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APPPHYS 100B: The Questions of Cloth: Weaving, Pattern Complexity and Structures of Fabric (ARTSINST 100B)

Students will learn to weave on a table loom while examining textile structures from historic, artistic and scientific perspectives. Emphasis on analyzing patterns and structures generated by weaving, with elementary introductions to information-scientific notions of algorithmic complexity, image compression, and source coding. This class is primarily intended for non-STEM majors with little or no prior experience in working with textiles. Limited enrollment. Prerequisites: Instructor permission.
Last offered: Winter 2023 | Units: 4 | UG Reqs: WAY-FR

ARTSINST 100B: The Questions of Cloth: Weaving, Pattern Complexity and Structures of Fabric (APPPHYS 100B)

Students will learn to weave on a table loom while examining textile structures from historic, artistic and scientific perspectives. Emphasis on analyzing patterns and structures generated by weaving, with elementary introductions to information-scientific notions of algorithmic complexity, image compression, and source coding. This class is primarily intended for non-STEM majors with little or no prior experience in working with textiles. Limited enrollment. Prerequisites: Instructor permission.
Last offered: Winter 2023 | Units: 4 | UG Reqs: WAY-FR

BIO 143: Quantitative Methods for Marine Ecology and Conservation (BIO 243, CEE 164, CEE 264H, EARTHSYS 143H, EARTHSYS 243H, OCEANS 143)

NOTE: This course will be taught in-person on main campus, in hybrid format with Zoom options. The goal of this course is to learn the foundations of ecological modeling with a specific (but not exclusive) focus on marine conservation and sustainable exploitation of renewable resources. Students will be introduced to a range of methods - from basic to advanced - to characterize population structure, conduct demographic analyses, estimate extinction risk, identify temporal trends and spatial patterns, quantify the effect of environmental determinants and anthropogenic pressures on the dynamics of marine populations, describe the potential for adaptation to climate change. This course will emphasize learning by doing, and will rely heavily on practical computer laboratories, in R and/or Phyton, based on data from our own research activities or peer reviewed publications. Students with a background knowledge of statistics, programming and calculus will be most welcome. Formally BIOHOPK 143H and 243H.
Terms: Win | Units: 4 | UG Reqs: WAY-AQR, WAY-FR

BIODS 48N: Riding the Data Wave (STATS 48N)

Imagine collecting a bit of your saliva and sending it in to one of the personalized genomics company: for very little money you will get back information about hundreds of thousands of variable sites in your genome. Records of exposure to a variety of chemicals in the areas you have lived are only a few clicks away on the web; as are thousands of studies and informal reports on the effects of different diets, to which you can compare your own. What does this all mean for you? Never before in history humans have recorded so much information about themselves and the world that surrounds them. Nor has this data been so readily available to the lay person. Expression as "data deluge'' are used to describe such wealth as well as the loss of proper bearings that it often generates. How to summarize all this information in a useful way? How to boil down millions of numbers to just a meaningful few? How to convey the gist of the story in a picture without misleading oversimplifications? To answer these questions we need to consider the use of the data, appreciate the diversity that they represent, and understand how people instinctively interpret numbers and pictures. During each week, we will consider a different data set to be summarized with a different goal. We will review analysis of similar problems carried out in the past and explore if and how the same tools can be useful today. We will pay attention to contemporary media (newspapers, blogs, etc.) to identify settings similar to the ones we are examining and critique the displays and summaries there documented. Taking an experimental approach, we will evaluate the effectiveness of different data summaries in conveying the desired information by testing them on subsets of the enrolled students.
Last offered: Autumn 2020 | Units: 3 | UG Reqs: WAY-AQR, WAY-FR

BIOE 80: Introduction to Bioengineering (Engineering Living Matter) (ENGR 80)

Students completing BIOE 80 should have a working understanding for how to approach the systematic engineering of living systems to benefit all people and the planet. Our main goals are (1) to help students learn ways of thinking about engineering living matter and (2) to empower students to explore the broader ramifications of engineering life. Specific concepts and skills covered include but are not limited to: capacities of natural life on Earth; scope of the existing human-directed bioeconomy; deconstructing complicated problems; reaction & diffusion systems; microbial human anatomy; conceptualizing the engineering of biology; how atoms can be organized to make molecules; how to print DNA from scratch; programming genetic sensors, logic, & actuators; biology beyond molecules (photons, electrons, etc.); constraints limiting what life can do; and possible health challenges in 2030. And we explore questions like, how does what we want shape bioengineering, and who should choose and realize various competing bioengineering futures?
Terms: Spr | Units: 4 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CEE 164: Quantitative Methods for Marine Ecology and Conservation (BIO 143, BIO 243, CEE 264H, EARTHSYS 143H, EARTHSYS 243H, OCEANS 143)

NOTE: This course will be taught in-person on main campus, in hybrid format with Zoom options. The goal of this course is to learn the foundations of ecological modeling with a specific (but not exclusive) focus on marine conservation and sustainable exploitation of renewable resources. Students will be introduced to a range of methods - from basic to advanced - to characterize population structure, conduct demographic analyses, estimate extinction risk, identify temporal trends and spatial patterns, quantify the effect of environmental determinants and anthropogenic pressures on the dynamics of marine populations, describe the potential for adaptation to climate change. This course will emphasize learning by doing, and will rely heavily on practical computer laboratories, in R and/or Phyton, based on data from our own research activities or peer reviewed publications. Students with a background knowledge of statistics, programming and calculus will be most welcome. Formally BIOHOPK 143H and 243H.
Terms: Win | Units: 4 | UG Reqs: WAY-AQR, WAY-FR

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

CME 108: Introduction to Scientific Computing

Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floating-point arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness. Implementation of numerical methods in MATLAB programming assignments. Prerequisites: CME 100, 102 or MATH 51, 52, 53; prior programming experience (MATLAB or other language at level of CS 106A or higher).
Terms: Aut | Units: 3 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

CS 24: Minds and Machines (LINGUIST 35, PHIL 99, PSYCH 35, SYMSYS 1, SYMSYS 200)

(Formerly SYMSYS 100). An overview of the interdisciplinary study of cognition, information, communication, and language, with an emphasis on foundational issues: What are minds? What is computation? What are rationality and intelligence? Can we predict human behavior? Can computers be truly intelligent? How do people and technology interact, and how might they do so in the future? Lectures focus on how the methods of philosophy, mathematics, empirical research, and computational modeling are used to study minds and machines. Students must take this course before being approved to declare Symbolic Systems as a major. All students interested in studying Symbolic Systems are urged to take this course early in their student careers. The course material and presentation will be at an introductory level, without prerequisites. If you have any questions about the course, please email symsys1staff@gmail.com.
Terms: Aut, Win | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR

