Analysis and design of modern power and energy systems. Provides a holistic overview of how electricity is produced, delivered, and managed, including power electronics, machines and supply technologies, grid optimization and control, grid planning, markets, and regulation. Students apply the material covered to consider critical challenges associated with global energy systems, including electrification, decarbonization, resilience, and load growth. Students taking graduate version complete different assignments.

Analysis and design of modern power and energy systems. Provides a holistic overview of how electricity is produced, delivered, and managed, including power electronics, machines and supply technologies, grid optimization and control, grid planning, markets, and regulation. Students apply the material covered to consider critical challenges associated with global energy systems, including electrification, decarbonization, resilience, and load growth. Students taking graduate version complete additional assignments.

Introduces the design and construction of power electronic circuits and motor drives. Laboratory exercises include the construction of drive circuitry for an electric go-cart, flash strobes, computer power supplies, three-phase inverters for AC motors, and resonant drives for lamp ballasts and induction heating. Basic electric machines introduced include DC, induction, and permanent magnet motors, with drive considerations. Provides instruction in written and oral communication. Students taking independent inquiry version 6.2221 expand the scope of their laboratory project. Enrollment limited.

Introduces the design and construction of power electronic circuits and motor drives. Laboratory exercises include the construction of drive circuitry for an electric go-cart, flash strobes, computer power supplies, three-phase inverters for AC motors, and resonant drives for lamp ballasts and induction heating. Basic electric machines introduced include DC, induction, and permanent magnet motors, with drive considerations. Provides instruction in written and oral communication. To satisfy the independent inquiry component of this subject, students expand the scope of their laboratory project. Enrollment limited.

Introduction to the quantum mechanics needed to engineer quantum systems for computation, communication, and sensing. Topics include: motivation for quantum engineering, qubits and quantum gates, rules of quantum mechanics, mathematical background, quantum electrical circuits and other physical quantum systems, harmonic and anharmonic oscillators, measurement, the Schrödinger equation, noise, entanglement, benchmarking, quantum communication, and quantum algorithms. No prior experience with quantum mechanics is assumed.

Study of electromagnetics and electromagnetic energy conversion leading to an understanding of devices, including electromagnetic sensors, actuators, motors and generators. Quasistatic Maxwell's equations and the Lorentz force law. Studies of the quasistatic fields and their sources through solutions of Poisson's and Laplace's equations. Boundary conditions and multi-region boundary-value problems. Steady-state conduction, polarization, and magnetization. Charge conservation and relaxation, and magnetic induction and diffusion. Extension to moving materials. Electric and magnetic forces and force densities derived from energy, and stress tensors. Extensive use of engineering examples. Students taking graduate version complete additional assignments.

Advanced topics in analysis, design, manufacturing, and control of power electronics. Topics include: architectures and topologies for power electronics including soft-switched, resonant, high-frequency and switched-capacitor circuits; high-frequency power magnetics; advanced modeling and control methods; power components, devices, and materials; manufacturing methods; low-noise circuit design and filtering; circuit theory applications to power electronics; emerging technologies and applications in power conversion.

Treatment of electromechanical transducers, rotating and linear electric machines. Lumped-parameter electromechanics. Power flow using Poynting's theorem, force estimation using the Maxwell stress tensor and Principle of virtual work. Development of analytical techniques for predicting device characteristics: energy conversion density, efficiency; and of system interaction characteristics: regulation, stability, controllability, and response. Use of electric machines in drive systems. Problems taken from current research.

Explores electromagnetic phenomena in modern applications, including wireless and optical communications, circuits, computer interconnects and peripherals, microwave communications and radar, antennas, sensors, micro-electromechanical systems, and power generation and transmission. Fundamentals include quasistatic and dynamic solutions to Maxwell's equations; waves, radiation, and diffraction; coupling to media and structures; guided and unguided waves; modal expansions; resonance; acoustic analogs; and forces, power, and energy.

Introduction to fundamental concepts and techniques of optics, photonics, and fiber optics, aimed at developing skills for independent research. Topics include: Review of Maxwell's equations, light propagation, reflection and transmission, dielectric mirrors and filters. Scattering matrices, interferometers, and interferometric measurement. Fresnel and Fraunhoffer diffraction theory. Lenses, optical imaging systems, and software design tools. Gaussian beams, propagation and resonator design. Optical waveguides, optical fibers and photonic devices for encoding and detection. Discussion of research operations / funding and professional development topics. The course reviews and introduces mathematical methods and techniques, which are fundamental in optics and photonics, but also useful in many other engineering specialties.

Covers the fundamentals of optics and the interaction of light and matter, leading to devices such as light emitting diodes, optical amplifiers, and lasers. Topics include classical ray, wave, beam, and Fourier optics; Maxwell's electromagnetic waves; resonators; quantum theory of photons; light-matter interaction; laser amplification; lasers; and semiconductors optoelectronics. Students taking graduate version complete additional assignments.

Principles of operation and applications of optical imaging devices and systems (includes optical signal generation, transmission, detection, storage, processing and display). Topics include review of the basic properties of electromagnetic waves; coherence and interference; diffraction and holography; Fourier optics; coherent and incoherent imaging and signal processing systems; optical properties of materials; lasers and LEDs; electro-optic and acousto-optic light modulators; photorefractive and liquid-crystal light modulation; spatial light modulators and displays; near-eye and projection displays, holographic and other 3-D display schemes, photodetectors; 2-D and 3-D optical storage technologies; adaptive optical systems; role of optics in next-generation computers. Requires a research paper on a specific contemporary optical imaging topic. Recommended prerequisite: 8.03.

Elementary quantum mechanics and statistical physics. Introduces applied quantum physics. Emphasizes experimental basis for quantum mechanics. Applies Schrodinger's equation to the free particle, tunneling, the harmonic oscillator, and hydrogen atom. Variational methods. Elementary statistical physics; Fermi-Dirac, Bose-Einstein, and Boltzmann distribution functions. Simple models for metals, semiconductors, and devices such as electron microscopes, scanning tunneling microscope, thermonic emitters, atomic force microscope, and more. Some familiarity with continuous time Fourier transforms recommended.

Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.

Continuation of 6.6510 emphasizing applications-related physical issues in solids. Topics include: electronic structure and energy band diagrams of semiconductors, metals, and insulators; Fermi surfaces; dynamics of electrons under electric and magnetic fields; classical diffusive transport phenomena such as electrical and thermal conduction and thermoelectric phenomena; quantum transport in tunneling and ballistic devices; optical properties of metals, semiconductors, and insulators; impurities and excitons; photon-lattice interactions; Kramers-Kronig relations; optoelectronic devices based on interband and intersubband transitions; magnetic properties of solids; exchange energy and magnetic ordering; magneto-oscillatory phenomena; quantum Hall effect; superconducting phenomena and simple models.

Contemporary understanding of imaging is computational: encoding onto a form of radiation the information about a physical object, transferring the radiation through the imaging system, converting it to a digital signal, and computationally decoding and presenting the information to the user. This class introduces a unified formulation of computational imaging systems as a three-round "learning spiral": the first two rounds, instructors describe the physical and algorithmic parts in two exemplary imaging systems. The third round, students conduct themselves as the class project on an imaging system of their choice. The undergraduate and graduate versions share lectures but have different recitations. Throughout the term, we also conduct optional "clinics" to even out background knowledge of linear algebra, optimization, and computational imaging-related programming best practices for students of diverse disciplinary backgrounds.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.

