumfiel, G)
MATH 52H. Honors Multivariable Mathematics
Continuation of 51H. Prerequisite: 51H. GER:DB-Math
5 units, Win (Simon, L)
MATH 53. Ordinary Differential Equations with Linear Algebra
Linear ordinary differential equations, applications to oscillations, matrix methods including determinants, eigenvalues and eigenvectors, matrix exponentials, systems of linear differential equations with constant coefficients, stability of non-linear systems and phase plane analysis, numerical methods, Laplace transforms. Integrated with topics from linear algebra (103). Prerequisite: 51. GER:DB-Math
5 units, Aut (Lisi, S), Win (Iyer, G), Spr (Staff), Sum (Staff)
MATH 53H. Honors Multivariable Mathematics
Continuation of 52H. Prerequisite: 52H. GER:DB-Math
5 units, Spr (Brendle, S)
MATH 70SI. The Game of Go: Strategy, Theory, and History
Strategy and mathematical theories of the game of Go, with guest appearance by a professional Go player.
1 unit, Win (Bump, D)
MATH 80Q. Capillary Surfaces: Explored and Unexplored Territory
(S,Sem) Stanford Introductory Seminar. Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles. Interested students may participate in ongoing investigations with affinity between mathematics and physics.
3 units, Win (Finn, R)
MATH 87Q. Mathematics of Knots, Braids, Links, and Tangles
(S,Sem) Stanford Introductory Seminar. Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.
3 units, Spr (Wieczorek, W)
MATH 88Q. The Mathematics of the Rubik's Cube
(S,Sem) Stanford Introductory Seminar. Preference to sophomores. Group theory through topics that can be illustrated with the Rubik's cube: subgroups, homomorphisms and quotient groups, the symmetric and alternating groups, conjugation, commutators, and Sylow subgroups.
3 units, Win (Bump, D)
MATH 100. Mathematics for Elementary School Teachers
Mathematics and pedagogical strategies. Core mathematical content in grades K-6, classroom presentation, how to handle student errors, and mathematical issues that come up during instruction.
4 units, Spr (Milgram, R)
MATH 104. Applied Matrix Theory
Linear algebra for applications in science and engineering: orthogonality, projections, the four fundamental subspaces of a matrix, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems. Prerequisites: MATH 51 and MATH 52 or 53. GER:DB-Math
3 units, Aut (Demanet, L), Win (Liu, T)
MATH 106. Functions of a Complex Variable
Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. Prerequisite: 52. GER:DB-Math
3 units, Win (Nedelec, L), Sum (Brumfiel, G)
MATH 108. Introduction to Combinatorics and Its Applications
Topics: graphs, trees (Cayley's Theorem, application to phylogony), eigenvalues, basic enumeration (permutations, Stirling and Bell numbers), recurrences, generating functions, basic asymptotics. Prerequisites: 51 or 103 or equivalent. GER:DB-Math
3 units, Spr (Kahle, M)
MATH 109. Applied Group Theory
Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. GER:DB-Math, WIM
3 units, Aut (Ionel, E)
MATH 110. Applied Number Theory and Field Theory
Number theory and its applications to modern cryptography. Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. GER:DB-Math
3 units, Spr (Lucianovic, M)
MATH 111. Computational Commutative Algebra
Introduction to the theory of commutative rings, ideals, and modules. Systems of polynomial equations in several variables from the algorithmic viewpoint. Groebner bases, Buchberger's algorithm, elimination theory. Applications to algebraic geometry and to geometric problems. GER:DB-Math
3 units, not given this year
MATH 113. Linear Algebra and Matrix Theory
Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization. GER:DB-Math
3 units, Aut (Staff), Win (Cohen, R), Spr (Tzou, L)
MATH 114. Linear Algebra and Matrix Theory II
Advanced topics in linear algebra such as: invariant subspaces; canonical forms of matrices; minimal polynomials and elementary divisors; vector spaces over arbitrary fields; inner products; Jordan normal forms; Hermitian and unitary matrices; multilinear algebra; and applications. Prerequisite: 51H or 113. GER:DB-Math
3 units, Spr (Katznelson, Y)
MATH 115. Functions of a Real Variable
The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 51. GER:DB-Math
3 units, Aut (Toussaint, A), Win (Katznelson, Y), Sum (Brumfiel, G)
MATH 116. Complex Analysis
Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. Prerequisites: 52, and 115 or 171. GER:DB-Math
3 units, Win (Li, J)
MATH 118. Mathematics of Computation
Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements. Prerequisites: MATH 51 and 53. GER:DB-Math
3 units, Win (Demanet, L)
MATH 120. Modern Algebra
Basic structures in algebra: groups, rings, and fields. Elements of group theory: permutation groups, finite Abelian groups, p-groups, Sylow theorems. Polynomial rings, principal ideal domains, unique factorization domains. GER:DB-Math, WIM
3 units, Aut (Vakil, R), Spr (Soundararajan, K)
MATH 121. Modern Algebra II
Continuation of 120. Fields of fractions. Solvable and simple groups. Elements of field theory and Galois theory. Prerequisite: 120. GER:DB-Math
3 units, Win (Staff)
MATH 131M. Partial Differential Equations I
More theoretical version of MATH 131P; suitable for Mathematics majors and mathematically inclined Physics majors . Topics include first order equations, physical and mathematical sources of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, and Fourier series. Introduction to PDE in multiple space dimensions. Emphasis is on a mathematically rigorous treatment of the subject. Students may not take both 131M and 131P. Prerequisite: 53. GER:DB-Math
3 units, Win (Schoen, R)
MATH 131P. Partial Differential Equations I
A more applied version of MATH 131M; suitable for non-Math majors. Topics include physical examples of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, Fourier series, Harmonic functions, Bessel functions, spherical harmonics. Students may not take both 131M and 131P. Prerequisite: 53. GER:DB-Math
3 units, Aut (Iyer, G)
MATH 132. Partial Differential Equations II
Laplace's equation and properties of harmonic functions. Green's functions. Distributions and Fourier transforms. Eigenvalue problems and generalized Fourier series. Numerical solutions. Prerequisite: 131M or 131P (formerly 131). GER:DB-Math
3 units, Spr (Nedelec, L)
MATH 135. Nonlinear Dynamics and Chaos
Topics: one- and two-dimensional flows, bifurcations, phase plane analysis, limit cycles and their bifurcations. Lorenz equations, fractals and strange attractors. Prerequisite: 51 and 53 or equivalent. GER:DB-Math
3 units, not given this year
MATH 136. Stochastic Processes
(Same as STATS 219.) Introduction to measure theory, Lp spaces and Hilbert spaces. Random variables, expectation, conditional expectation, conditional distribution. Uniform integrability, almost sure and Lp convergence. Stochastic processes: definition, stationarity, sample path continuity. Examples: random walk, Markov chains, Gaussian processes, Poisson processes, Martingales. Construction and basic properties of Brownian motion. Prerequisite: STATS 116 or MATH 151 or equivalent. Recommended: MATH 115 or equivalent. GER:DB-Math
3 units, Aut (Ross, K)
MATH 137. Mathematical Methods of Classical Mechanics
Newtonian mechanics. Lagrangian formalism. E. Noether's theorem. Oscillations. Rigid bodies. Introduction to symplectic geometry. Hamiltonian formalism. Legendre transform. Variational principles. Geometric optics. Introduction to the theory of integrable systems. Prerequisites: 51, 52, 53, or 51H, 52H, 53H. GER:DB-Math
3 units, Win (Eliashberg, Y)
MATH 138. Celestial Mechanics
Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171. GER:DB-Math
3 units, not given this year
MATH 143. Differential Geometry
Geometry of curves and surfaces in three-space and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces. GER:DB-Math
3 units, Win (Nedelec, L)
MATH 145. Algebraic Geometry
Real algebraic curves, Hilbert's nullstellensatz, complex affine and projective curves, Bezout's theorem, the degree/genus formula, Riemann surfaces, Riemann-Roch theorem. Prerequisites: 106 or 116, and 109 or 120. Recommended: familiarity with surfaces equivalent to 143, 146, 147, or 148. GER:DB-Math
3 units, Spr (Li, J)
MATH 146. Analysis on Manifolds
Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 52 or 52H. GER:DB-Math
3 units, Win (Ionel, E)
MATH 147. Differential Topology
Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171. GER:DB-Math
3 units, Spr (Staff), alternate years, not given this year
MATH 148. Algebraic Topology
Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120. GER:DB-Math
3 units, Spr (Diaconis, P), alternate years, not given next year
MATH 151. Introduction to Probability Theory
Counting; axioms of probability; conditioning and independence; expectation and variance; discrete and continuous random variables and distributions; joint distributions and dependence; central limit theorem and laws of large numbers. Prerequisite: 52 or consent of instructor. GER:DB-Math