CS 101: Introduction to Computing Principles

Introduces the essential ideas of computing: data representation, algorithms, programming "code", computer hardware, networking, security, and social issues. Students learn how computers work and what they can do through hands-on exercises. In particular, students will see the capabilities and weaknesses of computer systems so they are not mysterious or intimidating. Course features many small programming exercises, although no prior programming experience is assumed or required. CS101 is not a complete programming course such as CS106A. CS101 is effectively an alternative to CS105. A laptop computer is recommended for the in-class exercises.
Last offered: Autumn 2018 | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, 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

CS 105: Introduction to Computers

For non-technical majors. What computers are and how they work. Practical experience in development of websites and an introduction to programming. A survey of Internet technology and the basics of computer hardware. Students in technical fields and students looking to acquire programming skills should take 106A or 106X. Students with prior computer science experience at the level of 106 or above require consent of instructor. Prerequisite: minimal math skills.
Terms: Aut, Spr | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 106A: Programming Methodology

Introduction to the engineering of computer applications emphasizing modern software engineering principles: program design, decomposition, encapsulation, abstraction, and testing. Emphasis is on good programming style and the built-in facilities of respective languages. Uses the Python programming language. No prior programming experience required.
Terms: Aut, Win, Spr, Sum | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 106AX: Programming Methodologies in JavaScript and Python (Accelerated)

Introduction to the engineering of computer applications emphasizing modern software engineering principles: object-oriented design, decomposition, encapsulation, abstraction, and testing. This course targets an audience with prior programming experience, and that prior experience is leveraged so material can be covered in greater depth.
Terms: Aut | Units: 3-5 | UG Reqs: WAY-FR
Instructors: ; Cain, J. (PI); Gupta, A. (TA)

CS 106B: Programming Abstractions

Abstraction and its relation to programming. Software engineering principles of data abstraction and modularity. Object-oriented programming, fundamental data structures (such as stacks, queues, sets) and data-directed design. Recursion and recursive data structures (linked lists, trees, graphs). Introduction to time and space complexity analysis. Uses the programming language C++ covering its basic facilities. Prerequisite: 106A or equivalent.
Terms: Aut, Win, Spr, Sum | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 106X: Programming Abstractions (Accelerated)

Intensive version of 106B for students with a strong programming background interested in a rigorous treatment of the topics at an accelerated pace. Significant amount of additional advanced material and substantially more challenging projects. Some projects may relate to CS department research. Prerequisite: excellence in 106A or equivalent, or consent of instructor.
Last offered: Autumn 2019 | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 107: Computer Organization and Systems

Introduction to the fundamental concepts of computer systems. Explores how computer systems execute programs and manipulate data, working from the C programming language down to the microprocessor. Topics covered include: the C programming language, data representation, machine-level code, computer arithmetic, elements of code compilation, memory organization and management, and performance evaluation and optimization. Prerequisites: 106B or X, or consent of instructor.
Terms: Aut, Win, Spr, Sum | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 107E: Computer Systems from the Ground Up

Introduction to the fundamental concepts of computer systems through bare metal programming on the Raspberry Pi. Explores how five concepts come together in computer systems: hardware, architecture, assembly code, the C language, and software development tools. Students do all programming with a Raspberry Pi kit and several add-ons (LEDs, buttons). Topics covered include: the C programming language, data representation, machine-level code, computer arithmetic, compilation, memory organization and management, debugging, hardware, and I/O. Enrollment limited to 40. Check website for details: http://cs107e.stanford.edu on student selection process. Prerequisite: CS106B or CS106X, and consent of instructor. There is a $75 course lab fee.
Terms: Win, Spr | Units: 3-5 | UG Reqs: WAY-FR

CS 109: Introduction to Probability for Computer Scientists

Topics include: counting and combinatorics, random variables, conditional probability, independence, distributions, expectation, point estimation, and limit theorems. Applications of probability in computer science including machine learning and the use of probability in the analysis of algorithms. Prerequisites: 103, 106B or X, multivariate calculus at the level of MATH 51 or CME 100 or equivalent.
Terms: Aut, Win, Spr, Sum | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

CS 157: Computational Logic

Rigorous introduction to Symbolic Logic from a computational perspective. Encoding information in the form of logical sentences. Reasoning with information in this form. Overview of logic technology and its applications - in mathematics, science, engineering, business, law, and so forth. Topics include the syntax and semantics of Propositional Logic, Relational Logic, and Herbrand Logic, validity, contingency, unsatisfiability, logical equivalence, entailment, consistency, natural deduction (Fitch), mathematical induction, resolution, compactness, soundness, completeness.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 161: Design and Analysis of Algorithms

Worst and average case analysis. Recurrences and asymptotics. Efficient algorithms for sorting, searching, and selection. Data structures: binary search trees, heaps, hash tables. Algorithm design techniques: divide-and-conquer, dynamic programming, greedy algorithms, amortized analysis, randomization. Algorithms for fundamental graph problems: minimum-cost spanning tree, connected components, topological sort, and shortest paths. Possible additional topics: network flow, string searching. Prerequisite: 106B or 106X; 103 or 103B; 109 or STATS 116.
Terms: Aut, Win, Sum | Units: 3-5 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

CS 230: Deep Learning

Deep Learning is one of the most highly sought after skills in AI. We will help you become good at Deep Learning. In this course, you will learn the foundations of Deep Learning, understand how to build neural networks, and learn how to lead successful machine learning projects. You will learn about Convolutional networks, RNNs, LSTM, Adam, Dropout, BatchNorm, Xavier/He initialization, and more. You will work on case studies from healthcare, autonomous driving, sign language reading, music generation, and natural language processing. You will master not only the theory, but also see how it is applied in industry. You will practice all these ideas in Python and in TensorFlow, which we will teach. AI is transforming multiple industries. After this course, you will likely find creative ways to apply it to your work. This class is taught in the flipped-classroom format. You will watch videos and complete in-depth programming assignments and online quizzes at home, then come in to class for advanced discussions and work on projects. This class will culminate in an open-ended final project, which the teaching team will help you on. Prerequisites: Familiarity with programming in Python and Linear Algebra (matrix / vector multiplications). CS 229 may be taken concurrently.
Last offered: Spring 2023 | Units: 3-4 | UG Reqs: WAY-AQR, WAY-FR

CS 246: Mining Massive Data Sets

The availability of massive datasets is revolutionizing science and industry. This course discusses data mining and machine learning algorithms for analyzing very large amounts of data. Topics include: Big data systems (Hadoop, Spark); Link Analysis (PageRank, spam detection); Similarity search (locality-sensitive hashing, shingling, min-hashing); Stream data processing; Recommender Systems; Analysis of social-network graphs; Association rules; Dimensionality reduction (UV, SVD, and CUR decompositions); Algorithms for large-scale mining (clustering, nearest-neighbor search); Large-scale machine learning (decision tree ensembles); Multi-armed bandit; Computational advertising. Prerequisites: At least one of CS107 or CS145.
Terms: Win | Units: 3-4 | UG Reqs: WAY-FR

