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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: Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-AQR

CME 100: Vector Calculus for Engineers (ENGR 154)

Computation and visualization using MATLAB. Differential vector calculus: vector-valued functions, analytic geometry in space, functions of several variables, partial derivatives, gradient, linearization, unconstrained maxima and minima, Lagrange multipliers and applications to trajectory simulation, least squares, and numerical optimization. Introduction to linear algebra: matrix operations, systems of algebraic equations with applications to coordinate transformations and equilibrium problems. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green's, divergence, and Stokes' theorems. Numerous examples and applications drawn from classical mechanics, fluid dynamics and electromagnetism. Prerequisites: knowledge of single-variable calculus equivalent to the content of Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with Math 21, 5 on Calc AB with Math 21). Placement diagnostic (recommendation non-binding) at: https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.
Terms: Aut, Spr | Units: 5 | 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 are presented. For analytical methods students learn to solve linear and non-linear first order ODEs; linear second order ODEs; and Laplace transforms. Numerical methods using MATLAB programming tool kit are also introduced to solve various types of ODEs including: first and second order ODEs, higher order ODEs, systems of ODEs, initial and boundary value problems, finite differences, and multi-step methods. This also includes accuracy and linear stability analyses of various numerical algorithms which are essential tools for the modern engineer. This class is foundational for professional careers in engineering and as a preparation for more advanced classes at the undergraduate and graduate levels. Prerequisites: knowledge of single-variable calculus equivalent to the content of Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with Math 21, 5 on Calc AB with Math 21). Placement diagnostic (recommendation non-binding) at: https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.
Terms: Aut, Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

CME 104: Linear Algebra and Partial Differential Equations for Engineers (ENGR 155B)

Linear algebra: systems of algebraic equations, Gaussian elimination, undetermined and overdetermined systems, coupled systems of ordinary differential equations, LU factorization, eigensystem analysis, normal modes. Linear independence, vector spaces, subspaces and basis. Numerical analysis applied to structural equilibrium problems, electrical networks, and dynamic systems. Fourier series with applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations. Applications in heat and mass transport, mechanical vibration and acoustic waves, transmission lines, and fluid mechanics. Numerical methods for solution of partial differential equations: iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Examples and applications drawn from a variety of engineering fields. Prerequisite: CME102/ENGR155A.
Terms: Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

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. Numerical simulation using Monte Carlo techniques. Topics in mathematical statistics: random sampling, point estimation, confidence intervals, hypothesis testing, non-parametric tests, regression and correlation analyses. Numerous applications in engineering, manufacturing, reliability and quality assurance, medicine, biology, and other fields. Prerequisite: CME100/ENGR154 or Math 51 or 52.
Terms: Win, Sum | Units: 4 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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 first-order 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 first-order logic, set theory, binary relations, functions (injections, surjections, and bijections), cardinality, basic graph theory, the pigeonhole principle, mathematical induction, finite automata, regular expressions, the Myhill-Nerode theorem, context-free grammars, Turing machines, decidable and recognizable languages, self-reference 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, Sum | 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: Econ 1 or 1V, and Math 51 or Math 51A or CME 100 or CME 100A.
Terms: Aut, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR, WAY-SI
Instructors: ; Makler, C. (PI)

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:DB-Math, WAY-AQR, WAY-SI
Instructors: ; McKeon, S. (PI)

ENGR 108: Introduction to Matrix Methods

Formerly EE 103/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 k-means algorithm. Matrices, left and right inverses, QR factorization. Least-squares and model fitting, regularization and cross-validation. Constrained and nonlinear least-squares. Applications include time-series 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). ENGR 108 and Math 104 cover complementary topics in applied linear algebra. The focus of ENGR 108 is on a few linear algebra concepts, and many applications; the focus of Math 104 is on algorithms and concepts.
Terms: Aut, Sum | Units: 3-5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

ENGR 154: Vector Calculus for Engineers (CME 100)