Concepts, principles, and methods for planning with imperfect knowledge. Topics include state estimation, planning in information space, partially observable Markov decision processes, reinforcement learning and planning with uncertain models. Students will develop an understanding of how different planning algorithms and solutions techniques are useful in different problem domains. Previous coursework in artificial intelligence and state estimation strongly recommended.

Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval.

Introduction to fundamental principles and applications of artificial intelligence (AI) in medicine and medical research. Introduces foundational concepts in machine learning as they pertain to clinical decision support systems, personalized medicine, and advanced computational methods for drug optimization and protein folding. Stresses the interplay of clinical reasoning, risk stratification, and the design of novel therapeutics. Problem sets integrate theoretical knowledge and practical applications based on concrete problems in both medical and pharmaceutical science. Students taking graduate version complete additional assignments.

Introduction to fundamental principles and applications of artificial intelligence (AI) in medicine and medical research. Introduces foundational concepts in machine learning as they pertain to clinical decision support systems, personalized medicine, and advanced computational methods for drug optimization and protein folding. Stresses the interplay of clinical reasoning, risk stratification, and the design of novel therapeutics. Problem sets integrate theoretical knowledge and practical applications based on concrete problems in both medical and pharmaceutical science. Students taking graduate version complete additional assignments.

Introduction to fundamentals of modern data-driven decision-making frameworks, such as causal inference and hypothesis testing in statistics as well as supervised and reinforcement learning in machine learning. Explores how these frameworks are being applied in various societal contexts, including criminal justice, healthcare, finance, and social media. Emphasis on pinpointing the non-obvious interactions, undesirable feedback loops, and unintended consequences that arise in such settings. Enables students to develop their own principled perspective on the interface of data-driven decision making and society. Students taking graduate version complete additional assignments.

Introduction to computational theories of human cognition. Focus on principles of inductive learning and inference, and the representation of knowledge. Computational frameworks covered include Bayesian and hierarchical Bayesian models; probabilistic graphical models; nonparametric statistical models and the Bayesian Occam's razor; sampling algorithms for approximate learning and inference; and probabilistic models defined over structured representations such as first-order logic, grammars, or relational schemas. Applications to understanding core aspects of cognition, such as concept learning and categorization, causal reasoning, theory formation, language acquisition, and social inference. Graduate students complete a final project.

Surveys decision making methods used to create highly autonomous systems and decision aids. Applies models, principles and algorithms taken from artificial intelligence and operations research. Focuses on planning as state-space search, including uninformed, informed and sampling-based search, activity and motion planning, probabilistic and adversarial planning, Markov models and decision processes, Bayesian filtering, reinforcement learning, and machine learning. Includes methods for satisfiability and optimization of logical and finite domain constraints, graphical models, and methods for search, inference, and conflict-learning. Students taking graduate version complete additional assignments.

Introduces the fundamental algorithmic approaches for creating robot systems that can autonomously manipulate physical objects in unstructured environments such as homes and restaurants. Topics include perception (including approaches based on deep learning and approaches based on 3D geometry), planning (robot kinematics and trajectory generation, collision-free motion planning, task-and-motion planning, and planning under uncertainty), as well as dynamics and control (both model-based and learning-based). Students taking graduate version complete additional assignments. Students engage in extensive written and oral communication exercises.

Introduces the fundamental algorithmic approaches for creating robot systems that can autonomously manipulate physical objects in unstructured environments such as homes and restaurants. Topics include perception (including approaches based on deep learning and approaches based on 3D geometry), planning (robot kinematics and trajectory generation, collision-free motion planning, task-and-motion planning, and planning under uncertainty), as well as dynamics and control (both model-based and learning-based. Students taking graduate version complete additional assignments.

Introduces computational aspects of computer-aided design and manufacturing. Explores relevant methods in the context of additive manufacturing (e.g., 3D printing). Topics include computer graphics (geometry modeling, solid modeling, procedural modeling), physically-based simulation (kinematics, finite element method), 3D scanning/geometry processing, and an overview of 3D fabrication methods. Exposes students to the latest research in computational fabrication. Students taking the graduate version complete additional assignments.

Introduces the study of computational models of human language, covering classical statistical methods, representation learning, and modern deep network models through the lens of language modeling. Students complete a substantial final project, applying or extending these methods. Instruction and practice in oral and written communication provided.

Classical mechanics in a computational framework, Lagrangian formulation, action, variational principles, and Hamilton's principle. Conserved quantities, Hamiltonian formulation, surfaces of section, chaos, and Liouville's theorem. Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems. Invariant curves and cantori. Nonlinear resonances, resonance overlap and transition to chaos. Symplectic integration. Adiabatic invariants. Applications to simple physical systems and solar system dynamics. Extensive use of computation to capture methods, for simulation, and for symbolic analysis. Programming experience required.

Introduction to algorithms and computational complexity for high-dimensional statistical inference problems, with focus on provable polynomial-time guarantees. Covers modern algorithm design techniques via convex programming and Sum of Squares method; graphical models as a language to describe complex but tractable high-dimensional learning problems and associated learning algorithms; and basics of complexity for statistical problems, including statistical query and low-degree lower bounds and reductions. Mathematical maturity is recommended.

Introduces efficient deep learning computing techniques that enable powerful deep learning applications on resource-constrained devices. Topics include model compression, pruning, quantization, neural architecture search, distributed training, data/model parallellism, gradient compression, on-device fine-tuning. It also introduces application-specific acceleration techniques for video recognition, point cloud, and generative AI (diffusion model, LLM). Students will get hands-on experience accelerating deep learning applications with an open-ended design project.

Broad survey of numerical methods used in graphics, vision, robotics, machine learning, and scientific computing, with emphasis on incorporating these algorithms into downstream applications. Focuses on challenges that arise in applying/implementing numerical algorithms and recognizing which numerical methods are relevant to different applications. Topics include numerical linear algebra (QR, LU, SVD matrix factorizations; eigenvectors; conjugate gradients), ordinary and partial differential equations (divided differences, finite element method), and nonlinear systems and optimization (gradient descent, Newton/quasi-Newton methods, gradient-free optimization, constrained optimization). Examples and case studies drawn from the computer science and machine learning literatures.

Presents mathematical statistics as a formal language for reasoning about data and uncertainty. Introduction to the basic framework of statistical decision theory, along with core concepts such as sufficiency, Bayes and minimax optimality of statistical procedures, with applications to optimal estimation, hypothesis testing, and prediction. Discussion topics include causality, multiple hypothesis testing, nonparametric and semiparametric statistics, and results for model misspecification. Targeted to students interested in statistical and machine learning research, with an emphasis on proofs and fundamental understanding.