EARTHSYS 143H: Quantitative Methods for Marine Ecology and Conservation (BIO 143, BIO 243, CEE 164, CEE 264H, EARTHSYS 243H, OCEANS 143)

NOTE: This course will be taught in-person on main campus, in hybrid format with Zoom options. The goal of this course is to learn the foundations of ecological modeling with a specific (but not exclusive) focus on marine conservation and sustainable exploitation of renewable resources. Students will be introduced to a range of methods - from basic to advanced - to characterize population structure, conduct demographic analyses, estimate extinction risk, identify temporal trends and spatial patterns, quantify the effect of environmental determinants and anthropogenic pressures on the dynamics of marine populations, describe the potential for adaptation to climate change. This course will emphasize learning by doing, and will rely heavily on practical computer laboratories, in R and/or Phyton, based on data from our own research activities or peer reviewed publications. Students with a background knowledge of statistics, programming and calculus will be most welcome. Formally BIOHOPK 143H and 243H.
Terms: Win | Units: 4 | UG Reqs: WAY-AQR, 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 51: Economic Analysis II

Neoclassical analysis of general equilibrium, welfare economics, imperfect competition, externalities and public goods, risk and uncertainty, game theory, adverse selection, and moral hazard. Multivariate calculus is used. Prerequisite: ECON 50.
Terms: Aut, Spr | Units: 5 | UG Reqs: WAY-FR, WAY-SI
Instructors: ; Makler, C. (PI)

ECON 136: Market Design

Use of economic theory and analysis to design allocation mechanisms and market institutions. Course focuses on three areas: the design of matching algorithms to solve assignment problems, with applications to school choice, entry-level labor markets, and kidney exchanges; the design of auctions to solve general resource allocation problems, with applications to the sale of natural resources, financial assets, radio spectrum, and advertising; and the design of platforms and exchanges, with applications to internet markets. Emphasis on connecting economic theory to practical applications. Students must write term paper.
Terms: Spr | Units: 5 | UG Reqs: WAY-FR
Instructors: ; Milgrom, P. (PI)

ECON 137: Decision Modeling and Information

Effective decision models consider a decision maker's alternatives, information and preferences. The construction of such models in single-party situations with emphasis on the role of information. The course then evolves to two-party decision situations where one party has more information than the other. Models examined include: bidding exercises and the winner's curse, the Akerlof Model and adverse selection, the Principal-Agent model and risk sharing, moral hazard and contract design. Prerequisite: ECON 102A or equivalent. Recommended: Econ 50, Optimization and simulation in Excel.
Terms: Spr | Units: 5 | UG Reqs: WAY-AQR, WAY-FR
Instructors: ; McKeon, S. (PI)

ECON 160: Game Theory and Economic Applications

Introduction to game theory and its applications to economics. Topics: strategic and extensive form games, dominant strategies, Nash equilibrium, subgame-perfect equilibrium, and Bayesian equilibrium. The theory is applied to repeated games, oligopoly, auctions, and bargaining with examples from economics and political science. Prerequisites: Working knowledge of calculus and basic probability theory.
Last offered: Spring 2023 | Units: 5 | UG Reqs: WAY-FR, WAY-SI

ECON 180: Honors Game Theory

Rigorous introduction to game theory and applications. Topics include solution concepts for static and dynamic games of complete and incomplete information, signaling games, repeated games, bargaining, and elements of cooperative game theory. Applications mainly from economics, but also political science, biology, and computer science. Prerequisites: Experience with abstract mathematics and willingness to work hard. No background in economics required.
Last offered: Autumn 2019 | Units: 5 | UG Reqs: GER:DB-SocSci, WAY-FR, WAY-SI

EE 102A: Signals and Systems I

Concepts and tools for continuous- and discrete-time signal and system analysis with applications in signal processing, communications, and control. Mathematical representation of signals and systems. Linearity and time invariance. System impulse and step responses. System frequency response. Frequency-domain representations: Fourier series and Fourier transforms. Filtering and signal distortion. Time/frequency sampling and interpolation. Continuous-discrete-time signal conversion and quantization. Discrete-time signal processing. Prerequisites: MATH 53 or CME 102. EE 102A may be taken concurrently with either course, provided students have proficiency in complex numbers.
Terms: Win | Units: 4 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

EE 102B: Signals and Systems II

Continuation of EE 102A. Concepts and tools for continuous- and discrete-time signal and system analysis with applications in communications, signal processing and control. Analog and digital modulation and demodulation. Sampling, reconstruction, decimation and interpolation. Finite impulse response filter design. Discrete Fourier transforms, applications in convolution and spectral analysis. Laplace transforms, applications in circuits and feedback control. Z transforms, applications in infinite impulse response filter design. Prerequisite: EE 102A.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

EE 116: Semiconductor Devices for Energy and Electronics

The underpinnings of modern technology are the transistor (circuits), the capacitor (memory), and the solar cell (energy). EE 116 introduces the physics of their operation, their historical origins (including Nobel prize breakthroughs), and how they can be optimized for future applications. The class covers physical principles of semiconductors, including silicon and new material discoveries, quantum effects, band theory, operating principles, and device equations. Recommended (but not required) co-requisite: EE 65 or equivalent.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-EngrAppSci, WAY-FR, WAY-SMA

EE 142: Engineering Electromagnetics

Introduction to electromagnetism and Maxwell's equations in static and dynamic regimes. Electrostatics and magnetostatics: Gauss's, Coulomb's, Faraday's, Ampere's, Biot-Savart's laws. Electric and magnetic potentials. Boundary conditions. Electric and magnetic field energy. Electrodynamics: Wave equation; Electromagnetic waves; Phasor form of Maxwell's equations.Solution of the wave equation in 1D free space: Wavelength, wave-vector, forward and backward propagating plane waves.Poynting's theorem. Propagation in lossy media, skin depth. Reflection and refraction at planar boundaries, total internal reflection. Solutions of wave equation for various 1D-3D problems: Electromagnetic resonators, waveguides periodic media, transmission lines. Formerly EE 141. Prerequisites: an introductory course in electromagnetics (PHYSICS 43, PHYSICS 63, PHYSICS 81, or EE 42) and a solid background in vector calculus (CME 100, CME 102, or MATH 52, with MATH 52 being an ideal prerequisite)
Terms: Spr | Units: 3 | UG Reqs: GER:DB-EngrAppSci, WAY-FR, WAY-SMA
Instructors: ; Fan, J. (PI); Azzouz, M. (TA)

