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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: 3-5 | UG Reqs: GER:DB-Math, WAY-AQR
Instructors: ; Mukherjee, R. (PI)

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. Computer-based data analysis. Prerequisite: Biology core or consent of instructor.
Terms: Spr | Units: 3 | UG Reqs: GER: DB-NatSci, GER:DB-Math, WAY-AQR, WAY-FR
Instructors: ; Watanabe, J. (PI)

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: MATH 41 and 42, or 10 units AP credit. Note: Students enrolled in section 100-02 and 100A-02 are required to attend the discussion sections on Thursdays 5:15-6:45.
Terms: Aut, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

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: application at:http://soe.stanford.edu/current_students/edp/programs/ace.html
Terms: Aut, Spr | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR

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: CME 100/ENGR 154 or MATH 51.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

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: application at:http://soe.stanford.edu/current_students/edp/programs/ace.html
Terms: Aut, Win, Spr | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR

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. Matrices, left and right inverses, QR factorization. Least- squares and model fitting, regularization and cross-validation, time-series prediction, and other examples. Constrained least-squares; applications to least-norm reconstruction, optimal control, and portfolio optimization. Newton methods and nonlinear least-squares. Prerequisites: MATH 51 or CME 100.
Terms: Aut | Units: 4-5 | UG Reqs: GER:DB-Math, WAY-FR

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:DB-Math, WAY-FR
Instructors: ; Khayms, V. (PI)

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: application at:http://soe.stanford.edu/current_students/edp/programs/ace.html
Terms: Spr | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Khayms, V. (PI)

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, non-parametric tests, regression and correlation analyses; applications in engineering, industrial manufacturing, medicine, biology, and other fields. Prerequisite: CME 100/ENGR154 or MATH 51.
Terms: Win, Sum | Units: 3-4 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR
Instructors: ; Khayms, V. (PI)

CS 103: Mathematical Foundations of Computing

Mathematical foundations required for computer science, including propositional predicate logic, induction, sets, functions, and relations. Formal language theory, including regular expressions, grammars, finite automata, Turing machines, and NP-completeness. Mathematical rigor, proof techniques, and applications. Prerequisite: 106A or equivalent.
Terms: Aut, Win, Spr | Units: 3-5 | UG Reqs: GER:DB-Math, WAY-FR

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 taken for letter grades: Econ 1 or 1A or 1V, and Math 51 or CME 100 or CME 100A. Must be taken for a Letter grade if majoring/minoring in Economics.
Terms: Aut, Win, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR, WAY-SI

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 41 or equivalent.
Terms: Aut, Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-SI
Instructors: ; McKeon, S. (PI)

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. Matrices, left and right inverses, QR factorization. Least- squares and model fitting, regularization and cross-validation, time-series prediction, and other examples. Constrained least-squares; applications to least-norm reconstruction, optimal control, and portfolio optimization. Newton methods and nonlinear least-squares. Prerequisites: MATH 51 or CME 100.
Terms: Aut | Units: 4-5 | UG Reqs: GER:DB-Math, WAY-FR

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: MATH 41 and 42, or 10 units AP credit. Note: Students enrolled in section 100-02 and 100A-02 are required to attend the discussion sections on Thursdays 5:15-6:45.
Terms: Aut, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

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: CME 100/ENGR 154 or MATH 51.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

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:DB-Math, WAY-FR
Instructors: ; Khayms, V. (PI)

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, non-parametric tests, regression and correlation analyses; applications in engineering, industrial manufacturing, medicine, biology, and other fields. Prerequisite: CME 100/ENGR154 or MATH 51.
Terms: Win, Sum | Units: 3-4 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR
Instructors: ; Khayms, V. (PI)

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 health-related data, including linear and logistic regression, and will become sophisticated consumers of health-related statistical results.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-AQR
Instructors: ; Kurina, L. (PI)

MATH 16: Mathematics in the Real World (STATS 90)

