APHY 5060a, Basic Quantum Mechanics  John Sous

Basic concepts and techniques of quantum mechanics essential for solid state physics and quantum electronics. Topics include the Schrödinger treatment of the harmonic oscillator, atoms and molecules and tunneling, matrix methods, and perturbation theory.
MW 2:35pm-3:50pm

APHY 5120a, AI for Applied Physics Research  Logan Wright

Historically, the introduction of transformative tools—such as formal mathematics or the digital computer—fundamentally altered the methodology and scope of scientific inquiry. The computer, for instance, birthed simulations, computational optimization, and sophisticated curve-fitting. While now ubiquitous, these methods brought subtle but persistent challenges: numerical artifacts, convergence issues, and the potential loss of deep physical insight. Today, the rapid (and, frankly, surprising!) development of AI—from large language models to agentic systems—represents a similar “phase transition” in research. In disciplines like applied physics, where mathematics, code, and voluminous data intersect, this transition is both exciting and existential. This seminar examines how researchers are currently deploying these tools to advance the frontiers of their fields. We investigate parts of research where AI already or may soon exceed human performance, domains where humans are likely to retain a persistent advantage, identify common failure modes of AI-assisted research, and seek to critically develop “best practices” for rigorous, AI-augmented workflows.
M 1:30pm-3:25pm

APHY 5220b, Theory of Electromagnetic Waves, Radiation, and Scattering  Owen Miller

This is a graduate-level course on electromagnetic theory, focusing on electromagnetic wave phenomena in a variety of contexts. Electrostatics and magnetostatics are reviewed briefly and then the full time-dependent Maxwell equations are studied to derive the fundamentals of wave propagation, the wave equation, plane waves, polarization, energy and momentum flow in EM waves, conservation laws, gauge transformation, and Green functions for the wave equations. Dielectric media and Fresnel reflection and refraction, total internal reflection, group velocity, wave packets and dispersion. Beam propagation and gaussian optics, optical birefringence. Waves in confined structures, waveguides, optical fibers, resonant cavities. Radiating sources, electric and magnetic dipolar radiation, multipolar radiation, near and far-field solutions, radiating antennas. Scattering theory, scalar and vector diffraction, Rayleigh scattering, scattering matrix and temporal coupled mode theory, scattering resonances, multiple scattering, systems with gain and lasing. Time permitting: relativistic kinematics, covariance of Maxwell's equations, Lorentz transformation of electric and magnetic fields, relativistic mechanics of charged particles. Prerequisites: an undergraduate course on electricity and magnetism and graduate-level vector calculus and differential equations.
TTh 1:05pm-2:20pm

APHY 5480a / MSCI 5480a / PHYS 5480a, Solid State Physics I  Yu He

A two-term sequence (with APHY 5490) covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.
MW 11:35am-12:50pm

APHY 5490b / MSCI 5490b / PHYS 5490b, Solid State Physics II  Vidvuds Ozolins

A two-term sequence (with APHY 548) covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonons, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.
MW 11:35am-12:50pm

APHY 5750b, Physics of AI  John Sous

A introduction to current research in AI from a physics perspective, i.e. one that emphasizes the mechanisms through which various AI architectures learn. Topics include, linear regression, neural nets, transformers, diffusion, etc. Prerequisites: linear algebra, calculus.
MW 2:35pm-3:50pm

APHY 5760a, Topics in Applied Physics Research  Daniel Prober

The course introduces the fundamentals of applied physics research to graduate students in the Department of Applied Physics in order to introduce them to resources and opportunities for research activities. The content of the class includes overview presentations from faculty and other senior members of the department and related departments about their research and their career trajectories. The class also includes presentations from campus experts who offer important services that support Applied Physics graduate students in their successful degree completion.
F 3pm-4:20pm

APHY 5900b / PHYS 5900b, Responsible Conduct in Research for Physical Scientists  Sarah Demers

A review and discussion of best practices of conduct in research including scientific integrity and misconduct; mentorship; data management; and diversity, equity, and inclusion in science.
F 10am-11:15am

APHY 6070b, Modern Topics in Optics and Quantum Electronics  Peter Rakich

This course provides a survey of modern topics involving integrated photonics, optomechanics, nonlinear optics, and laser physics for students interested in contemporary experimental optics research. Subjects include nonlinear wave phenomena, optomechanical interactions, phonon physics, light scattering, light emission and detection, cavities, systems of cavities, traveling-wave devices and interactions, perturbation theory, reciprocal and nonreciprocal systems, parametric interactions, laser oscillators and related technologies. Students are encouraged to explore these and related research topics through independent study and classroom presentations.
MW 4pm-5:15pm

APHY 6100a / PHYS 6100a, Quantum Many-Body Theory  Yoram Alhassid

Identical particles and second quantization. Electron tunneling and spectral function. General linear response theory. Approximate methods of quantum many-body theory. Dielectric response, screening of long-range interactions, electric conductance, collective modes, and photon absorption spectra. Fermi liquid; Cooper and Stoner instabilities; notions of superconductivity and magnetism. BCS theory, Josephson effect, and Majorana fermions in condensed matter; superconducting qubits. Bose-Einstein condensation; Bogoliubov quasiparticles and solitons.
TTh 2:35pm-3:50pm

APHY 6280a / PHYS 6120a, Statistical Physics II  Nicholas Read

An advanced course in statistical mechanics. Topics may include mean field theory of and fluctuations at continuous phase transitions; critical phenomena, scaling, and introduction to the renormalization group ideas; topological phase transitions; dynamic correlation functions and linear response theory; quantum phase transitions; superfluid and superconducting phase transitions; cooperative phenomena in low-dimensional systems.
TTh 2:35pm-3:50pm

APHY 6500a / PHYS 6500a, Theory of Solids I  Leonid Glazman

A graduate-level introduction with focus on advanced and specialized topics. Knowledge of advanced quantum mechanics (Sakurai level) and solid state physics (Kittel and Ashcroft-Mermin level) is assumed. The course teaches advanced solid state physics techniques and concepts.
T 3:30pm-5:30pm

APHY 6600a / PHYS 6010a, Quantum Information and Computation  Aleksander Kubica

This course focuses on the theory of quantum information and computation. We cover the following tentative list of topics: overview of postulates of quantum mechanics and measurements, quantum circuits, physical implementation of quantum operations, introduction to computational complexity, quantum algorithms (DJ, Shor’s, Grover’s, and others as time permits), decoherence and noisy quantum channels, quantum error-correction and fault-tolerance, stabilizer formalism, error-correcting codes (Shor, Steane, surface-code, and others as time permits), quantum key distribution, quantum Shannon theory, entropy, and data compression.
TTh 11:35am-12:50pm

APHY 6700a, Statistical Methods with Applications in Science and Finance  Sohrab Ismail-Beigi

Introduction to key methods in statistical physics with examples drawn principally from the sciences (physics, chemistry, astronomy, statistics, biology) as well as added examples from finance. Students learn the fundamentals of Monte Carlo, stochastic random walks, and analysis of covariance analytically as well as via numerical exercises. Prerequisites: ENAS 1940, MATH 2220, ENAS 1300 or equivalents.
TTh 2:35pm-3:50pm

APHY 6750a / PHYS 6750a, Principles of Optics with Applications  Hui Cao

Introduction to the principles of optics and electromagnetic wave phenomena with applications to microscopy, optical fibers, laser spectroscopy, nanophotonics, plasmonics, and metamaterials. Topics include propagation of light, reflection and refraction, guiding light, polarization, interference, diffraction, scattering, Fourier optics, and optical coherence.
HTBA