EE 178: Probabilistic Systems Analysis

Introduction to probability and its role in modeling and analyzing real world phenomena and systems, including topics in statistics, machine learning, and statistical signal processing. Elements of probability, conditional probability, Bayes rule, independence. Discrete and continuous random variables. Signal detection. Functions of random variables. Expectation; mean, variance and covariance, linear MSE estimation. Conditional expectation; iterated expectation, MSE estimation, quantization and clustering. Parameter estimation. Classification. Sample averages. Inequalities and limit theorems. Confidence intervals. Prerequisites: Calculus at the level of MATH 51, CME 100 or equivalent and basic knowledge of computing at the level of CS106A.
Terms: Spr | Units: 3-4 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

ENERGY 120: Mass and Energy Transport in Porous Media (ENGR 120)

Engineering topics in mass and energy transport in porous media relevant to energy systems. Mass, momentum and energy conservation equations in porous structures. Single phase and multiphase flow through porous media. Gas laws. Introduction to thermodynamics. Chemical, physical, and thermodynamic properties of liquids and gases in the subsurface.
Terms: Win | Units: 3 | UG Reqs: GER:DB-EngrAppSci, WAY-FR, WAY-SMA

ENGLISH 184F: Literary Text Mining 2: Studies in Cultural Analytics

In this course, students will learn how to apply quantitative and computational methods for analyzing text to questions that are of significance to Literary Studies, and the humanities more broadly. Beginning with a series of readings and discussions on the theoretical implications of using quantitative methods for literary analysis, we will move to in-depth instruction in more advanced methods for computational text analysis, including topic models, word embeddings, and large language models. Students will not only become familiar with training and querying these models, but, more importantly, will gain hands-on experience in how to build these analytical techniques into humanities-based research.
Terms: Win | Units: 3-5 | UG Reqs: WAY-AQR, WAY-FR
Instructors: ; Algee-Hewitt, M. (PI)

ENGR 10: Introduction to Engineering Analysis

Integrated approach to the fundamental scientific principles that are the cornerstones of engineering analysis: conservation of mass, atomic species, charge, momentum, angular momentum, energy, production of entropy expressed in the form of balance equations on carefully defined systems, and incorporating simple physical models. Emphasis is on setting up analysis problems arising in engineering. Topics: simple analytical solutions, numerical solutions of linear algebraic equations, and laboratory experiences. Provides the foundation and tools for subsequent engineering courses. Prerequisite: AP Physics and AP Calculus or equivalent.
Terms: Sum | Units: 4 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR
Instructors: ; Cappelli, M. (PI)

ENGR 76: Information Science and Engineering

What is information? How can we measure and efficiently represent it? How can we reliably communicate and store it over media prone to noise and errors? How can we make sound decisions based on partial and noisy information? This course introduces the basic notions required to address these questions, as well as the principles and techniques underlying the design of modern information, communication, and decision-making systems with relations to and applications in machine-learning, through genomics, to neuroscience. Students will get a hands-on appreciation of the concepts via projects in small groups, where they will develop their own systems for streaming of multi-media data under human-centric performance criteria. Prerequisite: CS 106A.
Terms: Spr | Units: 5 | UG Reqs: WAY-AQR, WAY-FR

ENGR 80: Introduction to Bioengineering (Engineering Living Matter) (BIOE 80)

Students completing BIOE 80 should have a working understanding for how to approach the systematic engineering of living systems to benefit all people and the planet. Our main goals are (1) to help students learn ways of thinking about engineering living matter and (2) to empower students to explore the broader ramifications of engineering life. Specific concepts and skills covered include but are not limited to: capacities of natural life on Earth; scope of the existing human-directed bioeconomy; deconstructing complicated problems; reaction & diffusion systems; microbial human anatomy; conceptualizing the engineering of biology; how atoms can be organized to make molecules; how to print DNA from scratch; programming genetic sensors, logic, & actuators; biology beyond molecules (photons, electrons, etc.); constraints limiting what life can do; and possible health challenges in 2030. And we explore questions like, how does what we want shape bioengineering, and who should choose and realize various competing bioengineering futures?
Terms: Spr | Units: 4 | UG Reqs: GER:DB-EngrAppSci, WAY-FR

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 120: Mass and Energy Transport in Porous Media (ENERGY 120)

Engineering topics in mass and energy transport in porous media relevant to energy systems. Mass, momentum and energy conservation equations in porous structures. Single phase and multiphase flow through porous media. Gas laws. Introduction to thermodynamics. Chemical, physical, and thermodynamic properties of liquids and gases in the subsurface.
Terms: Win | Units: 3 | UG Reqs: GER:DB-EngrAppSci, WAY-FR, WAY-SMA

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

GEOPHYS 120: Geophysical Mechanics and Dynamics (GEOPHYS 220)

Introductory application of continuum mechanics to ice sheets and glaciers, water waves and tsunamis, and volcanoes. Emphasis on physical processes and mathematical description using balance of mass and momentum, combined with constitutive equations for fluids and solids. Designed for undergraduates with no prior geophysics background; also appropriate for beginning graduate students. Prerequisites: CME 100 or MATH 52 and PHYSICS 41 (or equivalent).
Terms: Win | Units: 3-5 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA
Instructors: ; Dunham, E. (PI); Ji, Q. (TA)

LINGUIST 30N: Linguistic Meaning and the Law

We will investigate how inherent properties of language, such as ambiguity, vagueness and context-dependence, play into the meaning of a legal text, and how the meaning of a law can remain invariant while its range of application can change with the facts and with our discovery of what the facts are. Our focus will be on the perspective linguistic analysis brings to legal theory, addressing current controversies surrounding different conceptions of `textualism¿ and drawing on well-known examples of legal reasoning about language in cases of identity fraud, obstruction of justice and genocide.
Terms: Win | Units: 3 | UG Reqs: WAY-FR
Instructors: ; Condoravdi, C. (PI)

LINGUIST 35: Minds and Machines (CS 24, PHIL 99, PSYCH 35, SYMSYS 1, SYMSYS 200)

(Formerly SYMSYS 100). An overview of the interdisciplinary study of cognition, information, communication, and language, with an emphasis on foundational issues: What are minds? What is computation? What are rationality and intelligence? Can we predict human behavior? Can computers be truly intelligent? How do people and technology interact, and how might they do so in the future? Lectures focus on how the methods of philosophy, mathematics, empirical research, and computational modeling are used to study minds and machines. Students must take this course before being approved to declare Symbolic Systems as a major. All students interested in studying Symbolic Systems are urged to take this course early in their student careers. The course material and presentation will be at an introductory level, without prerequisites. If you have any questions about the course, please email symsys1staff@gmail.com.
Terms: Aut, Win, Sum | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR

LINGUIST 110: Introduction to Phonology

Introduction to the sound systems of the world's languages, their similarities and differences. Theories that account for the tacit generalizations that govern the sound patterns of languages. Prerequisite: Linguist 1 or Linguist 105
Terms: Aut | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR
Instructors: ; Sanker, C. (PI); Yi, I. (TA)

LINGUIST 121A: The Syntax of English

A data-driven introduction to the study of generative syntax through an in-depth investigation of the sentence structure of English. Emphasis is on central aspects of English syntax, but the principles of theory and analysis extend to the study of the syntax of other languages. The course focuses on building up syntactic argumentation skills via the collective development of a partial formal theory of sentence structure, which attempts to model native speaker knowledge. Satisfies the WIM requirement for Linguistics and the WAY-FR requirement. Prerequisites: none (can be taken before or after Linguistics 121B). The discussion section is mandatory.
Terms: Win | Units: 4 | UG Reqs: WAY-FR

LINGUIST 121B: Crosslinguistic Syntax

A data-driven introduction to the study of syntax through the investigation of a diverse array of the world's languages, including but not limited to English. Emphasis is on understanding how languages are systematically alike and different in their basic sentence structure. The course focuses on building up syntactic argumentation skills via the collective development of a partial formal theory of sentence structure, which attempts to model native speaker knowledge. Satisfies the WIM requirement for Linguistics and the WAY-FR requirement. Prerequisites: none (can be taken before or after Linguistics 121A). The discussion section is mandatory.
Last offered: Autumn 2021 | Units: 4 | UG Reqs: WAY-FR

LINGUIST 130A: Introduction to Semantics and Pragmatics (LINGUIST 230A)

Linguistic meaning and its role in communication. Topics include logical semantics, conversational implicature, presupposition, and speech acts. Applications to issues in politics, the law, philosophy, advertising, and natural language processing. Those who have not taken logic, such as PHIL 150 or 151, should attend section. Prerequisites: LINGUIST 1, SYMSYS 1 (LINGUIST 35), consent of instructor, or graduate standing in Linguistics
Terms: Win | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR

LINGUIST 130B: Introduction to Lexical Semantics

Introduction to basic concepts and issues in the linguistic study of word meaning. We explore grammatical regularities in word meaning and the relation between word meaning and the conceptual realm. The questions we address include the following. How is the meaning of a word determined from its internal structure? How can simple words have complex meanings? What is a possible word? How does a word's meaning determine the word's syntactic distribution and what kind of reasoning does it support? What kind of information belongs to the lexical entry of a word? The course will show that the investigation¿of the linguistic and semantic structure of words draws on the full resources of linguistic theory and methodology. Prerequisites: SYMSYS1, LINGUIST1, LINGUIST35, or equivalent or permission of the instructor. LINGUIST 130A is not a prerequisite for this course.
Terms: Spr | Units: 3-4 | UG Reqs: GER:DB-SocSci, WAY-FR

LINGUIST 157: Sociophonetics (LINGUIST 257)

The study of phonetic aspects of sociolinguistic variation and the social significance of phonetic variation. Acoustic analysis of vowels, consonants, prosody, and voice quality. Hands-on work on collaborative research project. This course must be taken for a minimum of 3 units and a letter grade to be eligible for Ways credit. Prerequisite: 105, 110 or equivalent, or consent of instructor.
Last offered: Winter 2023 | Units: 1-4 | UG Reqs: WAY-FR

LINGUIST 160: Historical Linguistics

Principles of historical linguistics:, the nature of language change. Kinds and causes of change, variation and diffusion of changes through populations, differentiation of dialects and languages, determination and classification of historical relationships among languages, the reconstruction of ancestral languages and intermediate changes, parallels with cultural and genetic evolutionary theory, and implications of variation and change for the description and explanation of language in general. Prerequisite: introductory course in linguistics.
Terms: Aut | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR

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 56: Proofs and Modern Mathematics

How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the course develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This course may be paired with Math 51; though that course is not a pre- or co-requisite.
Terms: Aut, Win | Units: 4 | UG Reqs: 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 61DM: Modern Mathematics: Discrete Methods

This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. The linear algebra content is covered jointly with Math 61CM. Students should 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 Computer Science, Data Science, Economics, Mathematics, and most Natural Sciences and some Engineering majors. Those who plan to major in Physics or in Engineering, majors requiring Math 50's beyond Math 51, are recommended to take Math 60CM.
Terms: Aut | Units: 5 | UG Reqs: WAY-FR
Instructors: ; Fox, J. (PI); Xu, M. (TA)

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 62DM: Modern Mathematics: Discrete Methods

This is the second part of a theoretical (proof-based) sequence with a focus on discrete mathematics. The central objects discussed in this course are finite fields. These are beautiful structures in themselves, and very useful in large areas of modern mathematics, and beyond. Our goal will be to construct these, understand their structure, and along the way discuss unexpected applications in combinatorics and number theory. Highlights of the course include a complete proof of a polynomial time algorithm for primality testing, Sidon sets and finite projective planes, and an understanding of a lovely magic trick due to Persi Diaconis. Prerequisite: Math 61DM or 61CM.
Terms: Win | Units: 5 | UG Reqs: 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 63DM: Modern Mathematics: Discrete Methods

Third part of a proof-based sequence in discrete mathematics, though independent of the second part (62DM). The first half of the quarter gives a brisk-paced coverage of probability and random processes with an intensive use of generating functions and a rich variety of applications. The second half treats entropy, Bayesian inference, Markov chains, game theory, probabilistic methods in solving non-probabilistic problems. We use continuous calculus, e.g. in handling the Gaussian, but anything needed will be reviewed in a self-contained manner. Prerequisite: Math 61DM or 61CM
Terms: Spr | Units: 5 | UG Reqs: WAY-FR
Instructors: ; Tokieda, T. (PI)

MATH 77Q: Probability and gambling

One of the earliest probabilistic discussions was in 1654 between two French mathematicians, Pascal and Fermat, on the following question: 'If a pair of six-sided dice is thrown 24 times, should you bet even money on the occurrence of at least one `double six'?' Shortly after the discussion, Huygens, a Dutch scientist, published De Ratiociniis in Ludo Aleae (The Value of all Chances in Games of Fortune) in 1657; this is considered to be the first treatise on probability. Due to the inherent appeal of games of chance, probability theory soon became popular, and the subject underwent rapid development in the 18th century with contributions from mathematical giants, such as Bernoulli, de Moivre, and Laplace. There are two fairly different lines of thought associated with applications of probability: the solution of betting/gambling and the analysis of statistical data related to quantitative subjects such as mortality tables and insurance rates. In this Introsem, we will discuss poker and other games of chance, such as daily fantasy sports, from the perspective of risk analysis. This Introsem does not require any programming knowledge, but some experience with Excel, MATLAB, R, and/or Python will enhance your experience in our discussion of daily fantasy sports. Students should be familiar with all material from Math 51. No prior knowledge of sports and games of chance is required. Students must apply through the IntroSem application process.
Terms: Win, Spr | Units: 3 | UG Reqs: WAY-FR
Instructors: ; Kim, G. (PI)