Computation and visualization using MATLAB. Differential vector calculus: vector-valued functions, analytic geometry in space, functions of several variables, partial derivatives, gradient, linearization, unconstrained maxima and minima, Lagrange multipliers and applications to trajectory simulation, least squares, and numerical optimization. Introduction to linear algebra: matrix operations, systems of algebraic equations with applications to coordinate transformations and equilibrium problems. Integral vector calculus: multiple integrals in Cartesian, cylindrical, and spherical coordinates, line integrals, scalar potential, surface integrals, Green's, divergence, and Stokes' theorems. Numerous examples and applications drawn from classical mechanics, fluid dynamics and electromagnetism. Prerequisites: knowledge of single-variable calculus equivalent to the content of Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with Math 21, 5 on Calc AB with Math 21). Placement diagnostic (recommendation non-binding) at: https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.
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 are presented. For analytical methods students learn to solve linear and non-linear first order ODEs; linear second order ODEs; and Laplace transforms. Numerical methods using MATLAB programming tool kit are also introduced to solve various types of ODEs including: first and second order ODEs, higher order ODEs, systems of ODEs, initial and boundary value problems, finite differences, and multi-step methods. This also includes accuracy and linear stability analyses of various numerical algorithms which are essential tools for the modern engineer. This class is foundational for professional careers in engineering and as a preparation for more advanced classes at the undergraduate and graduate levels. Prerequisites: knowledge of single-variable calculus equivalent to the content of Math 19-21 (e.g., 5 on Calc BC, 4 on Calc BC with Math 21, 5 on Calc AB with Math 21). Placement diagnostic (recommendation non-binding) at: https://exploredegrees.stanford.edu/undergraduatedegreesandprograms/#aptext.
Terms: Aut, Win, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

ENGR 155B: Linear Algebra and Partial Differential Equations for Engineers (CME 104)

Linear algebra: systems of algebraic equations, Gaussian elimination, undetermined and overdetermined systems, coupled systems of ordinary differential equations, LU factorization, eigensystem analysis, normal modes. Linear independence, vector spaces, subspaces and basis. Numerical analysis applied to structural equilibrium problems, electrical networks, and dynamic systems. Fourier series with applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations. Applications in heat and mass transport, mechanical vibration and acoustic waves, transmission lines, and fluid mechanics. Numerical methods for solution of partial differential equations: iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Examples and applications drawn from a variety of engineering fields. Prerequisite: CME102/ENGR155A.
Terms: Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

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. Numerical simulation using Monte Carlo techniques. Topics in mathematical statistics: random sampling, point estimation, confidence intervals, hypothesis testing, non-parametric tests, regression and correlation analyses. Numerous applications in engineering, manufacturing, reliability and quality assurance, medicine, biology, and other fields. Prerequisite: CME100/ENGR154 or Math 51 or 52.
Terms: Win, Sum | Units: 4 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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: 4 | UG Reqs: GER:DB-Math, WAY-AQR

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

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: periodic trigonometric functions, 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: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR

MATH 20: Calculus

The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initial-value 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: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR

MATH 21: Calculus

This course addresses a variety of topics centered around the theme of "calculus with infinite processes", largely the content of BC-level AP Calculus that isn't in the AB-level syllabus. It is needed throughout probability and statistics at all levels, as well as to understand approximation procedures that arise in all quantitative fields (including economics and computer graphics). After an initial review of limit rules, the course goes on to discuss sequences of numbers and of functions, as well as limits "at infinity" for each (needed for any sensible discussion of long-term behavior of a numerical process, such as: iterative procedures and complexity in computer science, dynamic models throughout economics, and repeated trials with data in any field). Integration is discussed for rational functions (a loose end from Math 20) and especially (improper) integrals for unbounded functions and "to infinity": this shows up in contexts as diverse as escape velocity for a rocket, the present value of a perpetual yield asset, and important calculations in probability (including the famous "bell curve" and to understand why many statistical tests work as they do). The course then turns to infinite series (how to "sum" an infinite collection of numbers), some useful convergence and divergence rests for these, and the associated killer app: power series and their properties, as well as Taylor approximations, all of which provide the framework that underlies virtually all mathematical models used in any quantitative field. 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: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 51: Linear Algebra, Multivariable Calculus, and Modern Applications