APHY 6760a / PHYS 6760a, Introduction to Light-Matter Interactions  Peter Rakich

Optical properties of materials and a variety of coherent light-matter interactions are explored through the classical and quantum treatments. The role of electronic, phononic, and plasmonic interactions in shaping the optical properties of materials is examined using generalized quantum and classical coupled-mode theories. The dynamic response of media to strain, magnetic, and electric fields is also treated. Modern topics are explored, including optical forces, photonic crystals, and metamaterials; multi-photon absorption; and parametric processes resulting from electronic, optomechanical, and Raman interactions.
MW 4pm-5:15pm

APHY 6910a / PHYS 6910a, Quantum Optics  Shruti Puri

Quantization of the electromagnetic field, coherence properties and representation of the electromagnetic field, quantum phenomena in simple nonlinear optics, atom-field interaction, stochastic methods, master equation, Fokker-Planck equation, Heisenberg-Langevin equation, input-output formulation, cavity quantum electrodynamics, quantum theory of laser, trapped ions, light forces, quantum optomechanics, Bose-Einstein condensation, quantum measurement and control.
TTh 2:35pm-3:50pm

APHY 7250a / MENG 8850a / MSCI 7250a, Advanced Synchrotron Techniques and Electron Spectroscopy of Materials  Charles Ahn

This course provides descriptions of advanced concepts in synchrotron X-ray and electron-based methodologies for studies of a wide range of materials at atomic and nano-scales. Topics include X-ray and electron interactions with matter, X-ray scattering and diffraction, X-ray spectroscopy and inelastic methods, time-resolved applications, X-ray imaging and microscopy, photo-electron spectroscopy, electron microscopy and spectroscopy, among others. Emphasis is on applying the fundamental knowledge of these advanced methodologies to real-world materials studies in a variety of scientific disciplines.
HTBA

APHY 7270b, Circuit Quantum Electrodynamics  Michael Hatridge

Circuit quantum electrodynamics, or circuit QED, is the field of quantum optics with microwave photons, and is the basis of most solid-state quantum computing technologies. In the microwave domain, we can engineer the properties of “artificial atoms” constructed from circuits containing Josephson junctions and couple them strongly to stationary photons trapped in a cavity or resonant circuit and to traveling photons in a waveguide or transmission line. With this approach, one can access regions of strong coupling between “light” and “matter” that are not accessible in atomic physics. This course is an introduction to the concepts and techniques of circuit QED, aimed for the beginning graduate student or advanced undergraduate with a solid background in classical electricity and magnetism and also basic quantum mechanics. The course is intended to provide an important base of knowledge that can prepare the student for research in quantum information processing with superconducting circuits. Topics to be covered include the basics of superconducting qubits, the Jaynes-Cummings model and the strong dispersive regime, and the use of light-matter interactions for measurement, control, and quantum computing with quantum circuits. Extra work for graduate students: The last homework set is optional for undergrads and required for grads. Prerequisites: one semester QM and one semester E&M at advanced undergrad or higher.
TTh 2:35pm-3:50pm

BENG 5200a / C&MP 5500a / MCDB 5500a / PHAR 5500a / PTB 5500a, Physiological Systems  W. Mark Saltzman and Stuart Campbell

The course develops a foundation in human physiology by examining the homeostasis of vital parameters within the body, and the biophysical properties of cells, tissues, and organs. Basic concepts in cell and membrane physiology are synthesized through exploring the function of skeletal, smooth, and cardiac muscle. The physical basis of blood flow, mechanisms of vascular exchange, cardiac performance, and regulation of overall circulatory function are discussed. Respiratory physiology explores the mechanics of ventilation, gas diffusion, and acid-base balance. Renal physiology examines the formation and composition of urine and the regulation of electrolyte, fluid, and acid-base balance. Organs of the digestive system are discussed from the perspective of substrate metabolism and energy balance. Hormonal regulation is applied to metabolic control and to calcium, water, and electrolyte balance. The biology of nerve cells is addressed with emphasis on synaptic transmission and simple neuronal circuits within the central nervous system. The special senses are considered in the framework of sensory transduction. Weekly discussion sections provide a forum for in-depth exploration of topics. Graduate students evaluate research findings through literature review and weekly meetings with the instructor.
MWF 9:25am-10:15am

BENG 5270b / C&MP 5560b / MCDB 5600b / PHAR 5600b, Cellular and Molecular Physiology: Molecular Machines in Human Disease  Emile Boulpaep

The course focuses on understanding the processes that transfer molecules across membranes at the cellular, molecular, biophysical, and physiological levels. Students learn about the different classes of molecular machines that mediate membrane transport, generate electrical currents, or perform mechanical displacement. Emphasis is placed on the relationship between the molecular structures of membrane proteins and their individual functions. The interactions among transport proteins in determining the physiological behaviors of cells and tissues are also stressed. Molecular motors are introduced and their mechanical relationship to cell function is explored. Students read papers from the scientific literature that establish the connections between mutations in genes encoding membrane proteins and a wide variety of human genetic diseases.
MWF 9:25am-10:15am

BENG 5340a / MSCI 5340, Biomaterials  Anjelica Gonzalez

Introduction to materials, classes of materials from atomic structure to physical properties. Major classes of materials: metals, ceramics and glasses, and polymers, addressing their specific characteristics, properties, and biological applications. Throughout the presentation of the synthesis, characterization, and properties of the classes of materials, a connection is made to the selection of materials for use in specific biological applications by matching the material’s properties to those necessary for success in the application. Case studies address the successes and failures of particular materials from each of the classes in biological applications.
MW 1:05pm-2:20pm

BENG 5350b / MSCI 5350b / PATH 5630b, Biomaterial-Tissue Interactions  Themis Kyriakides

Study of the interactions between tissues and biomaterials, with an emphasis on the importance of molecular- and cellular-level events in dictating the performance and longevity of clinically relevant devices. Attention to specific areas such as biomaterials for tissue engineering and the importance of stem/progenitor cells, as well as biomaterial-mediated gene and drug delivery.
TTh 9am-10:15am

BENG 5410a / MSCI 6410a, Physical and Chemical Basis of Bioimaging and Biosensing  Douglas Rothman, Daniel Coman, and Jason Bini

Basic principles and technologies for imaging and sensing the chemical, electrical, and structural properties of living tissues and biological macromolecules. Topics include magnetic resonance spectroscopy, MRI, positron emission tomography, and molecular imaging with MRI and fluorescent probes.
MW 1:05pm-2:20pm

BENG 5415a, Practical Applications of Bioimaging and Biosensing  Daniel Coman, Jason Bini, and Evelyn Lake

Detecting, measuring, and quantifying the structural and functional properties of tissue is of critical importance in both biomedical research and medicine. This course focuses on the practicalities of generating quantitative results from raw bioimaging and biosensing data to complement other courses focus on the theoretical foundations which enable the collection of these data. Participants in the course work with real, cutting-edge data collected here at Yale. They become familiar with an array of current software tools, denoising and processing techniques, and quantitative analysis methods that are used in the pursuit of extracting meaningful information from imaging data. The subject matter of this course ranges from bioenergetics, metabolic pathways, molecular processes, brain receptor kinetics, protein expression and interactions to wide spread functional networks, long-range connectivity, and organ-level brain organization. The course provides a unique hands-on experience with processing and analyzing in vitro and in vivo bioimaging and biosensing data that is relevant to current research topics. The specific imaging modes which are covered include in vivo magnetic resonance spectroscopy (MRS) and spectroscopic imaging (MRSI), functional, structural, and molecular imaging (MRI), wide-field fluorescent optical imaging, and positron emission tomography (PET). The course provides the necessary background in biochemistry, bioenergetics, and biophysics for students to motivate the image manipulations which they learn to perform. Prerequisites: Math through first order differential equations, PHYS 180/181, CHEM 161, BIOL 101/102, BENG 249 or other experience with scientific software like MATLAB, BENG 350 and BENG 410 (both of which can be taken at the same time as this course)  0 Course cr
F 1pm-2:15pm