Introduction to non-calculus applications of mathematical ideas and principles in real-world problems. Topics include probability and counting, basic statistical concepts, geometric series. Applications include insurance, gambler's ruin, false positives in disease testing, present value of money, and mortgages. No knowledge of calculus required. Enrollment limited to students who do not have Stanford credit for a high school or college course in calculus or statistics.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Poulson, J. (PI)

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

MATH 20: Calculus

Continuation of 19. Applications of differential calculus; introduction to integral calculus of functions of one variable, including: the definite integral, methods of symbolic and numerical integration, applications of the definite integral. Prerequisites: 19 or equivalent.
Terms: Win, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR

MATH 21: Calculus

Continuation of 20. Applications of integral calculus, introduction to differential equations, infinite series. Prerequisite: 20 or equivalent.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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

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
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

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

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

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

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
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

MATH 52H: Honors Multivariable Mathematics

Continuation of 51H. Prerequisite: 51H.
Terms: Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Eliashberg, Y. (PI)

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

MATH 53H: Honors Multivariable Mathematics

Continuation of 52H. Prerequisite: 52H.
Terms: Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Li, J. (PI)

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
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
Instructors: ; Gu, Y. (PI)

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:DB-Math
Instructors: ; Lemke Oliver, R. (PI)

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: Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Li, Z. (PI)

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: Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Wilson, J. (PI)

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.
| Units: 3 | UG Reqs: GER:DB-Math

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

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
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
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
Instructors: ; Bump, D. (PI); Vakil, R. (PI)

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:DB-Math
Instructors: ; Soundararajan, K. (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
Instructors: ; Ford, A. (PI); Ying, L. (PI)

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: 131P.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Gu, Y. (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
Instructors: ; Zheng, T. (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
Instructors: ; Mazzeo, R. (PI)

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.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Schoen, R. (PI)

MATH 145: Algebraic Geometry

Hilbert's nullstellensatz, complex affine and projective curves, Bezout's theorem, the degree/genus formula, blow-up, Riemann-Roch theorem. Prerequisites: 120, and 121 or knowledge of fraction fields. Recommended: familiarity with surfaces equivalent to 143, 146, 147, or 148.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Vakil, R. (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
Instructors: ; Ionel, E. (PI)

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.
Last offered: Spring 2014 | Units: 3 | UG Reqs: GER:DB-Math

MATH 148: Algebraic Topology

Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Berwick-Evans, D. (PI)

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:DB-Math
Instructors: ; Dembo, A. (PI)

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:DB-Math
Instructors: ; Soundararajan, K. (PI)

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 and 121, especially modules over principal ideal domains and Galois theory of finite fields.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Conrad, B. (PI)

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.
Last offered: Spring 2014 | Units: 3 | UG Reqs: GER:DB-Math

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
Instructors: ; Sommer, R. (PI)

MATH 162: Philosophy of Mathematics (PHIL 162, PHIL 262)

(Graduate students register for PHIL 262.) General survey of the philosophy of mathematics, focusing on epistemological issues. Includes survey of some basic concepts (proof, axiom, definition, number, set); mind-bending theorems about the limits of our current mathematical knowledge, such as Gödel's Incompleteness Theorems, and the independence of the continuum hypothesis from the current axioms of set theory; major philosophical accounts of mathematics: Logicism, Intuitionism, Hilbert's program, Quine's empiricism, Field's program, Structuralism; concluding with a discussion of Eugene Wigner's `The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. Students won't be expected to prove theorems or complete mathematical exercises. However, includes some material of a technical nature. Prerequisite: PHIL150 or consent of instructor.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math
Instructors: ; Donaldson, T. (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

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.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Vasy, A. (PI)

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
Instructors: ; Ford, A. (PI)

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.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Menz, G. (PI)

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.
| Units: 3 | UG Reqs: GER:DB-Math

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.
| Units: 3 | UG Reqs: GER:DB-Math

