BIO 141:
Biostatistics (STATS 141)
Introductory statistical methods for biological data: describing data (numerical and graphical summaries); introduction to probability; and statistical inference (hypothesis tests and confidence intervals). Intermediate statistical methods: comparing groups (analysis of variance); analyzing associations (linear and logistic regression); and methods for categorical data (contingency tables and odds ratio). Course content integrated with statistical computing in R.
Terms: Aut

Units: 35

UG Reqs: GER:DBMath, WAYAQR

Grading: Letter or Credit/No Credit
BIOHOPK 174H:
Experimental Design and Probability (BIOHOPK 274H)
(Graduate students register for 274H.) Variability is an integral part of biology. Introduction to probability and its use in designing experiments to address biological problems. Focus is on analysis of variance, when and how to use it, why it works, and how to interpret the results. Design of complex, but practical, asymmetrical experiments and environmental impact studies, and regression and analysis of covariance. Computerbased data analysis. Prerequisite: Biology core or consent of instructor.
Terms: Win, Spr

Units: 3

UG Reqs: GER: DBNatSci, GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
CME 100:
Vector Calculus for Engineers (ENGR 154)
Computation and visualization using MATLAB. Differential vector calculus: analytic geometry in space, functions of several variables, partial derivatives, gradient, unconstrained maxima and minima, Lagrange multipliers. Introduction to linear algebra: matrix operations, systems of algebraic equations, methods of solution and applications. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green¿s, divergence, and Stokes¿ theorems. Examples and applications drawn from various engineering fields. Prerequisites: 10 units of AP credit (Calc BC with 5, or Calc AB with 5 or placing out of the single variable math placement test: https://exploredegreesnextyear.stanford.edu/undergraduatedegreesandprograms/#aptextt), or Math 1921.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
CME 100A:
Vector Calculus for Engineers, ACE
Students attend CME100/ENGR154 lectures with additional recitation sessions; two to four hours per week, emphasizing engineering mathematical applications and collaboration methods. Enrollment by department permission only. Prerequisite: must be enrolled in the regular CME10001 or 02. Application at: https://engineering.stanford.edu/students/programs/engineeringdiversityprograms/additionalcalculusengineers
Terms: Aut, Win, Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
CME 102:
Ordinary Differential Equations for Engineers (ENGR 155A)
Analytical and numerical methods for solving ordinary differential equations arising in engineering applications: Solution of initial and boundary value problems, series solutions, Laplace transforms, and nonlinear equations; numerical methods for solving ordinary differential equations, accuracy of numerical methods, linear stability theory, finite differences. Introduction to MATLAB programming as a basic tool kit for computations. Problems from various engineering fields. Prerequisite: 10 units of AP credit (Calc BC with 5, or Calc AB with 5 or placing out of the single variable math placement test: https://exploredegreesnextyear.stanford.edu/undergraduatedegreesandprograms/#aptextt),), or Math 1921. Recommended: CME100.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
CME 102A:
Ordinary Differential Equations for Engineers, ACE
Students attend CME102/ENGR155A lectures with additional recitation sessions; two to four hours per week, emphasizing engineering mathematical applications and collaboration methods. Prerequisite: students must be enrolled in the regular section (CME102) prior to submitting application at:nhttps://engineering.stanford.edu/students/programs/engineeringdiversityprograms/additionalcalculusengineers
Terms: Aut, Win, Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
CME 103:
Introduction to Matrix Methods (EE 103)
Introduction to applied linear algebra with emphasis on applications. Vectors, norm, and angle; linear independence and orthonormal sets; applications to document analysis. Clustering and the kmeans algorithm. Matrices, left and right inverses, QR factorization. Leastsquares and model fitting, regularization and crossvalidation. Constrained and nonlinear leastsquares. Applications include timeseries prediction, tomography, optimal control, and portfolio optimization. Undergraduate students should enroll for 5 units, and graduate students should enroll for 3 units. Prerequisites:MATH 51 or CME 100, and basic knowledge of computing (CS 106A is more than enough, and can be taken concurrently). EE103/CME103 and Math 104 cover complementary topics in applied linear algebra. The focus of EE103 is on a few linear algebra concepts, and many applications; the focus of Math 104 is on algorithms and concepts.
Terms: Aut