BENG 5450a, Biomedical Image Processing and Analysis  James Duncan and Lawrence Staib

This course is an introduction to biomedical image processing and analysis, covering image processing basics and techniques for image enhancement, feature extraction, compression, segmentation, registration, and motion analysis including traditional and machine-learning techniques. Students learn the fundamentals behind image processing and analysis methods and algorithms with an emphasis on biomedical applications.
TTh 11:35am-12:50pm

BENG 5482a, Physics of Magnetic Resonance Spectroscopy in Vivo  Graeme Mason

The physics of chemical measurements performed with nuclear magnetic resonance spectroscopy, with special emphasis on applications to measurement studies in living tissue. Concepts that are common to magnetic resonance imaging are introduced. Topics include safety, equipment design, techniques of spectroscopic data analysis, and metabolic modeling of dynamic spectroscopic measurements.
MW 11:35am-12:50pm

BENG 5485b / INP 9585b, Fundamentals of Neuroimaging  Fahmeed Hyder, Elizabeth Goldfarb, and Douglas Rothman

The neuroenergetic and neurochemical basis of several dominant neuroimaging methods, including fMRI. Topics range from technical aspects of different methods to interpretation of the neuroimaging results. Controversies and/or challenges for application of fMRI and related methods in medicine are identified.
W 3:30pm-5:20pm

BENG 5550b / MSCI 5550b, Vascular Mechanics  Jay Humphrey

This course is designed to enable students to apply methods of continuum biomechanics to study diverse vascular conditions and treatments, including aging, atherosclerosis, aneurysms, effects of hypertension, design of tissue-engineered constructs, and vein grafts from an engineering perspective. Emphasis is placed on ensuring that the mechanics is driven by advances in the vascular mechanobiology.
TTh 1:05pm-2:20pm

BENG 5570b / MENG 8370b, Computational Mechanics  Martin Pfaller

This course integrates fundamental concepts from nonlinear continuum mechanics and finite element methods applied to solid and fluid mechanics, focusing on theoretical understanding and numerical techniques. Topics covered are fundamentals of tensor calculus, kinematics, balance equations, constitutive relationships, geometric and material nonlinearities, nonlinear solution strategies, stability, nonlinear dynamics, errors, convergence, and adaptivity. Applications in biomedical engineering are stressed throughout the course. Prerequisites: fundamentals in calculus, differential equations, and linear algebra.
TTh 9am-10:15am

BENG 5580b, Multiscale Models of Biomechanical Systems  Stuart Campbell

Current methods for simulating biomechanical function across biological scales, from molecules to organ systems of the human body. Theory and numerical methods; case studies exploring recent advances in multiscale biomechanical modeling. Includes computer laboratory sessions that introduce relevant software packages.
MW 1:05pm-2:20pm

BENG 5630a, Immunoengineering  Tarek Fahmy

An advanced class that introduces immunology principles and methods to engineering students. The course focuses on biophysical principles and biomaterial applications in understanding and engineering immunity. The course is divided into three parts. The first part introduces the immune system: organs, cells, and molecules. The second part introduces biophysical characterization and quantitative modeling in understanding immune system interactions. The third part focuses on intervention, modulation, and techniques for studying the immune system with emphasis on applications of biomaterials for intervention and diagnostics.
TTh 11:35am-12:50pm

BENG 5680b, Topics in Immunoengineering  Tarek Fahmy

This course addresses the intersection of immunobiology with engineering and biophysics. It invokes engineering tools, such as biomaterials, solid-state devices, nanotechnology, biophysical chemistry, and chemical engineering, toward developing newer and effective solutions to cancer immunotherapy, autoimmune therapy, vaccine design, transplantation, allergy, asthma, and infections. The central theme is that dysfunctional immunity is responsible for a wide range of disease states and that engineering tools and methods can forge a link between the basic science and clinically translatable solutions that will potentially be “modern cures” to disease. This course is a follow-up to ENAS 553 and focuses more on the clinical translation aspect as well as new understandings in immunology and how they can be translated to the clinic and eventually to the market. Prerequisites: ENAS 553, differential equations, and advanced calculus.
MW 4pm-5:15pm

BENG 5724a, Topics in Computational and Systems Biology  Purushottam Dixit

This course covers topics related to modeling biological networks across time and length scales. Specifically, the course covers models of intracellular signaling networks, transcriptional regulation networks, cellular metabolic networks, and ecological networks in microbial consortia. For each type of network, we cover the biological basics, standard mathematical treatments including deterministic and stochastic modeling, methods to infer model parameters from data, and new machine-learning based inference approaches. The required mathematical methods are briefly covered. The course assignments involve coding in MATLAB.
MW 2:35pm-3:50pm

BENG 5767b, Systems Biology of Cell Signaling  Andre Levchenko

This course designed for graduate and advanced undergraduate students is focused on systems biology approaches to the fundamental processes underlying the sensory capability of individual cells and cell-cell communication in health and disease. The course is designed to provide deep treatment of both the biological underpinnings and mathematical modeling of the complex events involved in signal transduction. As such, it can be attractive to students of biology, bioengineering, biophysics, computational biology, and applied math. The class is part of the planned larger track in systems biology, being one of its final, more specialized courses. In spite of this, each lecture has friendly introduction to the specific topic of interest, aiming to provide sufficient refreshment of the necessary knowledge. The topics have been selected to represent both cutting-edge directions in systems analysis of signaling processes and exciting settings to explore, making learning complex notions more enjoyable. Prerequisites: basic knowledge of biochemistry and cell biology, as well as programming experience and basic notions from probability theory and differential equations.
MW 4pm-5:15pm

BENG 5900a or b, Research Seminars in Biomedical Engineering  Kathryn Miller-Jensen

The purpose of this course is to introduce graduate students to state-of-the-art research in all areas of biomedical engineering (BME), as well as related disciplines, so that students understand the range of current research questions that are being addressed. An important goal is to encourage students to explore research topics beyond their particular field of study and develop the ability to contextualize their work in terms of larger research questions in BME. We therefore require that BME Ph.D. students enrolled in this course attend at least eight research seminars during the term: six must be part of the BME seminar and symposia series, and two can be from any other relevant Yale graduate department/program seminar series. This course is graded Sat/Unsat with sign-in sheets used to monitor attendance. Required of first- and second-year BME Ph.D. students.  0 Course cr
HTBA

BENG 5910b, Effective Fellowship Grant Writing: From Concept to Submission  Fadi Akar

This course is designed to equip participants with the essential skills and strategies needed to prepare successful fellowship grant applications. It covers the entire grant writing process, from understanding funding opportunities and requirements to developing a compelling proposal. Participants learn how to clearly define research goals, align their projects with funding criteria, and craft persuasive narratives that effectively communicate their ideas. The course also emphasizes the importance of strong supporting documents, including CVs, budgets, and letters of recommendation. Through practical exercises, peer reviews, and expert feedback, participants refine their writing techniques, enhance proposal clarity, and increase their chances of securing funding. The course is intended for seniors and graduate students in biomedical engineering or a related biomedical sciences department.
Th 1pm-2:50pm