Introduction to computational aspects of statistical inference via probabilistic graphical models. Directed and undirected graphical models, and factor graphs, over discrete and Gaussian distributions; hidden Markov models, linear dynamical systems. Sum-product and junction tree algorithms; forward-backward algorithm, Kalman filtering and smoothing. Min-sum and Viterbi algorithms. Variational methods, mean-field theory, and loopy belief propagation. Sampling methods; Glauber dynamics and mixing time analysis. Parameter structure learning for graphical models; Baum-Welch and Chow-Liu algorithms. Selected topics such as causal inference, particle filtering, restricted Boltzmann machines, and graph neural networks.

Principles, techniques, and algorithms in machine learning from the point of view of statistical inference; representation, generalization, and model selection; and methods such as linear/additive models, active learning, boosting, support vector machines, non-parametric Bayesian methods, hidden Markov models, Bayesian networks, and convolutional and recurrent neural networks. Recommended prerequisite: 6.3900 or other previous experience in machine learning. Enrollment may be limited.

Covers foundations and recent advances in statistical machine learning theory, with the dual goals of providing students with the theoretical knowledge to use machine learning and preparing more advanced students to contribute to progress in the field. The content is roughly divided into three parts. The first part is about classical regularization, margin, stochastic gradient methods, overparametrization, implicit regularization, and stability. The second part is about deep networks: approximation and optimization theory plus roots of generalization. The third part is about the connections between learning theory and the brain. Occasional talks by leading researchers on advanced research topics. Emphasis on current research topics.

Fundamentals of deep learning, including both theory and applications. Topics include neural net architectures (MLPs, CNNs, RNNs, graph nets, transformers), geometry and invariances in deep learning, backpropagation and automatic differentiation, learning theory and generalization in high-dimensions, and applications to computer vision, natural language processing, and robotics.

Examines the foundations of multi-agent systems from a combined game-theoretic, optimization, and learning-theoretic perspective. Studies settings involving multiple learning agents with distinct objectives, where standard training methods may induce nonstationary or unstable behaviors. Introduces fundamental models including matrix games, stochastic games, and imperfect-information games. Studies modern settings deviating from standard models wherein strategies and utilities are represented using deep neural networks. Presents equilibrium concepts, equilibrium computation methods, and equilibrium learning methods. Studies the complexity of solution concepts. Discusses applications in machine learning, including multi-agent reinforcement learning, adversarial learning, AI-AI interactions, and strategic decision-making systems underlying game-playing agents in domains such as Go, poker, Diplomacy, and Stratego.

Explores the problem of designing a general-purpose intelligent robot, addressing challenges in robot design, such as long-horizon decision making, continuous action spaces, partial observability, and nonstationary and adversarial environments. Students read foundational papers on each challenge as well as state-of-the-art research results. Instruction emphasizes cutting across disciplinary boundaries. Topics include control theory, machine learning, planning, decision making, and hardware design. Students present papers, engage in discussion, and complete reading assignments, concluding with an open-ended research project. One of 6.4100, 6.4120, 6.8110 or 6.7920 is recommended but not required.

Introduces computational aspects of computer-aided design and manufacturing. Explores relevant methods in the context of additive manufacturing (e.g., 3D printing). Topics include computer graphics (geometry modeling, solid modeling, procedural modeling), physically-based simulation (kinematics, finite element method), 3D scanning/geometry processing, and an overview of 3D fabrication methods. Exposes students to the latest research in computational fabrication. Students taking graduate version complete additional assignments.

Introduces the study of human language from a computational perspective, including syntactic, semantic and discourse processing models. Emphasizes machine learning methods and algorithms. Uses these methods and models in applications such as syntactic parsing, information extraction, statistical machine translation, dialogue systems. Instruction and practice in oral and written communication provided.

Examines the practical considerations for operationalizing machine learning in healthcare settings, with a focus on robust, private, and fair modeling using real retrospective healthcare data. Explores the pre-modeling creation of dataset pipeline to the post-modeling "implementation science," which addresses how models are incorporated at the point of care. Students complete three homework assignments (one each in machine learning, visualization, and implementation), followed by a project proposal and presentation. Students gain experience in dataset creation and curation, machine learning training, visualization, and deployment considerations that target utility and clinical value. Students partner with computer scientists, engineers, social scientists, and clinicians to better appreciate the multidisciplinary nature of data science.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.

Introduction to computational theories of human cognition. Focuses on principles of inductive learning and inference, and the representation of knowledge. Computational frameworks include Bayesian and hierarchical Bayesian models, probabilistic graphical models, nonparametric statistical models and the Bayesian Occam's razor, sampling algorithms for approximate learning and inference, and probabilistic models defined over structured representations such as first-order logic, grammars, or relational schemas. Applications to understanding core aspects of cognition, such as concept learning and categorization, causal reasoning, theory formation, language acquisition, and social inference. Graduate students complete a final project.

Introduction to fundamental principles and applications of artificial intelligence (AI) in medicine and medical research. Introduces foundational concepts in machine learning as they pertain to clinical decision support systems, personalized medicine, and advanced computational methods for drug optimization and protein folding. Stresses the interplay of clinical reasoning, risk stratification, and the design of novel therapeutics. Problem sets integrate theoretical knowledge and practical applications based on concrete problems in both medical and pharmaceutical science. Students taking graduate version complete additional assignments.

Introduction to fundamental principles and applications of artificial intelligence (AI) in medicine and medical research. Introduces foundational concepts in machine learning as they pertain to clinical decision support systems, personalized medicine, and advanced computational methods for drug optimization and protein folding. Stresses the interplay of clinical reasoning, risk stratification, and the design of novel therapeutics. Problem sets integrate theoretical knowledge and practical applications based on concrete problems in both medical and pharmaceutical science. Students taking graduate version complete additional assignments.

Application of the principles of energy and mass flow to major human organ systems. Anatomical, physiological and clinical features of the cardiovascular, respiratory and renal systems. Mechanisms of regulation and homeostasis. Systems, features and devices that are most illuminated by the methods of physical sciences and engineering models. Required laboratory work includes animal studies. Students taking graduate version complete additional assignments.

Application of the principles of energy and mass flow to major human organ systems. Anatomical, physiological and clinical features of the cardiovascular, respiratory and renal systems. Mechanisms of regulation and homeostasis. Systems, features and devices that are most illuminated by the methods of physical sciences and engineering models. Required laboratory work includes animal studies. Students taking graduate version complete additional assignments.

Molecular diffusion, diffusion-reaction, conduction, convection in biological systems; fields in heterogeneous media; electrical double layers; Maxwell stress tensor, electrical forces in physiological systems. Fluid and solid continua: equations of motion useful for porous, hydrated biological tissues. Case studies of membrane transport, electrode interfaces, electrical, mechanical, and chemical transduction in tissues, convective-diffusion/reaction, electrophoretic, electroosmotic flows in tissues/MEMs, and ECG. Electromechanical and physicochemical interactions in cells and biomaterials; musculoskeletal, cardiovascular, and other biological and clinical examples. Prior undergraduate coursework in transport recommended.

See description for 6.8701. Additionally examines recent publications in the areas covered, with research-style assignments. A more substantial final project is expected, which can lead to a thesis and publication.