This course provides unified coverage of linear algebra and multivariable differential calculus, and the free course e-text connects the material to many fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion includes orthogonality, linear independence, matrix algebra, and eigenvalues with applications such as least squares, linear regression, and Markov chains (relevant to population dynamics, molecular chemistry, and PageRank); the singular value decomposition (essential in image compression, topic modeling, and data-intensive work in many fields) is introduced in the final chapter of the e-text. The multivariable calculus portion includes unconstrained optimization via gradients and Hessians (used for energy minimization), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underlie many machine learning algorithms, such as backpropagation), and Newton's method (an ingredient in GPS and robotics). The course emphasizes computations alongside an intuitive understanding of key ideas. The widespread use of computers makes it important for users of math to understand concepts: novel users of quantitative tools in the future will be those who understand ideas and how they fit with examples and applications. This is the only course at Stanford whose syllabus includes nearly all the math background for CS 229, which is why CS 229 and CS 230 specifically recommend it (or other courses resting on it). For frequently asked questions about the differences between Math 51 and CME 100, see the FAQ on the placement page on the Math Department website. Prerequisite: Math 21 or equivalent (e.g. 5 on the AP Calculus BC test or suitable score on certain international exams: https://studentservices.stanford.edu/my-academics/earn-my-degree/undergraduate-degree-progress/test-transfer-credit/external-test-2). 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: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 51A: Linear Algebra, Multivariable Calculus, and Modern Applications, ACE

Students attend one of the regular MATH 51 lectures with a longer discussion section of four hours per week instead of two. Active mode: students in small groups discuss and work on problems from a worksheet distributed 2 or 3 days in advance, with a TA providing guidance and answering questions. Application required: https://forms.gle/ruykWBk6zJMgXRB49
Last offered: Spring 2022 | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR

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: Math 21 and Math 51 or equivalents.
Terms: Win, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 53: Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications

Ordinary differential equations and initial value problems, linear systems of such equations with an emphasis on second-order constant-coefficient equations, stability analysis for non-linear systems (including phase portraits and the role of eigenvalues), and numerical methods. Partial differential equations and boundary-value problems, Fourier series and initial conditions, and Fourier transform for non-periodic phenomena. Throughout the development we harness insights from linear algebra, and software widgets are used to explore course topics on a computer (no coding background is needed). The free e-text provides motivation from applications across a wide array of fields (biology, chemistry, computer science, economics, engineering, and physics) described in a manner not requiring any area-specific expertise, and it has an appendix on Laplace transforms with many worked examples as a complement to the Fourier transform in the main text. Prerequisite: Math 21 and Math 51, or equivalents.
Terms: Aut, Win, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 61CM: Modern Mathematics: Continuous Methods

This is the first part of a theoretical (i.e., proof-based) 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 as local graphs, integration on Euclidean space, and many examples. The linear algebra content is covered jointly with Math 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor. This series provides the necessary mathematical background for majors in all Computer Science, Data Science, Economics, Mathematics, Natural Sciences, and Engineering.
Terms: Aut | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 62CM: Modern Mathematics: Continuous Methods

A proof-based introduction to manifolds and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisites: Math 61CM.
Terms: Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 63CM: Modern Mathematics: Continuous Methods

A proof-based course on ordinary differential equations. Topics include the inverse and implicit function theorems, implicitly-defined submanifolds of Euclidean space, 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, behavior of solutions near an equilibrium point, and Sturm-Liouville theory. Prerequisite: Math 61CM or Math 61DM.
Terms: Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Ryzhik, L. (PI)

MATH 104: Applied Matrix Theory

Linear algebra for applications in science and engineering. The course introduces the key mathematical ideas in matrix theory, which are used in modern methods of data analysis, scientific computing, optimization, and nearly all quantitative fields of science and engineering. While the choice of topics is motivated by their use in various disciplines, the course will emphasize the theoretical and conceptual underpinnings of this subject. Topics include orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares methods, and algorithms for solving systems of linear equations; applications include clustering, principal component analysis and dimensionality reduction, regression. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and ENGR 108 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of ENGR 108 is on a few linear algebra concepts, and many applications. Prerequisites: MATH 51 and programming experience on par with CS 106A.
Terms: Aut, Win, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Vondrak, J. (PI)

MATH 109: Groups and Symmetry

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. Prerequisite: Math 51.
Terms: Aut | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 110: Number Theory for Cryptography