BENG 5990a or b, Special Investigations  Staff

Faculty-supervised individual projects with emphasis on research, laboratory, or theory. Students must define the scope of the proposed project with the faculty member who has agreed to act as supervisor, and submit a brief abstract to the director of graduate studies for approval.
HTBA

CENG 5210a / MSCI 5210a, Classical and Statistical Thermodynamics  Mingjiang Zhong

A unified approach to bulk-phase equilibrium thermodynamics, bulk-phase irreversible thermodynamics, and interfacial thermodynamics in the framework of classical thermodynamics, and an introduction to statistical thermodynamics. Both the activity coefficient and the equations of state are used in the description of bulk phases. Emphasis on classical thermodynamics of multicomponents, including concepts of stability and criticality, curvature effect, and gravity effect. The choice of Gibbs free energy function covers applications to a broad range of problems in chemical, environmental, biomedical, and petroleum engineering. The introduction includes theory of Gibbs canonical ensembles and the partition functions, fluctuations; Boltzmann statistics; Fermi-Dirac and Bose-Einstein statistics. Application to ideal monatomic and diatomic gases is covered.
MW 9am-10:15am

CENG 6020b, Chemical Reaction Engineering  Staff

Applications of physical-chemical and chemical-engineering principles to the design of chemical process reactors. Ideal reactors treated in detail in the first half of the course, practical homogeneous and catalytic reactors in the second.
TTh 1:05pm-2:20pm

CENG 6030b, Energy, Mass, and Momentum Processes  Amir Haji-Akbari

Application of continuum mechanics approach to the understanding and prediction of fluid flow systems that may be chemically reactive, turbulent, or multiphase.
TTh 1:05pm-2:20pm

CENG 6140b, Surface and Thin Film Characterization  Staff

Fundamental and practical aspects of spectroscopy, diffraction, and microscopy related to the structural and chemical characterization of surfaces and thin films. Emphasis on identification of adsorbed species by vibrational spectroscopy, determination of the chemical state of the surface by photoelectron spectroscopy, quantitative methods in surface analysis, determination of surface structure by scanning probe microscopy techniques and diffraction methods, and recent advances in surface characterization.
TTh 9am-10:15am

CENG 6150b / MSCI 6150b, Synthesis of Nanomaterials  Staff

This course focuses on the synthesis and engineering of nanomaterials. We also introduce different types of nanomaterials, unique properties at the nanoscale, measurement, and important applications of nanomaterials (including biomedical, electronic, and energy applications). Synthesis methods covered include gas phase and high vacuum techniques (CVD, MOCVD) as well as wet chemistry techniques such as reduction of metal salts, sonochemistry, and sol gel methods. Taking sample applications, we discuss the properties necessary for each, and how to control these properties through synthesis control, such as by using templating methods.
TTh 9am-10:15am

CENG 9000a or b, Seminar in Chemical and Environmental Engineering  Staff

The purpose of this course is to introduce graduate students to state-of-the-art research and engineering practice across chemical and environmental engineering so that students develop an appreciation of the breadth of academic and applied research questions being addressed at Yale, including by senior Ph.D. students and postdoctoral researchers, and at other universities and organizations. A key component of the course is to encourage students to develop the ability to contextualize their own work in relation to broader research and applications questions in CEE. The course centers on participation in seminars during the term. The course also encourages students to organize journal clubs, enhance their research literacy, and network with invited speakers. The course requires extensive participation by students as speakers, moderators, and active audience participation. Finally, it helps foster a sense of community within CEE. Attendance is monitored as appropriate. The course is graded Satisfactory/Unsatisfactory. All CEE master’s and Ph.D. students are encouraged to enroll and participate.   0 Course cr
HTBA

CPSC 5190a, Full Stack Web Programming  Alan Weide

This course introduces students to a variety of advanced software engineering and programming techniques in the context of full-stack web programming. The focus of the course includes both client- and server-side programming (and database programming), client/server communication, user interface programming, and parallel programming.
MW 9am-10:15am

CPSC 5260a, Building Distributed Systems  Y. Richard Yang

Ubiquitous services such as Google, Facebook, and Amazon run on the back of massive distributed systems. This course covers the fundamental principles, abstractions, and mechanisms that inform the design of such systems, as well as the practical details of real-world implementations. Technical topics covered include properties such as consistency, availability, durability, isolation, and failure atomicity; as well as protocols such as RPC, consensus, consistent hashing, and distributed transactions. The final project involves implementing a real-world distributed service.
HTBA

CPSC 5310a, Computer Music: Algorithmic and Heuristic Composition  Scott Petersen

Study of the theoretical and practical fundamentals of computer-generated music. Music and sound representations, acoustics and sound synthesis, scales and tuning systems, algorithmic and heuristic composition, and programming languages for computer music. Theoretical concepts are supplemented with pragmatic issues expressed in a high-level programming language.
MW 11:35am-12:50pm

CPSC 5370a, Database Systems  Avi Silberschatz

An introduction to database systems. Data modeling. The relational model and the SQL query language. Relational database design, integrity constraints, functional dependencies, and natural forms. Object-oriented databases. Implementation of databases: file structures, indexing, query processing, transactions, concurrency control, recovery systems, and security.
TTh 9am-10:15am

CPSC 5371a, Database Design and Implementation  Robert Soule

This course covers advanced topics in Database Systems, expanding on the material covered in CPSC 437/537. Topics covered include complex data types, application development, big data, data analytics, parallel and distributed storage, parallel and distributed query processing, advanced indexing techniques, advanced relational database design, and object-based databases.
TTh 9am-10:15am

CPSC 5380a, Big Data Systems: Trends and Challenges  Anurag Khandelwal

Today’s Internet-scale applications and cloud services generate massive amounts of data. At the same time, the availability of inexpensive storage has made it possible for these services and applications to collect and store every piece of data they generate, in the hopes of improving their services by analyzing the collected data. This introduces interesting new opportunities and challenges designing systems for collecting, analyzing, and serving the so-called big data. This course looks at technology trends that have paved the way for big data applications, surveys state-of-the-art systems for storage and processing of big data, and considers future research directions driven by open research problems. Our discussions span topics such as cluster architecture, big data analytics stacks, scheduling and resource management, batch and stream analytics, graph processing, ML/AI frameworks, and serverless platforms and disaggregated architectures.
MW 4pm-5:15pm

CPSC 5390a, Software Engineering  Timos Antonopoulos

Introduction to building a large software system in a team. Learning how to collect requirements and write a specification. Project planning and system design. Increasing software reliability: debugging, automatic test generation. Introduction to type systems, static analysis, and model checking.
TTh 11:35am-12:50pm

CPSC 5410a, Verifiable, Private, Decentralized Computing in the Age of AI  Ben Fisch