Covers the algorithmic and machine learning foundations of computational biology, combining theory with practice. Principles of algorithm design, influential problems and techniques, and analysis of large-scale biological datasets. Topics include (a) genomes: sequence analysis, gene finding, RNA folding, genome alignment and assembly, database search; (b) networks: gene expression analysis, regulatory motifs, biological network analysis; (c) evolution: comparative genomics, phylogenetics, genome duplication, genome rearrangements, evolutionary theory. These are coupled with fundamental algorithmic techniques including: dynamic programming, hashing, Gibbs sampling, expectation maximization, hidden Markov models, stochastic context-free grammars, graph clustering, dimensionality reduction, Bayesian networks.

Introduces the basics of synthetic biology, including quantitative cellular network characterization and modeling. Considers the discovery and genetic factoring of useful cellular activities into reusable functions for design. Emphasizes the principles of biomolecular system design and diagnosis of designed systems. Illustrates cutting-edge applications in synthetic biology and enhances skills in analysis and design of synthetic biological applications. Students taking graduate version complete additional assignments.

Introduces the basics of synthetic biology, including quantitative cellular network characterization and modeling. Considers the discovery and genetic factoring of useful cellular activities into reusable functions for design. Emphasizes the principles of biomolecular system design and diagnosis of designed systems. Illustrates cutting-edge applications in synthetic biology and enhances skills in analysis and design of synthetic biological applications. Students taking graduate version complete additional assignments.

Studies information processing performance of the human auditory system in relation to current physiological knowledge. Examines mathematical models for the quantification of auditory-based behavior and the relation between behavior and peripheral physiology, reflecting the tono-topic organization and stochastic responses of the auditory system. Mathematical models of psychophysical relations, incorporating quantitative knowledge of physiological transformations by the peripheral auditory system.

Examines the practical considerations for operationalizing machine learning in healthcare settings, with a focus on robust, private, and fair modeling using real retrospective healthcare data. Explores the pre-modeling creation of dataset pipeline to the post-modeling "implementation science," which addresses how models are incorporated at the point of care. Students complete three homework assignments (one each in machine learning, visualization, and implementation), followed by a project proposal and presentation. Students gain experience in dataset creation and curation, machine learning training, visualization, and deployment considerations that target utility and clinical value. Students partner with computer scientists, engineers, social scientists, and clinicians to better appreciate the multidisciplinary nature of data science.

Lab-intensive subject that investigates digital systems with a focus on FPGAs. Lectures and labs cover logic, flip flops, counters, timing, synchronization, finite-state machines, digital signal processing, communication protocols, and modern sensors. Prepares students for the design and implementation of a large-scale final project of their choice: games, music, digital filters, wireless communications, video, or graphics. Extensive use of System/Verilog for describing and implementing and verifying digital logic designs.

Fosters deep understanding and intuition that is crucial in innovating analog circuits and optimizing the whole system in bipolar junction transistor (BJT) and metal oxide semiconductor (MOS) technologies. Covers both theory and real-world applications of basic amplifier structures, operational amplifiers, temperature sensors, bandgap references. Covers topics such as noise, linearity and stability. Homework and labs give students access to CAD/EDA tools to design and analyze analog circuits. Provides practical experience through lab exercises, including a broadband amplifier design and characterization. Students taking graduate version complete additional assignments.

Fosters deep understanding and intuition that is crucial in innovating analog circuits and optimizing the whole system in bipolar junction transistor (BJT) and metal oxide semiconductor (MOS) technologies. Covers both theory and real-world applications of basic amplifier structures, operational amplifiers, temperature sensors, bandgap references. Covers topics such as noise, linearity and stability. Homework and labs give students access to CAD/EDA tools to design and analyze analog circuits. Provides practical experience through lab exercises, including a broadband amplifier design and characterization. Students taking graduate version complete additional assignments.

 Hands-on introduction to the design and construction of power electronic circuits and motor drives. Laboratory exercises (shared with 6.131 and 6.1311) include the construction of drive circuitry for an electric go-cart, flash strobes, computer power supplies, three-phase inverters for AC motors, and resonant drives for lamp ballasts and induction heating. Basic electric machines introduced including DC, induction, and permanent magnet motors, with drive considerations. Students taking graduate version complete additional assignments and an extended final project. Enrollment limited.

A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression, and identification). Concepts are introduced with lectures and online problems, and then mastered during weekly labs. In lab, students model, design, test, and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g., optimizing thrust-driven positioners or stabilizing magnetic levitators). Students taking graduate version complete additional problems and labs.

A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression and identification). Concepts are introduced with lectures and on-line problems, and then mastered during weekly labs. In lab, students model, design, test and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g. optimizing thrust-driven positioners or stabilizing magnetic levitators). Students in the graduate version complete additional problems and labs.

Device and circuit level optimization of digital building blocks. Circuit design styles for logic, arithmetic, and sequential blocks. Estimation and minimization of energy consumption. Interconnect models and parasitics, device sizing and logical effort, timing issues (clock skew and jitter), and active clock distribution techniques. Memory architectures, circuits (sense amplifiers), and devices. Evaluation of how design choices affect tradeoffs across key metrics including energy consumption, speed, robustness, and cost. Extensive use of modern design flow and EDA/CAD tools for the analysis and design of digital building blocks and digital VLSI design for labs and design projects

Principles and techniques of high-speed integrated circuits used in wireless/wireline data links and remote sensing. On-chip passive component design of inductors, capacitors, and antennas. Analysis of distributed effects, such as transmission line modeling, S-parameters, and Smith chart. Transceiver architectures and circuit blocks, which include low-noise amplifiers, mixers, voltage-controlled oscillators, power amplifiers, and frequency dividers. Involves IC/EM simulation and laboratory projects.

Highlights common principles that permeate the functioning of diverse technological, economic and social networks. Utilizes three sets of tools for analyzing networks -- random graph models, optimization, and game theory -- to study informational and learning cascades; economic and financial networks; social influence networks; formation of social groups; communication networks and the Internet; consensus and gossiping; spread and control of epidemics; control and use of energy networks; and biological networks. Students taking graduate version complete additional assignments.

Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.

Covers communications by progressing through signal representation, sampling, quantization, compression, modulation, coding and decoding, medium access control, and queueing and principles of protocols. By providing simplified proofs, seeks to present an integrated, systems-level view of networking and communications while laying the foundations of analysis and design. Lectures are offered online; in-class time is dedicated to recitations, exercises, and weekly group labs. Homework exercises are based on theoretical derivation and software implementation. Students taking graduate version complete additional assignments.

Introduction to modern heterogeneous networks and the provision of heterogeneous services. Architectural principles, analysis, algorithmic techniques, performance analysis, and existing designs are developed and applied to understand current problems in network design and architecture. Begins with basic principles of networking. Emphasizes development of mathematical and algorithmic tools; applies them to understanding network layer design from the performance and scalability viewpoint. Concludes with network management and control, including the architecture and performance analysis of interconnected heterogeneous networks. Provides background and insight to understand current network literature and to perform research on networks with the aid of network design projects.

Introduction to design, analysis, and fundamental limits of wireless transmission systems. Wireless channel and system models; fading and diversity; resource management and power control; multiple-antenna and MIMO systems; space-time codes and decoding algorithms; multiple-access techniques and multiuser detection; broadcast codes and precoding; cellular and ad-hoc network topologies; OFDM and ultrawideband systems; architectural issues.