3 units, Win (Carlsson, G)
MATH 152. Elementary Theory of Numbers
Euclid's algorithm, fundamental theorems on divisibility; prime numbers, congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; Lagrange's theorem and its applications; quadratic residues; introduction to the theory of binary quadratic forms. GER:DB-Math
3 units, Aut (Soundararajan, K)
MATH 154. Algebraic Number Theory
Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations; introduction to elliptic curves. Prerequisites: 120, 121. GER:DB-Math
3 units, Win (Conrad, B), alternate years, not given next year
MATH 155. Analytic Number Theory
Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity. GER:DB-Math
3 units, alternate years, not given this year
MATH 156. Group Representations
Group representations and their characters, classification of permutation group representations using partitions and Young tableaux, group actions on sets and the Burnside ring, and spherical space forms. Applications to geometric group actions and to combinatorics. Prerequisites: linear algebra (51 and 53, or 103 or 113) and group theory (109 or 120). GER:DB-Math
3 units, Spr (Staff)
MATH 161. Set Theory
Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents. Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. GER:DB-Math
3 units, Spr (Kahle, M)
MATH 162. Philosophy of Mathematics
(Same as PHIL 162, PHIL 262.) (Graduate students register for PHIL 262.) 20th-century approaches to the foundations and philosophy of mathematics. The background in mathematics, set theory, and logic. Schools and programs of logicism, predicativism, platonism, formalism, and constructivism. Readings from leading thinkers. Prerequisite: PHIL151 or consent of instructor.
4 units, not given this year
MATH 171. Fundamental Concepts of Analysis
Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisites: 51 and 52, or 51H and 52H. WIM GER:DB-Math, WIM
3 units, Aut (Ryzhik, L), Spr (Vasy, A)
MATH 172. Lebesgue Integration and Fourier Analysis
Similar to 205A, but for undergraduate Math majors and graduate students in other disciplines. Topics include Lebesgue measure on Euclidean space, Lebesgue integration, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor. GER:DB-Math
3 units, Spr (Iyer, G)
MATH 174A. Topics in Analysis and Differential Equations with Applications
For students planning graduate work in mathematics or physics, and for honors math majors and other students at ease with rigorous proofs and qualitative discussion. Topics may include: geometric theory of ODE's with applications to dynamics; mathematical foundations of classical mechanics including variational principles, Lagrangian and Hamiltonian formalisms, theory of integrable systems; theorems of existence and uniqueness; Sturm-Liouville theory. Prerequisite: 53H or 171, or consent of instructor. GER:DB-Math
3 units, not given this year
MATH 174B. Honors Analysis
Continuation of 174A. Topics may include: introduction to PDEs including transport equations, Laplace, wave, and heat equations; techniques of solution including separation of variables and Green's functions; Fourier series and integrals; introduction to the theory of distributions; mathematical foundations of quantum mechanics. Prerequisite: 174A. GER:DB-Math
3 units, not given this year
MATH 175. Elementary Functional Analysis
Linear operators on Hilbert space. Spectral theory of compact operators; applications to integral equations. Elements of Banach space theory. Prerequisite: 115 or 171. GER:DB-Math
3 units, Spr (Simon, L)
MATH 180. Introduction to Financial Mathematics
Financial derivatives: contracts and options. Hedging and risk management. Arbitrage, interest rate, and discounted value. Geometric random walk and Brownian motion as models of risky assets. 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. Optimal stopping and American options. Prerequisite: 53. Corequisites: 131, 151 or STATS 116. GER:DB-Math
3 units, Aut (Toussaint, A)
MATH 197. Senior Honors Thesis
1-6 units, Aut (Staff), Win (Staff), Spr (Staff)
MATH 199. Independent Work
Undergraduates pursue a reading program; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for math majors. Approval of Undergraduate Affairs Committee is required to use credit for honors majors area requirement.
1-3 units, Aut (Staff), Win (Staff), Spr (Staff)
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