You type your question into ChatGPT and get back a response. How do you trust its accuracy? Perhaps you have reviewed the latest published benchmark results for GPT-4, or trust that others have. But how do you know the response you are getting from OpenAI’s servers is the true output of GPT-4? Perhaps due to a bug or system overload your question was handled by a weaker AI model. Or worse, perhaps the servers were hacked by someone maliciously giving you incorrect results. And, how do you trust that the sensitive questions you are sending to ChatGPT will not be leaked or used against you? This is a course in cryptographic proof systems. In the digital world today, we trust services to perform many kinds of computations on our data, from managing financial ledgers and databases to complex analytics, such as large-language model (LLM) inference. We trust these services not only to operate correctly, but also to keep our information private. Cryptographic systems allow us to remove this trust. A “succinct” cryptographic proof enables a service to attach a small certificate on the correctness of its computation which can be verified easily on a low-power device, even if the original computation was extremely complex. The proof of some ML computation that was run for hours on a GPU farm can fit in an email and take just milliseconds to verify on a mobile device! Beyond correctness, a “zero-knowledge” proof system enables us to prove knowledge of secret information, including hidden inputs to a computation that achieves a certain output. For instance, OpenAI could prove that the response is the true output of GPT-4 (a proprietary model) without revealing sensitive details about the model itself. In industry, the market for cryptographic proofs is currently around $75 million and projected to reach $10 billion by 2030, according to some estimates. Cryptographic proofs have become the leading technology for scaling blockchains and achieving privacy in cryptocurrencies. Verifiable and zero-knowledge computing also create an important foundation for decentralizing AI services. Training and serving large models currently require vast resources, leading to centralization. Decentralized ML networks offer a compelling alternative—letting many independent operators contribute incremental work to the overall task using their own resources big or small, from a single GPU to a server cluster, and earn a share of the payments clients make to use the service. In such a setting it is critical to verify that operators contribute incremental work correctly. Or, they may use private data to jointly train an ML model. Succinct zero-knowledge proofs would enable these operators to prove correctness of their work without revealing sensitive data. We cover some of the other challenges and directions in decentralized model training and inference, such as reducing the amount of data that needs to be communicated between physically distributed islands of hardware and the potential role of reinforcement learning.
MW 1:05pm-2:20pm

CPSC 5420a, Theory of Computation  Dylan McKay

This course first introduces core, traditional ideas from the theory of computation with more modern ideas used in the process, including basic ideas of languages and automata. Building on the core ideas, the course then covers a breadth of topics in modular units, where each unit examines a new model and potentially a new perspective on computation. Topics may include: basic notions of Complexity Theory, provability and logic, circuits and non-uniform computation, randomized computation, quantum computation, query-based computation, notions of machine learning, compression, and algebraic models of computation. Additional topics might be introduced in lectures or student projects, according to student interests, including mechanism design, voting schemes, cryptography, biological computation, distributed computation, and pseudorandomness. Prerequisite: One of CPSC 365, 366, or 368 is required. This course is a proof-based theory course and mathematical maturity is expected.
TTh 4pm-5:15pm

CPSC 5470a, Introduction to Quantum Computing  Yongshan Ding

This course introduces the fundamental concepts in the theory and practice of quantum computing. Topics covered include information processing, quantum programming, quantum compilation, quantum algorithms, and error correction. The objective of the course is to engage students in applying fresh thinking to what computers can do. We establish an understanding of how quantum computers store and process data, and we discover how they differ from conventional digital computers. We anticipate this course will be of interest to students working in computer science, electrical engineering, physics, or mathematics. Students must be comfortable with programming, discrete probability, and linear algebra. Prior experience in quantum computing is useful but not required.
MW 11:35am-12:50pm

CPSC 5540a, Software Analysis and Verification  Ruzica Piskac

Introduction to concepts, tools, and techniques used in the formal verification of software. State-of-the-art tools used for program verification; detailed insights into algorithms and paradigms on which those tools are based, including model checking, abstract interpretation, decision procedures, and SMT solvers.
TTh 2:35pm-3:50pm

CPSC 5570a, Sensitive Information in a Connected World  Michael Fischer

Issues of ownership, control, privacy, and accuracy of the huge amount of sensitive information about people and organizations that is collected, stored, and used by today’s ubiquitous information systems. Readings consist of research papers that explore both the power and the limitations of existing privacy-enhancing technologies such as encryption and “trusted platforms.”
MW 2:35pm-3:50pm

CPSC 5585a, Probabilistic Programming  Alex Lew

Introduction to probabilistic programming, a field at the intersection of programming languages, probability theory, and artificial intelligence. The central idea in probabilistic programming is to represent probabilistic models as programs, and use special metaprograms (e.g., compilers and interpreters) to implement inference, learning, and prediction. Topics include the mathematical foundations of probabilistic programming (including the basics of synthetic probability theory); probabilistic modeling (including Bayesian approaches to regression, clustering, topic modeling, and structure learning); and exact and approximate algorithms for inference (including variable elimination, knowledge compilation, Markov chain Monte Carlo, sequential Monte Carlo, and variational inference). Coursework includes biweekly programming assignments, in-class quizzes, and a final project completed in groups of up to three students. Prerequisites: CPSC 2020, or equivalent background in probability. CPSC 2230, or equivalent programming maturity (with permission of instructor). Facility with Python programming and functional programming is expected. Although assignments are programming-oriented rather than proof-oriented, a degree of mathematical maturity (e.g., having taken at least one proof-based math course) is recommended.
TTh 4pm-5:15pm

CPSC 5590a, Building Interactive Machines  Marynel Vazquez

This advanced course brings together methods from machine learning, computer vision, robotics, and human-computer interaction to enable interactive machines to perceive and act in a variety of environments. Part of the course examines approaches for perception with different sensing devices and algorithms; the other part focuses on methods for decision-making and applied machine learning for control. The course is a combination of lectures, state-of-the-art reading, presentations and discussions, programming assignments, and a final team project. Prerequisites: CPSC 570 and understanding of probability, differential calculus, linear algebra, and planning (in Artificial Intelligence). Programming assignments require proficiency in Python and high-level familiarity with C++. Students who do not fit this profile may be allowed to enroll with the permission of the instructor.
MW 11:35am-12:50pm

CPSC 5610a, Statistics and Learning Theory for Computer Scientists  Manolis Zampetakis

This course explores the basic mathematical foundation of learning theory and statistics from a computer science perspective. This is a proof based course and mathematical maturity is necessary to follow the material of the course. The course is divided in three parts. The first part covers the basic mathematical theory of that is used in statistical analysis and includes: (a) log-likelihood estimation for parametric models, (b) density estimation, and (c) hypothesis testing, p-values and confidence intervals. The second part covers statistical learning theory: (a) definition of PAC learning and agnostic learning, (b) uniform convergence, (c) learnability via VC dimension and Rademacher complexity, and (d) computational aspects of learning theory. The final part is related to modern challenges in statistics and learning theory from a computational perspective and includes: (a) statistical analysis with corrupted data, (b) missing data and causal inference, (c) computationally efficient methods for learning theory, and (d) statistical analysis while preserving privacy, i.e., the notion of differential privacy and its applications. Prerequisites: This is an advanced course, which requires mathematical maturity and comfort with multivariate calculus, linear algebra, and probability theory. The course also assumes prior knowledge of discrete mathematics and algorithms (CPSC 2020 and CPSC 3650 or equivalent).
TTh 2:35pm-3:50pm