Provides an introduction to data networks with an analytic perspective, using wireless networks, satellite networks, optical networks, the internet and data centers as primary applications. Presents basic tools for modeling and performance analysis. Draws upon concepts from stochastic processes, queuing theory, and optimization.

Mathematical definitions of information measures, convexity, continuity, and variational properties. Lossless source coding; variable-length and block compression; Slepian-Wolf theorem; ergodic sources and Shannon-McMillan theorem. Hypothesis testing, large deviations and I-projection. Fundamental limits of block coding for noisy channels: capacity, dispersion, finite blocklength bounds. Coding with feedback. Joint source-channel problem. Rate-distortion theory, vector quantizers. Advanced topics include Gelfand-Pinsker problem, multiple access channels, broadcast channels (depending on available time).

Introduces fundamentals of information theory and its applications to contemporary problems in statistics, machine learning, and computer science. A thorough study of information measures, including Fisher information, f-divergences, their convex duality, and variational characterizations. Covers information-theoretic treatment of inference, hypothesis testing and large deviations, universal compression, channel coding, lossy compression, and strong data-processing inequalities. Methods are applied to deriving PAC-Bayes bounds, GANs, and regret inequalities in machine learning, parametric and non-parametric estimation in statistics, communication complexity, and computation with noisy gates in computer science. Fast-paced journey through a recent textbook with the same title. For a communication-focused version, consider 6.7470.

Introduction to probability theory. Probability spaces and measures. Discrete and continuous random variables. Conditioning and independence. Multivariate normal distribution. Abstract integration, expectation, and related convergence results. Moment generating and characteristic functions. Bernoulli and Poisson process. Finite-state Markov chains. Convergence notions and their relations. Limit theorems. Familiarity with elementary probability and real analysis is desirable.

Introduction to computational aspects of statistical inference via probabilistic graphical models. Directed and undirected graphical models, and factor graphs, over discrete and Gaussian distributions; hidden Markov models, linear dynamical systems. Sum-product and junction tree algorithms; forward-backward algorithm, Kalman filtering and smoothing. Min-sum and Viterbi algorithms. Variational methods, mean-field theory, and loopy belief propagation. Sampling methods; Glauber dynamics and mixing time analysis. Parameter structure learning for graphical models; Baum-Welch and Chow-Liu algorithms. Selected topics such as causal inference, particle filtering, restricted Boltzmann machines, and graph neural networks.

Introduction to the methods and applications of optimization. Topics include linear optimization, duality, non-linear optimization, integer optimization, and optimization under uncertainty. Instruction provided in modeling techniques to address problems arising in practice, mathematical theory to understand the structure of optimization problems, computational algorithms to solve complex optimization problems, and practical applications. Covers several examples and in-depth case studies based on real-world data to showcase impactful applications of optimization across management and engineering. Computational exercises based on the Julia-based programming language JuMP. Includes a term project. Basic competency in computational programming and linear algebra recommended. Students taking graduate version complete additional assignments. This subject was previously listed as 15.093/6.7200/IDS.200. One section primarily reserved for Sloan students; check syllabus for details.

Introduction to the methods and applications of optimization. Topics include linear optimization, duality, non-linear optimization, integer optimization, and optimization under uncertainty. Instruction provided in modeling techniques to address problems arising in practice, mathematical theory to understand the structure of optimization problems, computational algorithms to solve complex optimization problems, and practical applications. Covers several examples and in-depth case studies based on real-world data to showcase impactful applications of optimization across management and engineering. Computational exercises based on the Julia-based programming language JuMP. Includes a term project. Basic competency in computational programming and linear algebra recommended. Students taking graduate version complete additional assignments. This subject was previously listed as 6.7201.

Provides design-focused instruction on how to build complex software applications. Presents key skills in product design and human-computer interaction (identifying authentic demand, prototyping, visual design, and user testing), software concept design (achieving clarity, modularity, and focus in structuring functionality), and analyzing social and ethical implications. Implementation topics include client/server web apps, reactive front-ends, and databases. Students work on both individual projects and a larger team project, and are taught to make effective and intentional use of LLMs in design and coding.

Studies the design and implementation of modern, dynamic programming languages. Topics include fundamental approaches for parsing, semantics and interpretation, virtual machines, garbage collection, just-in-time machine code generation, and optimization. Includes a semester-long, group project that delivers a virtual machine that spans all of these topics.

Fundamental notions and big ideas for achieving security in computer systems. Topics include cryptographic foundations (pseudorandomness, collision-resistant hash functions, authentication codes, signatures, authenticated encryption, public-key encryption), systems ideas (isolation, non-interference, authentication, access control, delegation, trust), and implementation techniques (privilege separation, fuzzing, symbolic execution, runtime defenses, side-channel attacks). Case studies of how these ideas are realized in deployed systems. Lab assignments apply ideas from lectures to learn how to build secure systems and how they can be attacked.

Design and implementation of operating systems, and their use as a foundation for systems programming. Topics include virtual memory, file systems, threads, context switches, kernels, interrupts, system calls, interprocess communication, coordination, and interaction between software and hardware. A multi-processor operating system for RISC-V, xv6, is used to illustrate these topics. Individual laboratory assignments involve extending the xv6 operating system, for example to support sophisticated virtual memory features and networking.

Explores the impact of computer systems on individual humans, society, and the environment. Examines large- and small-scale power structures that stem from low-level technical design decisions, the consequences of those structures on society, and how they can limit or provide access to certain technologies. Students assess design decisions within an ethical framework and consider the impact of their decisions on non-users. Case studies of working systems and readings from the current literature provide comparisons and contrasts. Possible topics include the implications of hierarchical designs (e.g., DNS) for scale; how layered models influence what parts of a network have the power to take certain actions; and the environmental impact of proof-of-work-based systems such as Bitcoin. Students taking graduate version complete additional assignments. Enrollment may be limited.

Explores the impact of computer systems on individual humans, society, and the environment. Examines large- and small-scale power structures that stem from low-level technical design decisions, the consequences of those structures on society, and how they can limit or provide access to certain technologies. Students assess design decisions within an ethical framework and consider the impact of their decisions on non-users. Case studies of working systems and readings from the current literature provide comparisons and contrasts. Possible topics include the implications of hierarchical designs (e.g., DNS) for scale; how layered models influence what parts of a network have the power to take certain actions; and the environmental impact of proof-of-work-based systems such as Bitcoin. Students taking graduate version complete different assignments. Enrollment may be limited.

Lab-intensive subject that investigates digital systems with a focus on FPGAs. Lectures and labs cover logic, flip flops, counters, timing, synchronization, finite-state machines, digital signal processing, communication protocols, and modern sensors. Prepares students for the design and implementation of a large-scale final project of their choice: games, music, digital filters, wireless communications, video, or graphics. Extensive use of System/Verilog for describing and implementing and verifying digital logic designs.

Highlights common principles that permeate the functioning of diverse technological, economic and social networks. Utilizes three sets of tools for analyzing networks -- random graph models, optimization, and game theory -- to study informational and learning cascades; economic and financial networks; social influence networks; formation of social groups; communication networks and the Internet; consensus and gossiping; spread and control of epidemics; control and use of energy networks; and biological networks. Students taking graduate version complete additional assignments.