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: Aut | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Pekelis, L. (PI)

PHIL 50: Introductory Logic

Propositional and predicate logic; emphasis is on translating English sentences into logical symbols and constructing derivations of valid arguments.
Terms: Aut, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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:DB-Math, WAY-FR
Instructors: ; Icard, T. (PI)

PHIL 150E: Logic in Action: A New Introduction to Logic

A new introduction to logic, covering propositional, modal, and first-order logic, with special attention to major applications in describing information and information-driven 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.
Last offered: Spring 2014 | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

PHIL 151: Metalogic (PHIL 251)

(Formerly 160A.) The syntax and semantics of sentential and first-order logic. Concepts of model theory. Gödel's completeness theorem and its consequences: the Löwenheim-Skolem theorem and the compactness theorem. Prerequisite: 150 or consent of instructor.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Icard, T. (PI)

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.
Last offered: Winter 2013 | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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:DB-Math
Instructors: ; Sommer, R. (PI)

PHIL 154: Modal Logic (PHIL 254)

(Graduate students register for 254.) Syntax and semantics of modal logic, and technical topics like completeness and correspondence theory, including both classical and recent developments. Applications to topics in philosophy, computer science, and other fields. Prerequisite: 150 or preferably 151.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; van Benthem, J. (PI)

PHIL 162: Philosophy of Mathematics (MATH 162, PHIL 262)

(Graduate students register for PHIL 262.) General survey of the philosophy of mathematics, focusing on epistemological issues. Includes survey of some basic concepts (proof, axiom, definition, number, set); mind-bending theorems about the limits of our current mathematical knowledge, such as Gödel's Incompleteness Theorems, and the independence of the continuum hypothesis from the current axioms of set theory; major philosophical accounts of mathematics: Logicism, Intuitionism, Hilbert's program, Quine's empiricism, Field's program, Structuralism; concluding with a discussion of Eugene Wigner's `The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. Students won't be expected to prove theorems or complete mathematical exercises. However, includes some material of a technical nature. Prerequisite: PHIL150 or consent of instructor.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math
Instructors: ; Donaldson, T. (PI)

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.
Last offered: Spring 2013 | Units: 4 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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, t-tests, correlation, and regression. Possible topics: analysis of variance and chi-square tests, computer statistical packages.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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: Aut | Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Pekelis, L. (PI)

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, t-tests, correlation, and regression. Possible topics: analysis of variance and chi-square tests, computer statistical packages.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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: 4-5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR
Instructors: ; He, H. (PI); Taylor, J. (PI)

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: 3-5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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: 3-5 | UG Reqs: GER:DB-Math, WAY-AQR
Instructors: ; Mukherjee, R. (PI)

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.
Last offered: Spring 2013 | Units: 4 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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 cross-validation. Emphasis is on conceptual rather than theoretical understanding. Applications to social/biological sciences. Student assignments/projects require use of the software package R. Recommended: 60, 110, or 141.
Terms: Win | Units: 3-4 | UG Reqs: GER:DB-Math, WAY-AQR
Instructors: ; Taylor, J. (PI)

STATS 90: Mathematics in the Real World (MATH 16)

Introduction to non-calculus applications of mathematical ideas and principles in real-world problems. Topics include probability and counting, basic statistical concepts, geometric series. Applications include insurance, gambler's ruin, false positives in disease testing, present value of money, and mortgages. No knowledge of calculus required. Enrollment limited to students who do not have Stanford credit for a high school or college course in calculus or statistics.
| Units: 3 | UG Reqs: GER:DB-Math
Instructors: ; Poulson, J. (PI)

UGXFER GER2C1: GER 2C SUBSTITUTION (1ST)

| Units: 0-99 | UG Reqs: GER:DB-Math | Repeatable for credit

UGXFER GER2C2: GER 2C SUBSTITUTION (2ND)

| Units: 0 | UG Reqs: GER:DB-Math | Repeatable for credit
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