Units: 35

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
CME 104:
Linear Algebra and Partial Differential Equations for Engineers (ENGR 155B)
Linear algebra: matrix operations, systems of algebraic equations, Gaussian elimination, undetermined and overdetermined systems, coupled systems of ordinary differential equations, eigensystem analysis, normal modes. Fourier series with applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations. Numerical methods for solution of partial differential equations: iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Examples and applications from various engineering fields. Prerequisite: CME 102/ENGR 155A.
Terms: Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
CME 104A:
Linear Algebra and Partial Differential Equations for Engineers, ACE
Students attend CME104/ENGR155B lectures with additional recitation sessions; two to four hours per week, emphasizing engineering mathematical applications and collaboration methods. Prerequisite: students must be enrolled in the regular section (CME104) prior to submitting application at: https://engineering.stanford.edu/students/programs/engineeringdiversityprograms/additionalcalculusengineers
Terms: Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
CME 106:
Introduction to Probability and Statistics for Engineers (ENGR 155C)
Probability: random variables, independence, and conditional probability; discrete and continuous distributions, moments, distributions of several random variables. Topics in mathematical statistics: random sampling, point estimation, confidence intervals, hypothesis testing, nonparametric tests, regression and correlation analyses; applications in engineering, industrial manufacturing, medicine, biology, and other fields. Prerequisite: CME 100/ENGR154 or MATH 51 or 52.
Terms: Win, Sum

Units: 4

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
CS 103:
Mathematical Foundations of Computing
What are the theoretical limits of computing power? What problems can be solved with computers? Which ones cannot? And how can we reason about the answers to these questions with mathematical certainty? This course explores the answers to these questions and serves as an introduction to discrete mathematics, computability theory, and complexity theory. At the completion of the course, students will feel comfortable writing mathematical proofs, reasoning about discrete structures, reading and writing statements in firstorder logic, and working with mathematical models of computing devices. Throughout the course, students will gain exposure to some of the most exciting mathematical and philosophical ideas of the late nineteenth and twentieth centuries. Specific topics covered include formal mathematical proofwriting, propositional and firstorder logic, set theory, binary relations, functions (injections, surjections, and bijections), cardinality, basic graph theory, the pigeonhole principle, mathematical induction, finite automata, regular expressions, the MyhillNerode theorem, contextfree grammars, Turing machines, decidable and recognizable languages, selfreference and undecidability, verifiers, and the P versus NP question. Students with significant proofwriting experience are encouraged to instead take CS154. Students interested in extra practice and support with the course are encouraged to concurrently enroll in CS103A. Prerequisite: CS106B or equivalent. CS106B may be taken concurrently with CS103.
Terms: Aut, Win, Spr

Units: 35

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
ECON 50:
Economic Analysis I
Individual consumer and firm behavior under perfect competition. The role of markets and prices in a decentralized economy. Monopoly in partial equilibrium. Economic tools developed from multivariable calculus using partial differentiation and techniques for constrained and unconstrained optimization. Prerequisites: Econ 1 or 1V, and Math 51 or Math 51A or CME 100 or CME 100A.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYFR, WAYSI

Grading: Letter or Credit/No Credit
ECON 102A:
Introduction to Statistical Methods (Postcalculus) for Social Scientists
Probabilistic modeling and statistical techniques relevant for economics. Concepts include: probability trees, conditional probability, random variables, discrete and continuous distributions, correlation, central limit theorems, point estimation, hypothesis testing and confidence intervals for both one and two populations. Prerequisite: MATH 20 or equivalent.
Terms: Aut, Win

Units: 5

UG Reqs: GER:DBMath, WAYAQR, WAYSI

Grading: Letter or Credit/No Credit
EE 103:
Introduction to Matrix Methods (CME 103)
Introduction to applied linear algebra with emphasis on applications. Vectors, norm, and angle; linear independence and orthonormal sets; applications to document analysis. Clustering and the kmeans algorithm. Matrices, left and right inverses, QR factorization. Leastsquares and model fitting, regularization and crossvalidation. Constrained and nonlinear leastsquares. Applications include timeseries prediction, tomography, optimal control, and portfolio optimization. Undergraduate students should enroll for 5 units, and graduate students should enroll for 3 units. Prerequisites:MATH 51 or CME 100, and basic knowledge of computing (CS 106A is more than enough, and can be taken concurrently). EE103/CME103 and Math 104 cover complementary topics in applied linear algebra. The focus of EE103 is on a few linear algebra concepts, and many applications; the focus of Math 104 is on algorithms and concepts.
Terms: Aut