MATH 87Q: Mathematics of Knots, Braids, Links, and Tangles

Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.
Terms: Spr | Units: 3 | UG Reqs: WAY-FR
Instructors: ; Wieczorek, W. (PI)

MATH 101: Math Discovery Lab

MDL is a discovery-based project course in mathematics. Students work independently in small groups to explore open-ended mathematical problems and discover original mathematics. Students formulate conjectures and hypotheses; test predictions by computation, simulation, or pure thought; and present their results to classmates. WIM. Admission is by application. Motivated students with a mathematical background of at least Math 51 or 61CM or 61DM (or equivalent) are encouraged to apply. Please visit https://mathematics.stanford.edu/math-101 for more information about the course and application.
Terms: Win | Units: 4 | UG Reqs: WAY-FR

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 107: Graph Theory

An introductory course in graph theory establishing fundamental concepts and results in variety of topics. Topics include: basic notions, connectivity, cycles, matchings, planar graphs, graph coloring, matrix-tree theorem, conditions for hamiltonicity, Kuratowski's theorem, Ramsey and Turan-type theorem. Prerequisites: 51 or equivalent and some familiarity with proofs is required.
Terms: Win | Units: 4 | UG Reqs: WAY-FR
Instructors: ; McKenzie, T. (PI); Li, Z. (TA)

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 117: Advanced Complex Analysis

Review of holomorphic and meromorphic 1-forms, product development, Gamma-function and Riemann zeta-function, Fourier series and integrals, Fourier and Laplace transforms, differential geometric and analytic approaches to Riemann surfaces and conformal mappings, introduction to hyperbolic geometry, Laplace and d-bar equations and their solvability, the Uniformization Theorem, divisors and line bundles, Riemann-Roch theorem, Abel Jacobi theory. Prerequisite: Math 116.
Terms: Win | Units: 4 | UG Reqs: 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 122: Modules and Group Representations

Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite: Math 113 and 120.
Terms: Spr | Units: 4 | UG Reqs: WAY-FR
Instructors: ; Taylor, R. (PI)

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 142: Hyperbolic Geometry

An introductory course in hyperbolic geometry. Topics may include: different models of hyperbolic geometry, hyperbolic area and geodesics, Isometries and Mobius transformations, conformal maps, Fuchsian groups, Farey tessellation, hyperbolic structures on surfaces and three manifolds, limit sets. Prerequisites: some familiarity with the basic concepts of differential geometryand the topology of surfaces and manifolds is strongly recommended
Last offered: Autumn 2021 | Units: 4 | UG Reqs: WAY-FR

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 144: Introduction to Topology and Geometry

Point set topology, including connectedness, compactness, countability and separation axioms. The inverse and implicit function theorems. Smooth manifolds, immersions and submersions, embedding theorems. Prerequisites: Math 61CM or both Math 113 and Math 171.
Terms: Win | Units: 4 | UG Reqs: WAY-FR

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 158: Probability and Stochastic Differential Equations for Applications (CME 298)

Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with some applications in science and/or engineering. Prerequisites: Math 53 and introductory probability (such as Stats 116 or Math 151).
Terms: Spr | Units: 4 | UG Reqs: WAY-FR
Instructors: ; Adhikari, A. (PI)

MATH 159: Discrete Probabilistic Methods

Modern discrete probabilistic methods suitable for analyzing discrete structures of the type arising in number theory, graph theory, combinatorics, computer science, information theory and molecular sequence analysis. Prerequisite: STATS 116/MATH 151 or equivalent. Typically in alternating years.
Terms: Aut | Units: 4 | UG Reqs: WAY-FR

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 173: Theory of Partial Differential Equations

A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods. The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had 172 or 205A. In years when Math 173 is not offered, Math 220 is a recommended alternative (with similar content but a different emphasis). Prerequisite: 171 or equivalent.
Terms: Spr | Units: 4 | UG Reqs: WAY-FR
Instructors: ; Sussman, E. (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

MATH 177: Geometric Methods in the Theory of Ordinary Differential Equations

Hamiltonian systems and their geometry. First order PDE and Hamilton-Jacobi equation. Structural stability and hyperbolic dynamical systems. Completely integrable systems. Perturbation theory.
Last offered: Spring 2018 | Units: 4 | UG Reqs: WAY-FR

MS&E 20: Discrete Probability Concepts And Models

Fundamental concepts and tools for the analysis of problems under uncertainty, focusing on structuring, model building, and analysis. Examples from legal, social, medical, and physical problems. Topics include axioms of probability, probability trees, belief networks, random variables, conditioning, and expectation. The course is fast-paced, but it has no prerequisites.
Terms: Sum | Units: 4 | UG Reqs: WAY-FR
Instructors: ; Shachter, R. (PI)

MS&E 120: Introduction to Probability

Probability is the foundation behind many important disciplines including statistics, machine learning, risk analysis, stochastic modeling and optimization. This course provides an in-depth undergraduate-level introduction to fundamental ideas and tools of probability. Topics include: the foundations (sample spaces, random variables, probability distributions, conditioning, independence, expectation, variance), a systematic study of the most important univariate and multivariate distributions (Normal, Multivariate Normal, Binomial, Poisson, etc...), as well as a peek at some limit theorems (basic law of large numbers and central limit theorem) and, time permitting, some elementary markov chain theory. Prerequisite: CME 100 or MATH 51.
Terms: Aut | Units: 4 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

MS&E 152: Introduction to Decision Analysis

How to make good decisions in a complex, dynamic, and uncertain world. People often make decisions that on close examination they regard as wrong. Decision analysis uses a structured conversation based on actional thought to obtain clarity of action in a wide variety of domains. Topics: distinctions, possibilities and probabilities, relevance, value of information and experimentation, relevance and decision diagrams, risk attitude. Prerequisites: high school algebra and basic spreadsheet skills.
Terms: Spr | Units: 3-4 | UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

OCEANS 143: Quantitative Methods for Marine Ecology and Conservation (BIO 143, BIO 243, CEE 164, CEE 264H, EARTHSYS 143H, EARTHSYS 243H)

