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MATH 19: Calculus

Introduction to differential calculus of functions of one variable. Topics: review of elementary functions including exponentials and logarithms, limits, rates of change, the derivative, and applications. Math 19, 20, and 21 cover the same material as Math 41 and 42, but in three quarters rather than two. Prerequisites: precalculus, including trigonometry, advanced algebra, and analysis of elementary functions.
Terms: Aut, Win, Sum | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

MATH 41: Calculus (accelerated)

Introduction to differential and integral calculus of functions of one variable. Topics: limits, rates of change, the derivative and applications, introduction to the definite integral and integration. Math 41 and 42 cover the same material as Math 19-20-21, but in two quarters rather than three. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions, including exponentials and logarithms.
Terms: Aut | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

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.
Terms: Aut | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)
Instructors: ; Berwick-Evans, D. (PI)

MATH 42: Calculus (Accelerated)

Continuation of 41. Methods of symbolic and numerical integration, applications of the definite integral, introduction to differential equations, infinite series. Prerequisite: 41 or equivalent.
Terms: Aut, Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

MATH 42A: Calculus ACE

Students attend MATH 42 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut, Win | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)

MATH 51: Linear Algebra and Differential Calculus of Several Variables

Geometry and algebra of vectors, systems of linear equations, matrices and linear transformations, diagonalization and eigenvectors, vector valued functions and functions of several variables, parametric curves, partial derivatives and gradients, the derivative as a matrix, chain rule in several variables, constrained and unconstrained optimization. Prerequisite: 21, or 42, or a score of 4 on the BC Advanced Placement exam or 5 on the AB Advanced Placement exam, or consent of instructor.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

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.
Terms: Aut, Win, Spr | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)

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.
Terms: Aut | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)
Instructors: ; Vasy, A. (PI)

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 and 42 or equivalents.
Terms: Aut, Win, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

MATH 53: Ordinary Differential Equations with Linear Algebra

Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of second-order equations to oscillations, matrix exponentials, Laplace transforms, stability of non-linear systems and phase plane analysis, numerical methods. Prerequisite: 51 and 42 or equivalents.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

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. (Math 113 offers a more theoretical treatment.) Prerequisites: MATH 51 and MATH 52 or 53.
Terms: Aut, Spr, Sum | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Ying, L. (PI); Zheng, T. (PI)

MATH 106: Functions of a Complex Variable

Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. (Math 116 offers a more theoretical treatment.) Prerequisite: 52.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Gu, Y. (PI)

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. (Math 104 offers a more application-oriented treatment.)
Terms: Aut, Win, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

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.
Terms: Aut, Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Lin, Y. (PI); Miller, J. (PI)

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. (Math 106 offers a less theoretical treatment.) Prerequisites: 52, and 115 or 171.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Kerckhoff, S. (PI)

MATH 120: Groups and Rings

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 109 but altered content and more theoretical orientation. Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM.
Terms: Aut, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit
Instructors: ; Bump, D. (PI); Vakil, R. (PI)

MATH 131P: Partial Differential Equations I

An introduction to PDE; particularly 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 who have taken MATH 171 should consider taking MATH 173 rather than 131p. Prerequisite: 53.
Terms: Aut, Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Ford, A. (PI); Ying, L. (PI)

MATH 136: Stochastic Processes (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.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Zheng, T. (PI)

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.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Ionel, E. (PI)

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. Prerequisite: students should be comfortable doing proofs.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Sommer, R. (PI)

MATH 163: The Greek Invention of Mathematics (CLASSICS 136)

(Formerly CLASSGEN 103.) How was mathematics invented? A survey of the main creative ideas of ancient Greek mathematics. Among the issues explored are the axiomatic system of Euclid's Elements, the origins of the calculus in Greek measurements of solids and surfaces, and Archimedes' creation of mathematical physics. We will provide proofs of ancient theorems, and also learn how such theorems are even known today thanks to the recovery of ancient manuscripts.
Terms: Aut | Units: 3-5 | UG Reqs: GER:DB-Hum | Grading: Letter or Credit/No Credit
Instructors: ; Netz, R. (PI)

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. Prerequisite: 51H or 115 or consent of the instructor. WIM
Terms: Aut, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

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.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Ford, A. (PI)

MATH 193: Polya Problem Solving Seminar

Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class. Open to anyone with an interest in mathematics.
Terms: Aut | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit
Instructors: ; Soundararajan, K. (PI)