Units: 35

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
ENGR 154:
Vector Calculus for Engineers (CME 100)
Computation and visualization using MATLAB. Differential vector calculus: analytic geometry in space, functions of several variables, partial derivatives, gradient, unconstrained maxima and minima, Lagrange multipliers. Introduction to linear algebra: matrix operations, systems of algebraic equations, methods of solution and applications. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green¿s, divergence, and Stokes¿ theorems. Examples and applications drawn from various engineering fields. Prerequisites: 10 units of AP credit (Calc BC with 5, or Calc AB with 5 or placing out of the single variable math placement test: https://exploredegreesnextyear.stanford.edu/undergraduatedegreesandprograms/#aptextt), or Math 1921.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
ENGR 155A:
Ordinary Differential Equations for Engineers (CME 102)
Analytical and numerical methods for solving ordinary differential equations arising in engineering applications: Solution of initial and boundary value problems, series solutions, Laplace transforms, and nonlinear equations; numerical methods for solving ordinary differential equations, accuracy of numerical methods, linear stability theory, finite differences. Introduction to MATLAB programming as a basic tool kit for computations. Problems from various engineering fields. Prerequisite: 10 units of AP credit (Calc BC with 5, or Calc AB with 5 or placing out of the single variable math placement test: https://exploredegreesnextyear.stanford.edu/undergraduatedegreesandprograms/#aptextt),), or Math 1921. Recommended: CME100.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
ENGR 155B:
Linear Algebra and Partial Differential Equations for Engineers (CME 104)
Linear algebra: matrix operations, systems of algebraic equations, Gaussian elimination, undetermined and overdetermined systems, coupled systems of ordinary differential equations, eigensystem analysis, normal modes. Fourier series with applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations. Numerical methods for solution of partial differential equations: iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Examples and applications from various engineering fields. Prerequisite: CME 102/ENGR 155A.
Terms: Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
ENGR 155C:
Introduction to Probability and Statistics for Engineers (CME 106)
Probability: random variables, independence, and conditional probability; discrete and continuous distributions, moments, distributions of several random variables. Topics in mathematical statistics: random sampling, point estimation, confidence intervals, hypothesis testing, nonparametric tests, regression and correlation analyses; applications in engineering, industrial manufacturing, medicine, biology, and other fields. Prerequisite: CME 100/ENGR154 or MATH 51 or 52.
Terms: Win, Sum

Units: 4

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
HUMBIO 88:
Introduction to Statistics for the Health Sciences
Students will learn the statistical tools used to describe and analyze data in the fields of medicine and epidemiology. This very applied course will rely on current research questions and publicly available data. Students will gain proficiency with Stata to do basic analyses of healthrelated data, including linear and logistic regression, and will become sophisticated consumers of healthrelated statistical results.
Terms: Win

Units: 4

UG Reqs: GER:DBMath, WAYAQR

Grading: Letter (ABCD/NP)
HUMBIO 89:
Introduction to Health Sciences Statistics
This course aims to provide a firm grounding in the foundations of probability and statistics, with a focus on analyzing data from the health sciences. Students will learn how to read, interpret, and critically evaluate the statistics in medical and biological studies. The course also prepares students to be able to analyze their own data, guiding them on how to choose the correct statistical test, avoid common statistical pitfalls, and perform basic functions in R deducer. Cardinal Course certified by the Haas Center.
Terms: Aut, Win, Spr

Units: 3

UG Reqs: GER:DBMath, WAYAQR

Grading: Letter or Credit/No Credit
MATH 19:
Calculus
Introduction to differential calculus of functions of one variable. Review of elementary functions (including exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions (including exponentials and logarithms). You must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Terms: Aut, Win

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 20:
Calculus
The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus, and the Mean Value Theorem for integrals. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initialvalue problems, exponential and logistic models, direction fields, and parametric curves. Prerequisite: Math 19 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Terms: Aut, Win, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 21:
Calculus
Review of limit rules. Sequences, functions, limits at infinity, and comparison of growth of functions. Review of integration rules, integrating rational functions, and improper integrals. Infinite series, special examples, convergence and divergence tests (limit comparison and alternating series tests). Power series and interval of convergence, Taylor polynomials, Taylor series and applications. Prerequisite: Math 20 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Terms: Aut, Win, Spr

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 51:
Linear Algebra and Differential Calculus of Several Variables
Geometry and algebra of vectors, matrices and linear transformations, eigenvalues of symmetric matrices, vectorvalued functions and functions of several variables, partial derivatives and gradients, derivative as a matrix, chain rule in several variables, critical points and Hessian, leastsquares, , constrained and unconstrained optimization in several variables, Lagrange multipliers. Prerequisite: 21, 42, or the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