Covers the theory and practice of algorithms and data structures. Topics include models of computation, algorithm design and analysis, and performance engineering of algorithm implementations. Presents the design and implementation of sequential, parallel, cache-efficient, and external-memory algorithms. Illustrates many of the principles of algorithm engineering in the context of parallel algorithms and graph problems.

Presents major principles and techniques for program analysis. Includes formal semantics, type systems and type-based program analysis, abstract interpretation and model checking and synthesis. Emphasis on Haskell and Ocaml, but no prior experience in these languages is assumed. Student assignments include implementing of techniques covered in class, including building simple verifiers.

Surveys techniques for rigorous mathematical reasoning about correctness of software, emphasizing commonalities across approaches. Introduces interactive computer theorem proving with the Coq proof assistant, which is used for all assignments, providing immediate feedback on soundness of logical arguments. Covers common program-proof techniques, including operational semantics, model checking, abstract interpretation, type systems, program logics, and their applications to functional, imperative, and concurrent programs. Develops a common conceptual framework based on invariants, abstraction, and modularity applied to state and labeled transition systems.

Fundamental design and implementation issues in the engineering of operating systems. Lectures based on the study of a symmetric multiprocessor version of UNIX version 6 and research papers. Topics include virtual memory; file system; threads; context switches; kernels; interrupts; system calls; interprocess communication; coordination, and interaction between software and hardware. Individual laboratory assignments accumulate in the construction of a minimal operating system (for an x86-based personal computer) that implements the basic operating system abstractions and a shell. Knowledge of programming in the C language is a prerequisite.

Presents topics on the engineering and analysis of network protocols and architecture, including: architectural principles for designing heterogeneous networks; transport protocols; Internet routing; router design; congestion control and network resource management; wireless networks; network security; naming; overlay; and peer-to-peer networks. Includes readings from original research papers and a term-long project and paper. Students taking graduate version complete additional assignments.

Presents topics on the engineering and analysis of network protocols and architecture, including: architectural principles for designing heterogeneous networks; transport protocols; Internet routing; router design; congestion control and network resource management; wireless networks; network security; naming; overlay; and peer-to-peer networks. Includes readings from original research papers and a term-long project and paper. Students taking graduate version complete additional assignments.

Introduction to the basic principles of computer systems with emphasis on the use of rigorous techniques as an aid to understanding and building modern computing systems. Particular attention paid to concurrent and distributed systems. Topics include: specification and verification, concurrent algorithms, synchronization, naming, Networking, replication techniques (including distributed cache management), and principles and algorithms for achieving reliability.

Introduction to the principles underlying modern computer architecture. Emphasizes the relationship among technology, hardware organization, and programming systems in the evolution of computer architecture. Topics include pipelined, out-of-order, and speculative execution; caches, virtual memory and exception handling, superscalar, very long instruction word (VLIW), vector, and multithreaded processors; on-chip networks, memory models, synchronization, and cache coherence protocols for multiprocessors.

Introduces efficient deep learning computing techniques that enable powerful deep learning applications on resource-constrained devices. Topics include model compression, pruning, quantization, neural architecture search, distributed training, data/model parallellism, gradient compression, on-device fine-tuning. It also introduces application-specific acceleration techniques for video recognition, point cloud, and generative AI (diffusion model, LLM). Students will get hands-on experience accelerating deep learning applications with an open-ended design project.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.

A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression, and identification). Concepts are introduced with lectures and online problems, and then mastered during weekly labs. In lab, students model, design, test, and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g., optimizing thrust-driven positioners or stabilizing magnetic levitators). Students taking graduate version complete additional problems and labs.

A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression and identification). Concepts are introduced with lectures and on-line problems, and then mastered during weekly labs. In lab, students model, design, test and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g. optimizing thrust-driven positioners or stabilizing magnetic levitators). Students in the graduate version complete additional problems and labs.

Introduces the fundamental algorithmic approaches for creating robot systems that can autonomously manipulate physical objects in unstructured environments such as homes and restaurants. Topics include perception (including approaches based on deep learning and approaches based on 3D geometry), planning (robot kinematics and trajectory generation, collision-free motion planning, task-and-motion planning, and planning under uncertainty), as well as dynamics and control (both model-based and learning-based). Students taking graduate version complete additional assignments. Students engage in extensive written and oral communication exercises.

Introduces the fundamental algorithmic approaches for creating robot systems that can autonomously manipulate physical objects in unstructured environments such as homes and restaurants. Topics include perception (including approaches based on deep learning and approaches based on 3D geometry), planning (robot kinematics and trajectory generation, collision-free motion planning, task-and-motion planning, and planning under uncertainty), as well as dynamics and control (both model-based and learning-based. Students taking graduate version complete additional assignments.

Computer-aided design methodologies for synthesis of multivariable feedback control systems. Performance and robustness trade-offs. Model-based compensators; Q-parameterization; ill-posed optimization problems; dynamic augmentation; linear-quadratic optimization of controllers; H-infinity controller design; Mu-synthesis; model and compensator simplification; nonlinear effects. Computer-aided (MATLAB) design homework using models of physical processes.

Examines reinforcement learning (RL) as a methodology for approximately solving sequential decision-making under uncertainty, with foundations in optimal control and machine learning. Provides a mathematical introduction to RL, including dynamic programming, statistical, and empirical perspectives, and special topics. Core topics include: dynamic programming, special structures, finite and infinite horizon Markov Decision Processes, value and policy iteration, Monte Carlo methods, temporal differences, Q-learning, stochastic approximation, and bandits. Also covers approximate dynamic programming, including value-based methods and policy space methods. Applications and examples drawn from diverse domains. Focus is mathematical, but is supplemented with computational exercises. An analysis prerequisite is suggested but not required; mathematical maturity is necessary.

Provides design-focused instruction on how to build complex software applications. Presents key skills in product design and human-computer interaction (identifying authentic demand, prototyping, visual design, and user testing), software concept design (achieving clarity, modularity, and focus in structuring functionality), and analyzing social and ethical implications. Implementation topics include client/server web apps, reactive front-ends, and databases. Students work on both individual projects and a larger team project, and are taught to make effective and intentional use of LLMs in design and coding.

Introduction to computer graphics algorithms, software and hardware. Topics include ray tracing, the graphics pipeline, transformations, texture mapping, shadows, sampling, global illumination, splines, animation and color.

Introduces computational aspects of computer-aided design and manufacturing. Explores relevant methods in the context of additive manufacturing (e.g., 3D printing). Topics include computer graphics (geometry modeling, solid modeling, procedural modeling), physically-based simulation (kinematics, finite element method), 3D scanning/geometry processing, and an overview of 3D fabrication methods. Exposes students to the latest research in computational fabrication. Students taking the graduate version complete additional assignments.

Provides instruction in building cutting-edge interactive technologies, explains the underlying engineering concepts, and shows how those technologies evolved over time. Students use a studio format (i.e., extended periods of time) for constructing software and hardware prototypes. Topics include interactive technologies, such as multi-touch, augmented reality, haptics, wearables, and shape-changing interfaces. In a group project, students build their own interactive hardware/software prototypes and present them in a live demo at the end of term. Enrollment may be limited.