CPSC 5640a, Algorithms and their Societal Implications  Nisheeth Vishnoi

Today’s society comprises humans living in an interconnected world that is intertwined with a variety of sensing, communicating, and computing devices. Human-generated data is being recorded at unprecedented rates and scales, and powerful AI and ML algorithms, which are capable of learning from such data, are increasingly controlling various aspects of modern society: from social interactions. These data-driven decision-making algorithms have a tremendous potential to change our lives for the better, but, via the ability to mimic and nudge human behavior, they also have the potential to be discriminatory, reinforce societal prejudices, violate privacy, polarize opinions, and influence democratic processes. Thus, designing effective tools to govern modern society which reinforce its cherished values such as equity, justice, democracy, health, privacy, etc. has become paramount and requires a foundational understanding of how humans, data, and algorithms interact. This course is for students who would like to understand and address some of the key challenges and emerging topics at the aforementioned interplay between computation and society. On the one hand, we study human decision-making processes and view them through the lens of computation, and on the other hand we study and address the limitations of artificial decision-making algorithms when deployed in various societal contexts. The focus is on developing solutions through a combination of foundational work such as coming up with the right definitions, modeling, algorithms, and empirical evaluation. The current focus is on bias and privacy, with additional topics including robustness, polarization, and democratic representation. The grade will be based on class participation and a project. The project grade will be determined by a midterm and endterm report/presentation. The course has four primary modules: (1) Data: human-generated data; data collection and aggregation; (2) Decision-Making Algorithms: human decision-making algorithms; traditional algorithmic decision-making models and methods; machine learning-based decision-making models and methods; (3) Bias: socio-technical contexts and underlying computational problems; definitions of fairness; interventions for ensuring fairness; human biases through the lens of computation; privacy; need for definitions of privacy; differential privacy; beyond differential privacy; (4) Other topics: robustness; polarization; elections and social choice. Solid mathematical and programming background is necessary to enroll in this course. CPSC 365 and S&DS 251 are recommended.
T 9:25am-11:20am

CPSC 5670a, Introduction to Cryptography  Charalampos Papamanthou

This course introduces modern symmetric and public-key cryptography as well as their broad applications, both from a theoretical and practical perspective. There is an initial emphasis on fundamental cryptographic primitives (e.g., block ciphers, pseudorandom functions, pseudorandom generators, one-way functions), their concrete efficiency and implementation, as well as their security definitions and proofs. Ways of combining such primitives that lead to more complex objects used to secure today’s internet (e.g., via TLS), such as key exchange, randomized encryption, message authentication codes, and digital signatures are also studied. The last part of the course is devoted to modern and more advanced applications of cryptography (some of which are deployed at scale today), such as authenticated data structures, zero-knowledge proofs, oblivious RAM, private information retrieval, secret sharing, distributed consensus, and cryptocurrencies. (e.g, Bitcoin).
TTh 11:35am-12:50pm

CPSC 5690a, Randomized Algorithms  James Aspnes

Beginning with an introduction to tools from probability theory including some inequalities like Chernoff bounds, the course covers randomized algorithms from several areas: graph algorithms, algorithms in algebra, approximate counting, probabilistically checkable proofs, and matrix algorithms.
MW 2:35pm-3:50pm

CPSC 5700a, Introduction to Artificial Intelligence  Tesca Fitzgerald

How can we enable computers to make rational, intelligent decisions? This course explores fundamental techniques for Artificial Intelligence (AI), covering topics such as search, planning, learning, and reasoning under uncertainty. Through hands-on programming projects, students learn conceptual, algorithmic, and practical considerations for implementing foundational AI algorithms. By the end of this class, students have an understanding of the history and breadth of AI problems and topics, and are prepared to undertake more advanced courses in robotics, computer vision, natural language processing, and machine learning.
TTh 1:05pm-2:20pm

CPSC 5710a, Trustworthy Deep Learning  Rex Ying

In recent years, deep learning has seen applications in many fields, from science and technology, to finance, humanity, and businesses. However, real-world, high-impact machine learning applications demand more than just model performance. In particular, deep learning models are often required to be “trustworthy,” so that domain experts can trust that the models consistently behave in a way that corresponds to their domain knowledge. For example, medical experts would expect a deep learning diagnosis model to be able to explicitly utilize medical domain knowledge in its prediction; an insurance company would expect a decision on insurance price to be explainable in terms of risk factors; a financial company would expect its fraud detection model to be robust to adversarial attacks; a physicist would expect models to provide consistency with the underlying laws. This course introduces various fields of trustworthy deep learning, including model robustness, defenses for adversarial attacks, interpretability, explainability, fairness, privacy, domain adaptation, rules, and constraints. The course covers some of these aspects in the context of graph neural networks but also covers many other ML models in general deep learning, natural language processing, and computer vision. Prerequisites: a course in linear algebra and multi-variable calculus and familiarity with PyTorch and other common Python libraries such as Numpy, Sklearn. Deep learning courses such as CPSC 452 or 453 are recommended.
TTh 4pm-5:15pm

CPSC 5720a, Intelligent Robotics  Brian Scassellati

Introduction to the construction of intelligent, autonomous systems. Sensory-motor coordination and task-based perception. Implementation techniques for behavior selection and arbitration, including behavior-based design, evolutionary design, dynamical systems, and hybrid deliberative-reactive systems. Situated learning and adaptive behavior.
MWF 10:30am-11:20am

CPSC 5740a, Computational Intelligence for Games  James Glenn

A seminar on current topics in computational intelligence for games, including developing agents for playing games, procedural content generation, and player modeling. Students read, present, and discuss recent papers and competitions, and complete a term-long project that applies some of the techniques discussed during the term to a game of their choice.
TTh 9am-10:15am

CPSC 5750a / ECE 5750a / INP 7575a, Computational Vision and Biological Perception  Steven Zucker

An overview of computational vision with a biological emphasis. Suitable as an introduction to biological perception for computer science and engineering students, as well as an introduction to computational vision for mathematics, psychology, and physiology students.
MW 2:35pm-3:50pm

CPSC 5791a, Building Game Engines  Michael Shah

This course teaches the fundamentals of building a reusable software architecture by building games. This is a programming-intensive course where the end product of this course is a data-driven game engine that students work in small teams to implement in a systems programming language (e.g. C, C++, D, etc.). Students apply data structures, algorithms, and systems programming skills in the domain of games. Discussion and implementation of the components of a game engine may include: resource management (allocators, resource managers, serialization), abstraction (design patterns, game objects, scripting, graphics layers), graphics management algorithms (scene graphs, level of detail), physics (linear algebra, collision detection and resolution algorithms), artificial intelligence (e.g. pathfinding,decision making), and performance (concurrency, parallelism, math). Students work on a final course project for their portfolio. Prerequisite: previous experience with data structures and systems programming comparable to CPSC 223 and CPSC 323.
MW 4pm-5:15pm

CPSC 5800a, Introduction to Computer Vision  Alex Wong

This course focuses on fundamental topics in computer vision. We begin with the image formation process and discuss the role of camera models and intrinsic calibration in perspective projection. Basic image processing techniques (i.e. filtering) is introduced. After which, we discuss techniques to describe an image, from edges to feature descriptors and methods to establish correspondences between different images of the same scene. The course additionally covers topics in recognition (i.e. image classification, segmentation, detection, etc.) and reconstruction (i.e. stereo, structure-from-motion, optical flow). Machine learning and deep learning based methods in a subset of the topics covered are also introduced. Students get hands-on experience in implementing the techniques covered in the class and applying them to real world datasets and applications. Students taking this course must have successfully passed courses in data structures and object-oriented programming (e.g. CPSC 223a or equivalent courses) and foundational mathematical tools such as discrete math and linear algebra (e.g. CPSC 202 or equivalent courses). It is recommended that students have taken or successfully passed calculus (e.g. MATH 112, MATH 115, MATH 120, or equivalent courses) and linear algebra (e.g. MATH 225, or equivalent courses). A background in statistics, machine learning and deep learning is useful but not required. Experience in programming with Python is preferable, as we use it for assignments and projects. Familiarity with Google Colab and numerical and image processing packages (i.e. NumPy, SciPy, and Sci-kit Image) is helpful throughout the course.
TTh 1:05pm-2:20pm