Number theory and its applications to modern cryptography. Topics include: congruences, primality testing and factorization, public key cryptography, and elliptic curves, emphasizing algorithms. Includes an introduction to proof-writing. This course develops math background useful in CS 255. WIM. Prerequisite: Math 51
Terms: Aut | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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; dual space and dual basis; eigenvectors and eigenvalues; diagonalization. Includes an introduction to proof-writing. (Math 104 offers a more application-oriented treatment.) Prerequisites: Math 51
Terms: Aut, Win, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 115: Functions of a Real Variable

The development of 1-dimensional real analysis (the logical framework for why calculus works): sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Includes introduction to proof-writing. Prerequisite: Math 51 or Math 56.
Terms: Aut, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 116: Complex Analysis

Holomorphic and analytic functions, power series, Cauchy integral and Cauchy integral formula, meromorphic functions and differential forms, calculus of residues and applications, analytic continuation, conformal mappings, Riemann mapping theorem, Laurent series and conformal classification of annuli, harmonic functions and Dirichlet problem, introduction to Riemann surfaces, theory of elliptic functions and integrals. ( Math 106 offers a less theoretical treatment). Prerequisites: 51,52 and 171, or 61cm and 62cm.
Terms: Aut | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Cortinovis, A. (PI)

MATH 120: Groups and Rings

Recommended for Mathematics majors and required of honors Mathematics majors. A more advanced treatment of group theory than in Math 109, also including ring theory. 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 course. Prerequisite: Math 51 and some prior proof-writing experience.
Terms: Aut, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Bump, D. (PI); Lopez, A. (TA)

MATH 131P: Partial Differential Equations

An introduction to techniques for solving PDE's. Topics include physical examples (such as the heat equation, wave equation, and Laplace's equation in 2 and 3 dimensions) and separation of variables with various coordinate systems to relate them to Sturm-Liouville problems using Fourier, Bessel, and Legendre series. Prerequisite: Math 53.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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/math-136/
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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: Math 53 and 147 or Math 62CM and 63CM.
Last offered: Spring 2019 | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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.
Last offered: Autumn 2014 | Units: 4 | UG Reqs: GER:DB-Math

MATH 143: Differential Geometry

Geometry of curves and surfaces in three-space. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces. Prerequisite: Math 52.
Terms: Aut | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Lai, Y. (PI); Cua, M. (TA)

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, point-set topology, or manifolds.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Zhang, Z. (PI)

MATH 147: Differential Topology

Introduction to smooth methods in topology including tranvsersality, intersection number, fixed point theorems, as well as differential forms and integration. Prerequisites: Math 144 or equivalent.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Chodosh, O. (PI)

MATH 148: Algebraic Topology

Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 151: Introduction to Probability Theory

A proof-oriented development of basic 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. CS majors can petition to use Math 151 in place of CS 109, provided they expect to take either CS 228 or CS 229 as well. Prerequisite: Math 61CM, or Math 52 and either Math 56 or Math 115 (or equivalent).
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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. Prerequisite: Math 51 and proof-writing experiences (e.g., Math 56).
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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.
Last offered: Spring 2023 | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 155: Analytic Number Theory

Introduction to Dirichlet series and Dirichlet characters, Poisson summation, Gauss sums, analytic continuation for Dirichlet L-functions, applications to prime numbers (e.g., prime number theorem, Dirichlet's theorem). Prerequisites: Complex analysis (Math 106 or 116), Math 152 (or comparable familiarity with the Euclidean algorithm, multiplicative group modulo n, and quadratic reciprocity), and experience with basic analysis arguments.
Terms: Aut | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Conrad, B. (PI); Rizk, K. (TA)

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: Math 56 or comfort with proof-writing.
Last offered: Autumn 2022 | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 171: Fundamental Concepts of Analysis

Recommended for Mathematics majors and required of honors Mathematics majors. A more advanced and general version of Math 115, introducing and using metric spaces. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: Math 61CM, or 61DM, or Math 51 and Math 115. WIM
Terms: Aut, Spr | Units: 4 | 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: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Sun, W. (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: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

OCEANS 174H: Experimental Design and Probability (OCEANS 274H)