Broad survey of numerical methods used in graphics, vision, robotics, machine learning, and scientific computing, with emphasis on incorporating these algorithms into downstream applications. Focuses on challenges that arise in applying/implementing numerical algorithms and recognizing which numerical methods are relevant to different applications. Topics include numerical linear algebra (QR, LU, SVD matrix factorizations; eigenvectors; conjugate gradients), ordinary and partial differential equations (divided differences, finite element method), and nonlinear systems and optimization (gradient descent, Newton/quasi-Newton methods, gradient-free optimization, constrained optimization). Examples and case studies drawn from the computer science and machine learning literatures.

Presents fundamentals and applications of hardware and software techniques used in digital and computational photography, with an emphasis on software methods. Provides sufficient background to implement solutions to photographic challenges and opportunities. Topics include cameras and image formation, image processing and image representations, high-dynamic-range imaging, human visual perception and color, single view 3-D model reconstruction, morphing, data-rich photography, super-resolution, and image-based rendering. Students taking graduate version complete additional assignments.

Presents fundamentals and applications of hardware and software techniques used in digital and computational photography, with an emphasis on software methods. Provides sufficient background to implement solutions to photographic challenges and opportunities. Topics include cameras and image formation, image processing and image representations, high-dynamic-range imaging, human visual perception and color, single view 3-D model reconstruction, morphing, data-rich photography, super-resolution, and image-based rendering. Students taking graduate version complete additional assignments.

Introduces mathematical, algorithmic, and statistical tools needed to analyze geometric data and to apply geometric techniques to data analysis, with applications to fields such as computer graphics, machine learning, computer vision, medical imaging, and architecture. Potential topics include applied introduction to differential geometry, discrete notions of curvature, metric embedding, geometric PDE via the finite element method (FEM) and discrete exterior calculus (DEC),; computational spectral geometry and relationship to graph-based learning, correspondence and mapping, level set method, descriptor, shape collections, optimal transport, and vector field design.

Introduces computational aspects of computer-aided design and manufacturing. Explores relevant methods in the context of additive manufacturing (e.g., 3D printing). Topics include computer graphics (geometry modeling, solid modeling, procedural modeling), physically-based simulation (kinematics, finite element method), 3D scanning/geometry processing, and an overview of 3D fabrication methods. Exposes students to the latest research in computational fabrication. Students taking graduate version complete additional assignments.

Transistors at the nanoscale. Quantization, wavefunctions, and Schrodinger's equation. Introduction to electronic properties of molecules, carbon nanotubes, and crystals. Energy band formation and the origin of metals, insulators and semiconductors. Ballistic transport, Ohm's law, ballistic versus traditional MOSFETs, fundamental limits to computation.

Meets with undergraduate subject 6.2530, but requires the completion of additional/different homework assignments and or projects. See subject description under 6.2530.

Covers the fundamentals of optics and the interaction of light and matter, leading to devices such as light emitting diodes, optical amplifiers, and lasers. Topics include classical ray, wave, beam, and Fourier optics; Maxwell's electromagnetic waves; resonators; quantum theory of photons; light-matter interaction; laser amplification; lasers; and semiconductors optoelectronics. Students taking graduate version complete additional assignments.

Elementary quantum mechanics and statistical physics. Introduces applied quantum physics. Emphasizes experimental basis for quantum mechanics. Applies Schrodinger's equation to the free particle, tunneling, the harmonic oscillator, and hydrogen atom. Variational methods. Elementary statistical physics; Fermi-Dirac, Bose-Einstein, and Boltzmann distribution functions. Simple models for metals, semiconductors, and devices such as electron microscopes, scanning tunneling microscope, thermonic emitters, atomic force microscope, and more. Some familiarity with continuous time Fourier transforms recommended.

Covers physics of microelectronic semiconductor devices for integrated circuit applications. Topics include semiconductor fundamentals, p-n junction, metal-oxide semiconductor structure, metal-semiconductor junction, MOS field-effect transistor, and bipolar junction transistor.  Emphasizes physical understanding of device operation through energy band diagrams and short-channel MOSFET device design and modern device scaling. Familiarity with MATLAB recommended.

Continuation of 6.6510 emphasizing applications-related physical issues in solids. Topics include: electronic structure and energy band diagrams of semiconductors, metals, and insulators; Fermi surfaces; dynamics of electrons under electric and magnetic fields; classical diffusive transport phenomena such as electrical and thermal conduction and thermoelectric phenomena; quantum transport in tunneling and ballistic devices; optical properties of metals, semiconductors, and insulators; impurities and excitons; photon-lattice interactions; Kramers-Kronig relations; optoelectronic devices based on interband and intersubband transitions; magnetic properties of solids; exchange energy and magnetic ordering; magneto-oscillatory phenomena; quantum Hall effect; superconducting phenomena and simple models.

Statistical modeling and control in manufacturing processes. Use of experimental design and response surface modeling to understand manufacturing process physics. Defect and parametric yield modeling and optimization. Forms of process control, including statistical process control, run by run and adaptive control, and real-time feedback control. Application contexts include semiconductor manufacturing, conventional metal and polymer processing, and emerging micro-nano manufacturing processes.

Introduction to linear optimization and its extensions emphasizing both methodology and the underlying mathematical structures and geometrical ideas. Covers classical theory of linear programming as well as some recent advances in the field. Topics: simplex method; duality theory; sensitivity analysis; network flow problems; decomposition; robust optimization; integer programming; interior point algorithms for linear programming; and introduction to combinatorial optimization and NP-completeness.

Introduction to fundamentals of game theory and mechanism design with motivations for each topic drawn from engineering applications (including distributed control of wireline/wireless communication networks, transportation networks, pricing). Emphasis on the foundations of the theory, mathematical tools, as well as modeling and the equilibrium notion in different environments. Topics include normal form games, supermodular games, dynamic games, repeated games, games with incomplete/imperfect information, mechanism design, cooperative game theory, and network games.

Introduction to computational techniques for modeling and simulation of a variety of large and complex engineering, science, and socio-economical systems. Prepares students for practical use and development of computational engineering in their own research and future work. Topics include mathematical formulations (e.g., automatic assembly of constitutive and conservation principles); linear system solvers (sparse and iterative); nonlinear solvers (Newton and homotopy); ordinary, time-periodic and partial differential equation solvers; and model order reduction. Students develop their own models and simulators for self-proposed applications, with an emphasis on creativity, teamwork, and communication. Prior basic linear algebra required and at least one numerical programming language (e.g., MATLAB, Julia, Python, etc.) helpful.

Advanced introduction to numerical analysis: accuracy and efficiency of numerical algorithms. In-depth coverage of sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) may also be surveyed. Final project involves some programming.

Covers the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic, and hyperbolic partial differential and integral equations. Topics include mathematical formulations; finite difference, finite volume, finite element, and boundary element discretization methods; and direct and iterative solution techniques. The methodologies described form the foundation for computational approaches to engineering systems involving heat transfer, solid mechanics, fluid dynamics, and electromagnetics. Computer assignments requiring programming.