MATH 197: Senior Honors Thesis

Honors math major working on senior honors thesis under an approved advisor carries out research and reading. Satisfactory written account of progress achieved during term must be submitted to advisor before term ends. May be repeated 3 times for a max of 9 units. Contact department student services specialist to enroll.
Terms: Aut, Win, Spr | Units: 1-6 | Repeatable for credit | Grading: Letter (ABCD/NP)

MATH 199: Independent Work

For math majors only. 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. Contact department student services specialist to enroll.
Terms: Aut, Win, Spr, Sum | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

MATH 210A: Modern Algebra I

Basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, introduction to representation theory. Prerequisite: 122 or equivalent.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Vakil, R. (PI)

MATH 215A: Complex Analysis, Geometry, and Topology

Analytic functions, complex integration, Cauchy's theorem, residue theorem, argument principle, conformal mappings, Riemann mapping theorem, Picard's theorem, elliptic functions, analytic continuation and Riemann surfaces.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ionel, E. (PI)

MATH 217A: Differential Geometry

Smooth manifolds and submanifolds, tensors and forms, Lie and exterior derivative, DeRham cohomology, distributions and the Frobenius theorem, vector bundles, connection theory, parallel transport and curvature, affine connections, geodesics and the exponential map, connections on the principal frame bundle. Prerequisite: 215C or equivalent.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Yang, T. (PI)

MATH 230A: Theory of Probability (STATS 310A)

Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116, MATH 171.
Terms: Aut | Units: 2-4 | Grading: Letter or Credit/No Credit
Instructors: ; Diaconis, P. (PI)

MATH 391: Research Seminar in Logic and the Foundations of Mathematics (PHIL 391)

Contemporary work. May be repeated a total of three times for credit. Math 391 students attend the logic colloquium in 380-381T.
Terms: Aut, Win, Spr | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Feferman, S. (PI)

MATH 51M: Introduction to MATLAB for Multivariable Mathematics

Corequisite: MATH 51.
Terms: Aut | Units: 1 | Grading: Letter or Credit/No Credit
Instructors: ; Montague, D. (PI)

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.
Terms: Aut | Units: 1 | Grading: Satisfactory/No Credit
Instructors: ; Bump, D. (PI)

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.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Mazzeo, R. (PI)

MATH 191: Research Project

Undergraduates pursue an individual or group research project under Math department faculty supervision. Math Department approval is required prior to enrolling. A maximum of 5 units of credit can be used towards the Math course requirement for the Math major. May be repeated for credit for up to 10 units per academic year.
Terms: Aut, Win, Spr | Units: 1-10 | Repeatable for credit | Grading: Satisfactory/No Credit
Instructors: ; Vakil, R. (PI)

MATH 196: Undergraduate Colloquium

Weekly lectures by different experts on topics in pure and applied mathematics that go beyond the standard curriculum. May be repeated for credit for up to 3 units. Does not count toward the math major or minor.
Terms: Aut, Win, Spr | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit

MATH 198: Practical Training

Only for students majoring in mathematics. Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students submit a concise report detailing work activities, problems worked on, and key results. May be repeated for credit up to 3 units. Prerequisite: qualified offer of employment and consent of department. Prior approval by Math Department is required; you must contact the Math Department's Student Services staff for instructions before being granted permission to enroll.
Terms: Aut, Win, Spr, Sum | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit
Instructors: ; Conrad, B. (PI)

MATH 205A: Real Analysis

Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ryzhik, L. (PI)

MATH 220: Partial Differential Equations of Applied Mathematics (CME 303)

First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: foundation in multivariable calculus and ordinary differential equations.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ryzhik, L. (PI)

MATH 248: Introduction to Ergodic Theory

Topics may include 1) subadditive and multiplicative ergodic theorems, 2) notions of mixing, weak mixing, spectral theory, 3) metric and topological entropy of dynamical systems, 4) measures of maximal entropy. Prerequisites: Solid background in "Measure and Integration" (Math 205A) and some functional analysis, including Riesz representation theorem and Hahn-Banach theorem (Math 205B).
Terms: Aut | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Mirzakhani, M. (PI)

MATH 285: Geometric Measure Theory

Hausdorff measures and dimensions, area and co-area formulas for Lipschitz maps, integral currents and flat chains, minimal surfaces and their singular sets.
Terms: Aut | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Simon, L. (PI)
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