Nature is inherently variable. Statistics gives us the tools to quantify the uncertainty of our measurements and draw conclusions from data. This course is an introduction to experimental design, probability, and data analysis. Topics include summary statistics, data visualization, probability distributions, statistical inference, and general linear models (e.g., t-tests, analysis of variance, regression). Students will use R to explore and analyze datasets relevant to the life and ocean sciences. No programming or statistical background is assumed. This course takes place in-person only at Hopkins Marine Station; for information on how to spend spring quarter in residence: https://hopkinsmarinestation.stanford.edu/undergraduate-studies/spring-courses-23-24 (Individual course registration also permitted.) Depending on enrollment numbers, a weekly shuttle to Hopkins or mileage reimbursements for qualifying carpools will be provided; terms and conditions apply. Graduate students register for OCEANS 274H.
Terms: Spr | Units: 4 | UG Reqs: GER: DB-NatSci, GER:DB-Math, WAY-AQR, WAY-FR

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 short-answer questions designed to test conceptual understanding.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Bassett, R. (PI)

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

PHIL 151: Metalogic (PHIL 251)

In this course we will go through some of the seminal ideas, constructions, and results from modern logic, focusing especially on classical first-order ("predicate") logic. After introducing general ideas of induction and recursion, we will study a bit of elementary (axiomatic) set theory before then covering basic definability theory, viz. assessing the theoretical limits of what can and cannot be expressed in a first-order language. The centerpiece result of the class is the completeness - and closely related compactness - of first-order logic, a result with a number of momentous consequences, some useful, some philosophically puzzling. We will then study a connection with game theory, whereby a certain type of game characterizes precisely the expressive power of first-order logic. Further topics may include: the 0-1 law in finite model theory, second-order logic, and the algebraic approach to logic. Prerequisite: 150 or consent of instructor.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

PHIL 152: Computability and Logic (PHIL 252)

Kurt G¿del's ground-breaking Incompleteness Theorems demonstrate fundamental limits on formal mathematical reasoning. In particular, the First Incompleteness Theorem says, roughly, that for any reasonable theory of the natural numbers there are statements in the language that are neither provable nor refutable in that theory. In this course, we will explore the expressive power of different axiomatizations of number theory, on our path to proving the Incompleteness Theorems. This study entails an exploration of models of computation, and the power and limitations of what is computable, leading to an introduction to elementary recursion theory. At the conclusion of the course, we will discuss technical and philosophical repercussions of these results. Prerequisite: 151/251.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math
Instructors: ; Sommer, R. (PI); Tan, J. (TA)

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 classical and recent topics in philosophy, computer science, mathematics, linguistics, and game theory. Prerequisite: 150 or preferably 151.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

PHIL 162: Philosophy of Mathematics (PHIL 262)

Prerequisite: PHIL150 or consent of instructor. This is a general overview of the philosophy of mathematics, focusing on the nature of mathematical truth and knowledge, the metaphysics of mathematical objects, and issues arising from mathematical practice. Topics to be discussed will include logicism, intuitionism, formalism, Goedel's incompleteness theorem, platonism, nominalism, fictionalism, structuralism, the nature of mathematical rigor, the role of diagrams in mathematics, and mathematical beauty.
Last offered: Winter 2023 | Units: 4 | UG Reqs: GER:DB-Math

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 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 100: Mathematics of Sports

This course will teach you how statistics and probability can be applied in sports, in order to evaluate team and individual performance, build optimal in-game strategies and ensure fairness between participants. Topics will include examples drawn from multiple sports such as basketball, baseball, soccer, football and tennis. The course is intended to focus on data-based applications, and will involve computations in R with real data sets via tutorial sessions and homework assignments. Prereqs: No statistical or programming background is assumed, but introductory courses, e.g, Stats 60,101 or 116, are recommended. A prior knowledge of Linear Algebra (e.g., Math 51) and basic probability is strongly recommended.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-AQR

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. Please note that students must enroll in one section in addition to the main lecture.
Terms: Aut | Units: 5 | UG Reqs: GER:DB-Math, WAY-AQR, WAY-FR

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. Undergraduate students enroll for 5 units, graduate students enroll for 4 units. Undergraduate students must enroll in one section in addition to the main lecture. Sections are optional for graduate students. Note: Autumn 2023-24 is the last time this course will be offered. It will be replaced by STATS 117 and STATS 118 in 2024-25.
Terms: Aut | Units: 4-5 | 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: Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-AQR

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. Prerequisite: introductory statistical methods course. Recommended: 60, 110, or 141.
Terms: Spr, Sum | Units: 3 | UG Reqs: GER:DB-Math, WAY-AQR
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