CPSC 5880a, Advances in Foundation Models  Arman Cohan

​​Foundation models are a recent class of AI models that are large-scale in terms of number of parameters and are trained on broad data (generally using self-supervision at scale). These models have demonstrated exceptional capabilities in natural language processing, computer vision, and other tasks. Examples of foundation models are GPT-4, ChatGPT, GPT-3, Dall-E, Stable Diffusion, etc. In this course, we discuss building blocks of foundation models, including transformers, self-supervised learning, transfer learning, learning from human feedback, power of scale, large language models, in-context learning, chain-of-thought prompting, parameter-efficient fine-tuning, vision transformers, diffusion models, generative modeling, safety, ethical and societal considerations, their impact, etc. While the course primarily focuses on advances on large language models, we also cover foundation models in computer vision, as well as multi-modal foundation models. Prerequisite: either CPSC 4770/5770 or CPSC 4800/5800, or permission of the instructor.
MW 2:35pm-3:50pm

CPSC 6110a, Topics in Computer Science and Global Affairs  Joan Feigenbaum and Ted Wittenstein

This course focuses on “socio-technical” problems in computing and international relations. These are problems that cannot be solved through technological progress alone but rather require legal, political, or cultural progress as well. Examples include but are not limited to cyber espionage, disinformation, ransomware attacks, and intellectual-property theft. This course is offered jointly by the SEAS Computer Science Department and the Jackson School of Global Affairs. It is addressed to graduate students who are interested in socio-technical issues but whose undergraduate course work may not have addressed them; it is designed to bring these students rapidly to the point at which they can do research on socio-technical problems. Prerequisites: Basics of cryptography and computer security (as covered in Yale’s CPSC 467), networks (as covered in Yale’s CPSC 433), and databases (as covered in Yale’s CPSC 437) helpful but not required.
W 4pm-5:55pm

CPSC 6400a / AMTH 6400a / MATH 6400a, Topics in Numerical Computation  Eric Michielssen

This course discusses several areas of numerical computing that often cause difficulties to non-numericists, from the ever-present issue of condition numbers and ill-posedness to the algorithms of numerical linear algebra to the reliability of numerical software. The course also provides a brief introduction to “fast” algorithms and their interactions with modern hardware environments. The course is addressed to Computer Science graduate students who do not necessarily specialize in numerical computation; it assumes the understanding of calculus and linear algebra and familiarity with (or willingness to learn) either C or FORTRAN. Its purpose is to prepare students for using elementary numerical techniques when and if the need arises.
MW 9am-10:15am

CPSC 6440a / MATH 7440a, Geometric and Topological Methods in Machine Learning  Smita Krishnaswamy

This course provides an introduction to geometric and topological methods in data science. Our starting point is the manifold hypothesis: that high dimensional data live on or near a much lower dimensional smooth manifold. We introduce tools to study the geometric and topological properties of this manifold in order to reveal relevant features and organization of the data. Topics include: metric space structures, curvature, geodesics, diffusion maps, eigenmaps, geometric model spaces, gradient descent, data embeddings and projections, and topological data analysis (TDA) in the form of persistence homology and their associated “barcodes.” We see applications of these methods in a variety of data types. Prerequisites: MATH 2250 or 2260; MATH 2550 or 2560; MATH 3020; and CPSC 1001. 
TTh 1:05pm-2:20pm

CPSC 6900a, Independent Project I  Staff

Independent Project I. By arrangement with faculty.
HTBA

CPSC 6910a, Independent Project II  Staff

By arrangement with faculty.
HTBA

CPSC 6920a, Independent Project  Holly Rushmeier

Individual research for students in the M.S. program. Requires a faculty supervisor and the permission of the director of graduate studies.
HTBA

CPSC 6930a, MS Thesis Research I  Staff

First term of thesis research for students in the two-year MS program in Computer Science None
HTBA

CPSC 6940a, MS Thesis Research II  Holly Rushmeier

Second term of thesis research for students in the two-year MS program in Computer Science.
HTBA

CPSC 7430a, Topics in Flow-Based Generative Modeling and Optimal Transport  Staff

Generative AI has had an immense impact on many scientific disciplines over the last several years. A driving force has been the development of scalable algorithms that allow practitioners to “flow” from simple distributions to complex ones. In this course, we explore the mathematical and algorithmic foundations of generative models through the lens of flow-based transport. Topics include flow matching, stochastic interpolants, diffusion models, optimal transport maps, Schrödinger bridges, and their applications and extensions. Prerequisites: probability theory, multivariate calculus, and linear algebra, some programming, and permission of the instructor.
HTBA

CPSC 7551b, The Economic Impacts of Generative AI  Nicole Immorlica

Generative AI is transforming how goods and information are produced, processed, and incorporated into economic activity. This graduate-level reading course examines the economic foundations and implications of this emerging technology, with a focus on theoretical frameworks drawn from economics and computer science. We cover a variety of topics, including the macroeconomic impact of AI on productivity and growth, the changing nature of work and expertise, how AI influences human decision-making and team collaboration, the design of incentives and mechanisms for AI alignment, and applications of AI to consumer markets and content ecosystems. Students read and discuss cutting-edge research papers. Each class is organized around a student presentation followed by group discussion. Over the course of the semester, students also develop a research proposal outlining a project related to the economics of generative AI and present it to the class during the final lectures of the course. By the end of the course, students have developed a broad understanding of the key economic questions raised by generative AI, the theoretical tools used to analyze them, and a promising novel research direction that could be further developed in independent projects. Expertise in microeconomics and algorithms helpful but not required.
HTBA

CPSC 9900a, Ethical Conduct of Research for Master’s Students  Staff

This course forms a vital part of research ethics training, aiming to instill moral research codes in graduate students of computer science, math, and applied math. By devling into case studies and real-life examples related to research misconduct, students will grasp core ethical principles in research and academia. The course also offers an opportunity to explore the societal impacts of research in computer science, math, and applied math. This course is designed specifically for first-year graduate students in computer science/applied math/math. Successful completion of the course necessitates in-person attendance on eight occasions; virtual participation will not fulfill this requirement. In cases where illness, job interviews, or unforeseen circumstances prevent attendance, makeup sessions will be offered. This course is 0 credits for YC students.  0 Course cr
HTBA

CPSC 9910a / MATH 9910a, Ethical Conduct of Research  Staff

This course forms a vital part of research ethics training, aiming to instill moral research codes in graduate students of computer science, math, and applied math. By delving into case studies and real-life examples related to research misconduct, students grasp core ethical principles in research and academia. The course also offers an opportunity to explore the societal impacts of research in computer science, math, and applied math. This course is designed specifically for first-year graduate students in computer science, applied math, and math. Successful completion of the course necessitates in-person attendance on eight occasions; virtual participation does not fulfill this requirement. In cases where illness, job interviews, or unforeseen circumstances prevent attendance, makeup sessions are offered.  0 Course cr
HTBA

CPSC 9920b, Writing and Presentations One-on-One  Janet Kayfetz

This is a unique class for Ph.D. students and postdocs. All meetings are conducted as one-on-one individual conferences. There are no assignments; participants work solely on their writing-in-progress and current presentation projects. Our goal is to produce excellent texts and presentations that meet specific deadlines. For writing, we look at content, rhetorical positioning and audience, organizational logic, knowledge claims, style, tone, sentence-level language, grammar, transitions, readability, clarity, and rigor. For presentations, we look at the presentation story, positioning and organization, fit and balance of the content into the allotted time, visuals, delivery, pronunciation, responding to questions, and personal style.  0 Course cr
HTBA