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: three hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see https://web.stanford.edu/dept/soe/osa/ace.fb
Terms: Aut, Win, Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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 or equivalents.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

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 secondorder equations to oscillations, matrix exponentials, Laplace transforms, stability of nonlinear systems and phase plane analysis, numerical methods. Prerequisite: 51 or equivalents.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 61CM:
Modern Mathematics: Continuous Methods
This is the first part of a theoretical (i.e., proofbased) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space, inverse and implicit function theorems, and many examples. The linear algebra content is covered jointly with Math 61DM. Students should know 1variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BClevel Advanced Placement calculus exam, or consent of the instructor.
Terms: Aut

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 62CM:
Modern Mathematics: Continuous Methods
A continuation of themes from Math 61CM, centered around: manifolds, multivariable integration, and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space and an introduction to general manifolds (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisite: Math 61CM.
Terms: Win

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter (ABCD/NP)
MATH 63CM:
Modern Mathematics: Continuous Methods
A proofbased course on ordinary differential equations, continuing themes from Math 61CM and Math 62CM. Topics include linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations with some applications to manifolds, behavior of solutions near an equilibrium point, and SturmLiouville theory. Prerequisites: Math 61CM and Math 62CM.
Terms: Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter (ABCD/NP)
MATH 104:
Applied Matrix Theory
Linear algebra for applications in science and engineering: orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, leastsquares, the condition number of a matrix, algorithms for solving linear systems. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and EE 103/CME 103 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of EE 103 is on a few linear algebra concepts, and many applications. Prerequisites: MATH 51 and programming experience on par with CS 106.
Terms: Aut, Win, Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 106:
Functions of a Complex Variable
Complex numbers, analytic functions, CauchyRiemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. (Math 116 offers a more theoretical treatment.) Prerequisite: 52.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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 equivalent.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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. Honors math majors and students who intend to do graduate work in mathematics should take 120. WIM.
Terms: Aut

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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. WIM.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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 applicationoriented treatment.)
Terms: Aut, Win, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

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: 21.
Terms: Aut, Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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:DBMath

Grading: Letter or Credit/No Credit
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 initialvalue and boundaryvalue ODE, finite and spectral elements. Prerequisites: MATH 51 and 53.
Terms: not given this year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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 nonPID. Unique factorization domains. WIM.
Terms: Aut, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 121:
Galois Theory
Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite: Math 120 and (also recommended) 113.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 131P:
Partial Differential Equations
An introduction to PDE; particularly suitable for nonMath 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: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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. http://statweb.stanford.edu/~adembo/math136/
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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 61CM, 62CM, 63CM.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 138:
Celestial Mechanics
Mathematically rigorous introduction to the classical Nbody 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; HamiltonJacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171.
Terms: not given this year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 143:
Differential Geometry
Geometry of curves and surfaces in threespace and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 145:
Algebraic Geometry
An introduction to the methods and concepts of algebraic geometry. The point of view and content will vary over time, but include: affine varieties, Hilbert basis theorem and Nullstellensatz, projective varieties, algebraic curves. Required: 120. Strongly recommended: additional mathematical maturity via further basic background with fields, pointset topology, or manifolds.
Terms: Aut, alternate years, not given next year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 146:
Analysis on Manifolds
Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.
Terms: Aut

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 147:
Differential Topology
Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, BorsukUlam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 148:
Algebraic Topology
Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: Win, alternate years, not given next year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 154:
Algebraic Number Theory
Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations. Prerequisites: 120 and 121, especially modules over principal ideal domains and Galois theory of finite fields.
Terms: Spr, alternate years, not given next year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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.
Terms: not given this year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 161:
Set Theory
Informal and axiomatic set theory: sets, relations, functions, and settheoretical operations. The ZermeloFraenkel axiom system and the special role of the axiom of choice and its various equivalents. Wellorderings 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: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 162:
Philosophy of Mathematics (PHIL 162)
Mathematics is a very peculiar human activity. It delivers a type of knowledge that is particularly stable, often conceived as a priori and necessary. Moreover, this knowledge is about abstract entities, which seem to have no connection to us, spatiotemporal creatures, and yet it plays a crucial role in our scientific endeavors. Many philosophical questions emerge naturally: What is the nature of mathematical objects? How can we learn anything about them? Where does the stability of mathematics comes from? What is the significance of results showing the limits of such knowledge, such as Gödel's incompleteness theorem? The first part of the course will survey traditional approaches to philosophy of mathematics ("the big Isms") and consider the viability of their answers to some of the previous questions: logicism, intuitionism, Hilbert's program, empiricism, fictionalism, and structuralism. The second part will focus on philosophical issues emerging from the actual practice of mathematics. We will tackle questions such as: Why do mathematicians reprove the same theorems? What is the role of visualization in mathematics? How can mathematical knowledge be effective in natural science? To conclude, we will explore the aesthetic dimension of mathematics, focusing on mathematical beauty. Prerequisite: PHIL150 or consent of instructor.
Terms: not given this year