Broad survey of numerical methods used in graphics, vision, robotics, machine learning, and scientific computing, with emphasis on incorporating these algorithms into downstream applications. Focuses on challenges that arise in applying/implementing numerical algorithms and recognizing which numerical methods are relevant to different applications. Topics include numerical linear algebra (QR, LU, SVD matrix factorizations; eigenvectors; conjugate gradients), ordinary and partial differential equations (divided differences, finite element method), and nonlinear systems and optimization (gradient descent, Newton/quasi-Newton methods, gradient-free optimization, constrained optimization). Examples and case studies drawn from the computer science and machine learning literatures.

Focuses on algorithms and techniques for writing and using modern technical software in a job, lab, or research group environment that may consist of interdisciplinary teams, where performance may be critical, and where the software needs to be flexible and adaptable. Topics include automatic differentiation, matrix calculus, scientific machine learning, parallel and GPU computing, and performance optimization with introductory applications to climate science, economics, agent-based modeling, and other areas. Labs and projects focus on performant, readable, composable algorithms, and software. Programming will be in Julia. Expects students to have some familiarity with Python, Matlab, or R. No Julia experience necessary.

Introduction to the methods and applications of optimization. Topics include linear optimization, duality, non-linear optimization, integer optimization, and optimization under uncertainty. Instruction provided in modeling techniques to address problems arising in practice, mathematical theory to understand the structure of optimization problems, computational algorithms to solve complex optimization problems, and practical applications. Covers several examples and in-depth case studies based on real-world data to showcase impactful applications of optimization across management and engineering. Computational exercises based on the Julia-based programming language JuMP. Includes a term project. Basic competency in computational programming and linear algebra recommended. Students taking graduate version complete additional assignments. This subject was previously listed as 15.093/6.7200/IDS.200. One section primarily reserved for Sloan students; check syllabus for details.

Introduction to the methods and applications of optimization. Topics include linear optimization, duality, non-linear optimization, integer optimization, and optimization under uncertainty. Instruction provided in modeling techniques to address problems arising in practice, mathematical theory to understand the structure of optimization problems, computational algorithms to solve complex optimization problems, and practical applications. Covers several examples and in-depth case studies based on real-world data to showcase impactful applications of optimization across management and engineering. Computational exercises based on the Julia-based programming language JuMP. Includes a term project. Basic competency in computational programming and linear algebra recommended. Students taking graduate version complete additional assignments. This subject was previously listed as 6.7201.

A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression, and identification). Concepts are introduced with lectures and online problems, and then mastered during weekly labs. In lab, students model, design, test, and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g., optimizing thrust-driven positioners or stabilizing magnetic levitators). Students taking graduate version complete additional problems and labs.

A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression and identification). Concepts are introduced with lectures and on-line problems, and then mastered during weekly labs. In lab, students model, design, test and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g. optimizing thrust-driven positioners or stabilizing magnetic levitators). Students in the graduate version complete additional problems and labs.

Representation, analysis, and design of discrete time signals and systems. Decimation, interpolation, and sampling rate conversion. Noise shaping. Flowgraph structures for DT systems. IIR and FIR filter design techniques. Parametric signal modeling, linear prediction, and lattice filters. Discrete Fourier transform, DFT computation, and FFT algorithms. Spectral analysis, time-frequency analysis, relation to filter banks. Multirate signal processing, perfect reconstruction filter banks, and connection to wavelets.

Presents mathematical statistics as a formal language for reasoning about data and uncertainty. Introduction to the basic framework of statistical decision theory, along with core concepts such as sufficiency, Bayes and minimax optimality of statistical procedures, with applications to optimal estimation, hypothesis testing, and prediction. Discussion topics include causality, multiple hypothesis testing, nonparametric and semiparametric statistics, and results for model misspecification. Targeted to students interested in statistical and machine learning research, with an emphasis on proofs and fundamental understanding.

Introduction to computational aspects of statistical inference via probabilistic graphical models. Directed and undirected graphical models, and factor graphs, over discrete and Gaussian distributions; hidden Markov models, linear dynamical systems. Sum-product and junction tree algorithms; forward-backward algorithm, Kalman filtering and smoothing. Min-sum and Viterbi algorithms. Variational methods, mean-field theory, and loopy belief propagation. Sampling methods; Glauber dynamics and mixing time analysis. Parameter structure learning for graphical models; Baum-Welch and Chow-Liu algorithms. Selected topics such as causal inference, particle filtering, restricted Boltzmann machines, and graph neural networks.

A more extensive and theoretical treatment of the material in 6.1400J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.

Covers the theory and practice of algorithms and data structures. Topics include models of computation, algorithm design and analysis, and performance engineering of algorithm implementations. Presents the design and implementation of sequential, parallel, cache-efficient, and external-memory algorithms. Illustrates many of the principles of algorithm engineering in the context of parallel algorithms and graph problems.

First-year graduate subject in algorithms. Emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Surveys a variety of computational models and the algorithms for them. Data structures, network flows, linear programming, computational geometry, approximation algorithms, online algorithms, parallel algorithms, external memory, streaming algorithms.

Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

Sublinear time algorithms understand parameters and properties of input data after viewing only a minuscule fraction of it. Tools from number theory, combinatorics, linear algebra, optimization theory, distributed algorithms, statistics, and probability are covered. Topics include: testing and estimating properties of distributions, functions, graphs, strings, point sets, and various combinatorial objects.

Design and analysis of algorithms, emphasizing those suitable for use in distributed networks. Covers various topics including distributed graph algorithms, locality constraints, bandwidth limitations and communication complexity, process synchronization, allocation of computational resources, fault tolerance, and asynchrony. No background in distributed systems required.

Introduction to algorithms and computational complexity for high-dimensional statistical inference problems, with focus on provable polynomial-time guarantees. Covers modern algorithm design techniques via convex programming and Sum of Squares method; graphical models as a language to describe complex but tractable high-dimensional learning problems and associated learning algorithms; and basics of complexity for statistical problems, including statistical query and low-degree lower bounds and reductions. Mathematical maturity is recommended.

A more extensive and theoretical treatment of the material in 6.1400J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Students in Course 18 must register for the undergraduate version, 18.404.

Introduction to quantum computational complexity theory, the study of the fundamental capabilities and limitations of quantum computers. Topics include complexity classes, lower bounds, communication complexity, proofs and advice, and interactive proof systems in the quantum world; classical simulation of quantum circuits. The objective is to bring students to the research frontier.

A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives such as public-key encryption, digital signatures, and pseudo-random number generation, as well as advanced cryptographic primitives such as zero-knowledge proofs, homomorphic encryption, and secure multiparty computation.

Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.

Examines the foundations of multi-agent systems from a combined game-theoretic, optimization, and learning-theoretic perspective. Studies settings involving multiple learning agents with distinct objectives, where standard training methods may induce nonstationary or unstable behaviors. Introduces fundamental models including matrix games, stochastic games, and imperfect-information games. Studies modern settings deviating from standard models wherein strategies and utilities are represented using deep neural networks. Presents equilibrium concepts, equilibrium computation methods, and equilibrium learning methods. Studies the complexity of solution concepts. Discusses applications in machine learning, including multi-agent reinforcement learning, adversarial learning, AI-AI interactions, and strategic decision-making systems underlying game-playing agents in domains such as Go, poker, Diplomacy, and Stratego.

Covers subject matter not offered in the regular curriculum. Consult department to learn of offerings for a particular term.