NOTE: This course will be taught in-person on main campus, in hybrid format with Zoom options. The goal of this course is to learn the foundations of ecological modeling with a specific (but not exclusive) focus on marine conservation and sustainable exploitation of renewable resources. Students will be introduced to a range of methods - from basic to advanced - to characterize population structure, conduct demographic analyses, estimate extinction risk, identify temporal trends and spatial patterns, quantify the effect of environmental determinants and anthropogenic pressures on the dynamics of marine populations, describe the potential for adaptation to climate change. This course will emphasize learning by doing, and will rely heavily on practical computer laboratories, in R and/or Phyton, based on data from our own research activities or peer reviewed publications. Students with a background knowledge of statistics, programming and calculus will be most welcome. Formally BIOHOPK 143H and 243H.
Terms: Win | Units: 4 | UG Reqs: WAY-AQR, 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 50S: Introduction to Formal Methods in Contemporary Philosophy

This course will serve as a first introduction to the formal tools and techniques of contemporary philosophy, including probability and formal logic. Traditionally, philosophy is an attempt to systematically tackle foundational problems related to value, inquiry, mind and reality. Contemporary philosophy continuesthis tradition of critical thinking with modern subject matter (often engaging with natural, social and mathematical science) and modern rigorous methods, including the methods of set theory, probability theory and formal logic. The aim of this course is to introduce such methods, along with various core philosophical distinctions and motivations. The focus will be on basic conceptual underpinnings and skills, not technical details. The material covered is also useful preparation for certain topics in mathematics, computer science, linguistics, economics and statistics. No previous philosophical or mathematical training is presupposed, though an appreciation of precise thinking is an advantage.
Last offered: Summer 2023 | Units: 3 | UG Reqs: WAY-FR

PHIL 99: Minds and Machines (CS 24, LINGUIST 35, PSYCH 35, SYMSYS 1, SYMSYS 200)

(Formerly SYMSYS 100). An overview of the interdisciplinary study of cognition, information, communication, and language, with an emphasis on foundational issues: What are minds? What is computation? What are rationality and intelligence? Can we predict human behavior? Can computers be truly intelligent? How do people and technology interact, and how might they do so in the future? Lectures focus on how the methods of philosophy, mathematics, empirical research, and computational modeling are used to study minds and machines. Students must take this course before being approved to declare Symbolic Systems as a major. All students interested in studying Symbolic Systems are urged to take this course early in their student careers. The course material and presentation will be at an introductory level, without prerequisites. If you have any questions about the course, please email symsys1staff@gmail.com.
Terms: Aut, Win | Units: 4 | UG Reqs: GER:DB-SocSci, 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

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 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

PHYSICS 14N: Quantum Information: Visions and Emerging Technologies

What sets quantum information apart from its classical counterpart is that it can be encoded non-locally, woven into correlations among multiple qubits in a phenomenon known as entanglement. We will discuss paradigms for harnessing entanglement to solve hitherto intractable computational problems or to push the precision of sensors to their fundamental quantum mechanical limits. We will also examine challenges that physicists and engineers are tackling in the laboratory today to enable the quantum technologies of the future.
Terms: Spr | Units: 3 | UG Reqs: WAY-FR, WAY-SMA
Instructors: ; Manoharan, H. (PI)

PHYSICS 61: Mechanics and Special Relativity

(First in a three-part series: PHYSICS 61, PHYSICS 71, PHYSICS 81.) This course covers Einstein's special theory of relativity and Newtonian mechanics at a level appropriate for students with a strong high school mathematics and physics background, who are contemplating a major in Physics or Engineering Physics or are interested in a rigorous treatment of physics. Postulates of special relativity, simultaneity, time dilation, length contraction, the Lorentz transformation, the space-time invariant, causality, relativistic momentum and energy, and invariant mass. Central forces, friction, contact forces, linear restoring forces. Momentum, work, energy, collisions. Angular momentum, torque, center of mass, moment of inertia, precession. Conserved quantities. Uses the language of vectors and multivariable calculus. Requirements to enroll in the course: Completion of Physics Placement Diagnostic and/or completion of at least one course in PHYSICS 20 or 40 series. Completion of or co-enrollment in MATH 51 or MATH 61CM or MATH 61DM. Prerequisites: mechanics at the level of PHYSICS 41 or score of 5 on AP Physics C Mechanics or equivalent; calculus at the level of MATH 21 or score of 5 on AP Calculus BC or equivalent.
Terms: Aut | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA

PHYSICS 71: Quantum and Thermal Physics

(Second in a three-part series: PHYSICS 61, PHYSICS 71, PHYSICS 81.) This course introduces the foundations of quantum mechanics and thermodynamics to students with a strong high school mathematics and physics background, who are contemplating a major in Physics or Engineering Physics or are interested in a rigorous treatment of physics. Topics related to quantum mechanics include atoms, electrons, and nuclei. Experimental evidence for physics that is not explained by classical mechanics and E&M. Quantization of light, Planck's constant. Photoelectric effect, Compton and Bragg scattering. Bohr model, atomic spectra. Matter waves, wave packets, interference. Fourier analysis and transforms Heisenberg uncertainty relationships. Particle-in-a-box, simple harmonic oscillator, barrier penetration, tunneling. Topics related to thermodynamics: limitations of classical mechanics in describing systems with a very large number of particles. Ideal gas, equipartition, heat capacity, the definition of temperature, entropy. A brief introduction to kinetic theory and statistical mechanics. Maxwell speed distribution, ideal gas in a box. Laws of thermodynamics. Cycles, heat engines, free energy. Prerequisites: Physics 61 and (Math 51 or Math 61CM). Corequisite: Physics 43 or equivalent (e.g. AP Physics C E&M), MATH 52 or 62CM. This course was offered as PHYSICS 65 prior to Academic Year 2022-2023.
Terms: Win | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA
Instructors: ; Manoharan, H. (PI)

PHYSICS 81: Electricity and Magnetism Using Special Relativity and Vector Calculus

(Third in a three-part series: PHYSICS 61, PHYSICS 71, PHYSICS 81.) This course recasts the foundations of electricity and magnetism in a way that will surprise, delight, and challenge students who have already encountered the subject at a college or AP level. Suitable for students contemplating a major in Physics or Engineering Physics, those interested in a rigorous treatment of physics as a foundation for other disciplines, or those curious about powerful concepts like transformations, symmetry, and conservation laws. Electrostatics and Gauss' law. Electric potential, electric field, conductors, image charges. Electric currents, DC circuits. Moving charges, magnetic field as a consequence of special relativity applied to electrostatics, Ampere's law. Solenoids, transformers, induction, AC circuits, resonance. Displacement current, Maxwell's equations. Electromagnetic waves. Throughout, we'll see the objects and theorems of vector calculus become manifest in charges, currents, and electromagnetic fields. Prerequisite: A score of 5 on the AP Physics C E&M exam or Physics 43; Physics 61; and Math 52 or Math 62CM. Recommended prerequisite: Physics 71. Corequisite: Math 53 or Math 63CM. This course was offered as PHYSICS 63 prior to Academic Year 2022-2023.
Terms: Spr | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA

PHYSICS 110: Advanced Mechanics (PHYSICS 210)

Lagrangian and Hamiltonian mechanics. Principle of least action, Euler-Lagrange equations. Small oscillations and beyond. Symmetries, canonical transformations, Hamilton-Jacobi theory, action-angle variables. Introduction to classical field theory. Selected other topics, including nonlinear dynamical systems, attractors, chaotic motion. Undergraduates register for Physics 110 (4 units). Graduates register for Physics 210 (3 units). Prerequisites: MATH 131P or PHYSICS 111. Recommended prerequisite: PHYSICS 130.
Terms: Aut | Units: 3-4 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA

PHYSICS 112: Mathematical Methods for Physics

The course will focus on the theory of functions of a complex variable - with broad implications in many areas of physics. As time allows, we will also cover the basics of group theory and the theory of group representations, with focus on symmetry groups that arise in various physical settings. Prerequisites: MATH 53 or equivalent and Physics 111 or the equivalent.
Terms: Spr | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-FR | Repeatable 3 times (up to 12 units total)

PHYSICS 113: Computational Physics

Numerical methods for solving problems in mechanics, astrophysics, electromagnetism, quantum mechanics, and statistical mechanics. Methods include numerical integration; solutions of ordinary and partial differential equations; solutions of the diffusion equation, Laplace's equation, and Poisson's equation with various methods; statistical methods including Monte Carlo techniques; matrix methods and eigenvalue problems. A short introduction to Python, which is used for class examples and active learning notebooks. Independent class projects allow deep explorations of course topics and make up a significant component of the course grade. No prerequisites but some previous programming experience is advisable.
Terms: Spr | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-AQR, WAY-FR

PHYSICS 120: Intermediate Electricity and Magnetism I

Vector analysis. Electrostatic fields, including boundary-value problems and multipole expansion. Dielectrics, static and variable magnetic fields, magnetic materials. Maxwell's equations. Prerequisites: PHYSICS 81; MATH 52 and MATH 53. Pre- or corequisite: PHYS 111 or MATH 131P or MATH 173 or Math 220.
Terms: Win | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA

PHYSICS 130: Quantum Mechanics I

The origins of quantum mechanics and wave mechanics. Schr¿dinger equation and solutions for one-dimensional systems. Commutation relations. Generalized uncertainty principle. Time-energy uncertainty principle. Separation of variables and solutions for three-dimensional systems; application to a hydrogen atom. Spherically symmetric potentials and angular momentum eigenstates. Spin angular momentum. Addition of angular momentum. Prerequisites: (PHYSICS 65 or PHYSICS 70 or PHYSICS 71) and (PHYSICS 111 or MATH 131P or MATH 173 or MATH 220) and PHYSICS 120.
Terms: Spr | Units: 4 | UG Reqs: GER: DB-NatSci, WAY-FR, WAY-SMA

POLISCI 153: Strategy: Introduction to Game Theory (POLISCI 354)

This course provides an introduction to strategic reasoning. We discuss ideas such as the commitment problem, credibility in signaling, cheap talk, moral hazard and adverse selection. Concepts are developed through games played in class, and applied to politics, business and everyday life.
Terms: Spr, Sum | Units: 4-5 | UG Reqs: WAY-FR

POLISCI 153Z: Strategy: Introduction to Game Theory

This course provides an introduction to strategic reasoning. We discuss ideas such as the commitment problem, credibility in signaling, cheap talk, moral hazard and adverse selection. Concepts are developed through games played in class, and applied to politics, business and everyday life.
Last offered: Summer 2022 | Units: 4 | UG Reqs: 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

PSYCH 35: Minds and Machines (CS 24, LINGUIST 35, PHIL 99, SYMSYS 1, SYMSYS 200)

(Formerly SYMSYS 100). An overview of the interdisciplinary study of cognition, information, communication, and language, with an emphasis on foundational issues: What are minds? What is computation? What are rationality and intelligence? Can we predict human behavior? Can computers be truly intelligent? How do people and technology interact, and how might they do so in the future? Lectures focus on how the methods of philosophy, mathematics, empirical research, and computational modeling are used to study minds and machines. Students must take this course before being approved to declare Symbolic Systems as a major. All students interested in studying Symbolic Systems are urged to take this course early in their student careers. The course material and presentation will be at an introductory level, without prerequisites. If you have any questions about the course, please email symsys1staff@gmail.com.
Terms: Aut, Win, Sum | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR

STATS 48N: Riding the Data Wave (BIODS 48N)

Imagine collecting a bit of your saliva and sending it in to one of the personalized genomics company: for very little money you will get back information about hundreds of thousands of variable sites in your genome. Records of exposure to a variety of chemicals in the areas you have lived are only a few clicks away on the web; as are thousands of studies and informal reports on the effects of different diets, to which you can compare your own. What does this all mean for you? Never before in history humans have recorded so much information about themselves and the world that surrounds them. Nor has this data been so readily available to the lay person. Expression as "data deluge'' are used to describe such wealth as well as the loss of proper bearings that it often generates. How to summarize all this information in a useful way? How to boil down millions of numbers to just a meaningful few? How to convey the gist of the story in a picture without misleading oversimplifications? To answer these questions we need to consider the use of the data, appreciate the diversity that they represent, and understand how people instinctively interpret numbers and pictures. During each week, we will consider a different data set to be summarized with a different goal. We will review analysis of similar problems carried out in the past and explore if and how the same tools can be useful today. We will pay attention to contemporary media (newspapers, blogs, etc.) to identify settings similar to the ones we are examining and critique the displays and summaries there documented. Taking an experimental approach, we will evaluate the effectiveness of different data summaries in conveying the desired information by testing them on subsets of the enrolled students.
Last offered: Autumn 2020 | Units: 3 | UG Reqs: 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 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

SYMSYS 1: Minds and Machines (CS 24, LINGUIST 35, PHIL 99, PSYCH 35, SYMSYS 200)

(Formerly SYMSYS 100). An overview of the interdisciplinary study of cognition, information, communication, and language, with an emphasis on foundational issues: What are minds? What is computation? What are rationality and intelligence? Can we predict human behavior? Can computers be truly intelligent? How do people and technology interact, and how might they do so in the future? Lectures focus on how the methods of philosophy, mathematics, empirical research, and computational modeling are used to study minds and machines. Students must take this course before being approved to declare Symbolic Systems as a major. All students interested in studying Symbolic Systems are urged to take this course early in their student careers. The course material and presentation will be at an introductory level, without prerequisites. If you have any questions about the course, please email symsys1staff@gmail.com.
Terms: Aut, Win, Spr, Sum | Units: 4 | UG Reqs: GER:DB-SocSci, WAY-FR
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