Units: 4

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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: 61CM or 61DM or 115 or consent of the instructor. WIM
Terms: Aut, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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 HardyLittlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

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: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MCS 100:
Mathematics of Sports (STATS 50)
The use of mathematics, statistics, and probability in the analysis of sports performance, sports records, and strategy. Topics include mathematical analysis of the physics of sports and the determinations of optimal strategies. New diagnostic statistics and strategies for each sport. Corequisite: STATS 60, 110 or 116.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
PHIL 49:
Survey of Formal Methods
Survey of important formal methods used in philosophy. The course covers the basics of propositional and elementary predicate logic, probability and decision theory, game theory, and statistics, highlighting philosophical issues and applications. Specific topics include the languages of propositional and predicate logic and their interpretations, rationality arguments for the probability axioms, Nash equilibrium and dominance reasoning, and the meaning of statistical significance tests. Assessment is through a combination of problems designed to solidify competence with the mathematical tools and shortanswer questions designed to test conceptual understanding.
Terms: Spr

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter (ABCD/NP)
PHIL 150:
Mathematical Logic (PHIL 250)
An introduction to the concepts and techniques used in mathematical logic, focusing on propositional, modal, and predicate logic. Highlights connections with philosophy, mathematics, computer science, linguistics, and neighboring fields.
Terms: Aut

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
PHIL 150E:
Logic in Action: A New Introduction to Logic
A new introduction to logic, covering propositional, modal, and firstorder logic, with special attention to major applications in describing information and informationdriven action. Highlights connections with philosophy, mathematics, computer science, linguistics, and neighboring fields. Based on the open source course 'Logic in Action,' available online at http://www.logicinaction.org/.nFulfills the undergraduate philosophy logic requirement.
Terms: not given this year

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
PHIL 151:
Metalogic (PHIL 251)
(Formerly 160A.) The syntax and semantics of sentential and firstorder logic. Concepts of model theory. Gödel's completeness theorem and its consequences: the LöwenheimSkolem theorem and the compactness theorem. Prerequisite: 150 or consent of instructor.
Terms: Win

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
PHIL 151A:
Recursion Theory (PHIL 251A)
Computable functions, Turing degrees, generalized computability and definability. "What does it mean for a function from the natural numbers to themselves to be computable?" and "How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". Theory of relative computability, reducibility notions and degree structures. Prerequisite is PHIL 150, or PHIL 151 or CS 103.
Terms: not given this year

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
PHIL 152:
Computability and Logic (PHIL 252)
Approaches to effective computation: recursive functions, register machines, and Turing machines. Proof of their equivalence, discussion of Church's thesis. Elementary recursion theory. These techniques used to prove Gödel's incompleteness theorem for arithmetic, whose technical and philosophical repercussions are surveyed. Prerequisite: 151.
Terms: Spr

Units: 4

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
PHIL 154:
Modal Logic (PHIL 254)
(Graduate students register for 254.) Syntax and semantics of modal logic and its basic theory: including expressive power, axiomatic completeness, correspondence, and complexity. Applications to topics in philosophy, computer science, mathematics, linguistics, and game theory. Prerequisite: 150 or preferably 151.
Terms: Spr

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
PHIL 162:
Philosophy of Mathematics (MATH 162)
Mathematics is a very peculiar human activity. It delivers a type of knowledge that is particularly stable, often conceived as a priori and necessary. Moreover, this knowledge is about abstract entities, which seem to have no connection to us, spatiotemporal creatures, and yet it plays a crucial role in our scientific endeavors. Many philosophical questions emerge naturally: What is the nature of mathematical objects? How can we learn anything about them? Where does the stability of mathematics comes from? What is the significance of results showing the limits of such knowledge, such as Gödel's incompleteness theorem? The first part of the course will survey traditional approaches to philosophy of mathematics ("the big Isms") and consider the viability of their answers to some of the previous questions: logicism, intuitionism, Hilbert's program, empiricism, fictionalism, and structuralism. The second part will focus on philosophical issues emerging from the actual practice of mathematics. We will tackle questions such as: Why do mathematicians reprove the same theorems? What is the role of visualization in mathematics? How can mathematical knowledge be effective in natural science? To conclude, we will explore the aesthetic dimension of mathematics, focusing on mathematical beauty. Prerequisite: PHIL150 or consent of instructor.
Terms: not given this year

Units: 4

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
PHIL 166:
Probability: Ten Great Ideas About Chance (PHIL 266, STATS 167, STATS 267)
Foundational approaches to thinking about chance in matters such as gambling, the law, and everyday affairs. Topics include: chance and decisions; the mathematics of chance; frequencies, symmetry, and chance; Bayes great idea; chance and psychology; misuses of chance; and harnessing chance. Emphasis is on the philosophical underpinnings and problems. Prerequisite: exposure to probability or a first course in statistics at the level of STATS 60 or 116.
Terms: not given this year

Units: 4

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
PSYCH 10:
Introduction to Statistical Methods: Precalculus (STATS 60, STATS 160)
Techniques for organizing data, computing, and interpreting measures of central tendency, variability, and association. Estimation, confidence intervals, tests of hypotheses, ttests, correlation, and regression. Possible topics: analysis of variance and chisquare tests, computer statistical packages.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
STATS 50:
Mathematics of Sports (MCS 100)
The use of mathematics, statistics, and probability in the analysis of sports performance, sports records, and strategy. Topics include mathematical analysis of the physics of sports and the determinations of optimal strategies. New diagnostic statistics and strategies for each sport. Corequisite: STATS 60, 110 or 116.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
STATS 60:
Introduction to Statistical Methods: Precalculus (PSYCH 10, STATS 160)
Techniques for organizing data, computing, and interpreting measures of central tendency, variability, and association. Estimation, confidence intervals, tests of hypotheses, ttests, correlation, and regression. Possible topics: analysis of variance and chisquare tests, computer statistical packages.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
STATS 110:
Statistical Methods in Engineering and the Physical Sciences
Introduction to statistics for engineers and physical scientists. Topics: descriptive statistics, probability, interval estimation, tests of hypotheses, nonparametric methods, linear regression, analysis of variance, elementary experimental design. Prerequisite: one year of calculus.
Terms: Aut, Sum

Units: 45

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
STATS 116:
Theory of Probability
Probability spaces as models for phenomena with statistical regularity. Discrete spaces (binomial, hypergeometric, Poisson). Continuous spaces (normal, exponential) and densities. Random variables, expectation, independence, conditional probability. Introduction to the laws of large numbers and central limit theorem. Prerequisites: MATH 52 and familiarity with infinite series, or equivalent.
Terms: Aut, Spr, Sum

Units: 35

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
STATS 141:
Biostatistics (BIO 141)
Introductory statistical methods for biological data: describing data (numerical and graphical summaries); introduction to probability; and statistical inference (hypothesis tests and confidence intervals). Intermediate statistical methods: comparing groups (analysis of variance); analyzing associations (linear and logistic regression); and methods for categorical data (contingency tables and odds ratio). Course content integrated with statistical computing in R.
Terms: Aut

Units: 35

UG Reqs: GER:DBMath, WAYAQR

Grading: Letter or Credit/No Credit
STATS 167:
Probability: Ten Great Ideas About Chance (PHIL 166, PHIL 266, STATS 267)
Foundational approaches to thinking about chance in matters such as gambling, the law, and everyday affairs. Topics include: chance and decisions; the mathematics of chance; frequencies, symmetry, and chance; Bayes great idea; chance and psychology; misuses of chance; and harnessing chance. Emphasis is on the philosophical underpinnings and problems. Prerequisite: exposure to probability or a first course in statistics at the level of STATS 60 or 116.
Terms: not given this year

Units: 4

UG Reqs: GER:DBMath, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
STATS 191:
Introduction to Applied Statistics
Statistical tools for modern data analysis. Topics include regression and prediction, elements of the analysis of variance, bootstrap, and crossvalidation. Emphasis is on conceptual rather than theoretical understanding. Applications to social/biological sciences. Student assignments/projects require use of the software package R. Prerequisite: introductory statistical methods course. Recommended: 60, 110, or 141.
Terms: Win

Units: 34

UG Reqs: GER:DBMath, WAYAQR

Grading: Letter or Credit/No Credit
UGXFER GER2C2:
GER 2C SUBSTITUTION (2ND)
Units: 0

UG Reqs: GER:DBMath

Repeatable for credit

Grading: Internal Transfer