ECE 5021b / S&DS 5510b, Stochastic Processes  Shuangping Li

Introduction to the study of random processes, including Markov chains, Markov random fields, martingales, random walks, Brownian motion, and diffusions. Techniques in probability such as coupling and large deviations. Applications chosen from image reconstruction, Bayesian statistics, finance, probabilistic analysis of algorithms, genetics, and evolution.
MW 1:05pm-2:20pm

ECE 5111b, Photonics and Optical Electronics  Jung Han

A survey of the enabling components and devices that constitute modern optical communication systems. Focus on the physics and principles of each functional unit, its current technological status, design issues relevant to overall performance, and future directions.
TTh 2:35pm-3:50pm

ECE 5750a / CPSC 5750a / INP 7575a, Computational Vision and Biological Perception  Steven Zucker

An overview of computational vision with a biological emphasis. Suitable as an introduction to biological perception for computer science and engineering students, as well as an introduction to computational vision for mathematics, psychology, and physiology students.
MW 2:35pm-3:50pm

ECE 7181b, Advanced Electronic Devices  Mengxia Liu

The science and technology of semiconductor electron devices. Topics include compound semiconductor material properties and growth techniques; heterojunction, quantum well, and superlattice devices; quantum transport; graphene and other 2-D material systems.
TTh 11:35am-12:50pm

ECE 8081b / MSCI 8081, Electrochemical Energy Storage Devices and Engineering  Liangbing Hu

The primary goal of this course is to introduce students to advanced energy storage devices and engineering. Through a combination of theoretical instruction, hands-on laboratory experiments, and practical demonstrations, a comprehensive picture of battery science and technology are illustrated, including electrochemistry basics, materials development, system designs and integration, fabrication, and manufacturing. Students learn skills such as how to measure impedance spectra and charge/discharge curves for different battery structures, and modify the battery materials/design accordingly to improve performance. By the end of the course, students are equipped with the knowledge and skills necessary to understand the design, fabrication, and application of batteries in various fields.
TTh 1:05pm-2:20pm

ECE 8400a, Detection and Estimation  Dionysis Kalogerias

Detection and Estimation refers to the development and study of statistical theory and methods in settings involving stochastic signals and, more generally, stochastic processes or stochastic data, where the goal is (optimal) testing of possibly multiple hypotheses regarding the generative model of the data, (optimal) signal estimation from potentially noisy measurements/observations, and parameter estimation whenever parametric signal/data models are available. Although these problems often come up in the context of signal processing and communications, the concepts are fundamental to the basic statistical methodologies used broadly across science, medicine, and engineering. The course has been designed from a contemporary perspective, and includes new and cutting-edge topics such as risk-aware statistical estimation and intrinsic links with stochastic optimization and statistical learning.
TTh 2:35pm-3:50pm

ECE 8680a, Emerging Materials and Technologies toward Sustainability  Liangbing Hu

The goal of this course is to demonstrate the role of new materials and emerging technologies in solving one of the most critical socio-economic issues of our time—sustainability. The course focuses on electrochemical, electrical, optical, thermal, and mechanically functional materials and their use in energy devices. Topics to be covered include electrochemical energy conversion and storage (fuel cells and batteries), catalysts and membrane separations (fossil fuel and biomass energy conversion), electrified heating (Joule, plasma, microwave), solar thermal and fuel, thermoelectrics, energy efficient lighting, and building energy savings (light, thermal).
TTh 1:05pm-2:20pm

ECE 8760a, Silicon Compilation  Rajit Manohar

A course for seniors and first-year graduate students on compiling computations into digital circuits using asynchronous design techniques. Emphasis is on the synthesis of circuits that are robust to uncertainties in gate and wire delays by the process of program transformations. Topics include circuits as concurrent programs, delay-insensitive design techniques, synthesis of circuits from programs, timing analysis and performance optimization, pipelining, and case studies of complex asynchronous designs.
MW 11:35am-12:50pm

ECE 8880a, FPGA-Based Accelerator Design and Implementation  Linghao Song

The goal of this course is to equip students with the skills to design and implement FPGA-based accelerators and deploy the generated bitstreams for real applications. Topics include FPGA programing using high level synthesis (HLS), pipeline and parallelism, dataflows, on-chip and off-chip memories, frequency tuning, and performance modeling and analysis of accelerator architectures. The course includes labs that cover both computation-intensive and memory-intensive workloads. This course includes a capstone project. It also introduces recent developments in FPGA technology and accelerator design. Prerequisites: coding skills and basic knowledge of computer systems.
TTh 11:35am-12:50pm

ECE 8900a, Design of Microwave Circuits and Systems  Arun Natarajan

This course provides an introduction to the analysis and design of microwave circuits and systems. Students develop a foundation in transmission lines, S-parameters, impedance matching, active microwave devices, noise and amplifier design, system-level performance analysis, and practical implementation issues in MMICs and RFICs. Emphasis is placed on connecting device- and circuit-level behavior to the design and characterization of modern high-frequency systems.
MW 2:35pm-3:50pm

ECE 9001b, Decisions and Computations across Networks  A Stephen Morse

Within the field of network science there has long been interest in distributed computation and distributed decision-making problems of many types. Among these are consensus and flocking problems, the multi-robot rendezvous problem, distributed averaging, distributed solutions to linear algebraic equations, social networking problems, localization of sensors in a multisensor network, and the distributed management of robotic formations. The aim of this course is to explain what these problems are and to discuss their solutions. Related concepts from spectral graph theory, rigid graph theory, non-homogeneous Markov chain theory, stability theory, and linear system theory are covered. Prerequisite: although most of the mathematics needed are covered in the lectures, students taking this course should have a working understanding of basic linear algebra.
MW 2:35pm-3:50pm

ECE 9100b / MATH 7120b, Topics in Denoising and Structure Recovery from Data  Boris Landa

Recovering signals and underlying structure from noisy observations is a fundamental problem in many areas of science and engineering. Over the past few decades, a rich body of work has emerged to address this challenge across diverse settings. A common guiding principle is to leverage structural assumptions—such as smoothness, sparsity, or low-rankness in the data—alongside models of the noise to enable effective recovery. This course explores both classical and modern approaches to denoising and structural data recovery, blending theoretical foundations with algorithmic and applied perspectives. Particular emphasis is placed on high-dimensional regimes relevant to modern data analysis, where noise can behave in counterintuitive ways, yet also exhibit predictable patterns that can be exploited for denoising. The course introduces tools and results from high-dimensional probability and random matrix theory that underpin many recent advances in this area.
TTh 2:35pm-3:50pm

ECE 9500a, How to Wirelessly Sense Almost Anything  Tara Boroushaki

This seminar course introduces graduate students and senior undergraduates in EE/CS to wireless sensing. We explore various signals and sensing modalities, including radio frequency, mmWave, acoustics, and visible light. We cover fundamental principles such as battery-free computing, wireless localization, sensor security, health sensing, and backscatter networking. In this course, we examine wireless sensing applications such as seeing through walls, contactless vital sign monitoring, smart homes, ocean IoT, and smart agriculture, as well as multi-modal sensing for augmented reality headsets and robotics. Lectures introduce fundamental technologies through research paper discussions and readings. Additionally, students undertake a semester-long research project.
T 1:30pm-3:25pm

ECE 9511b, Nanofabrication for Bioelectronic Devices  Claudia Cea

This course provides students with a comprehensive introduction to micro- and nanofabrication, with a strong emphasis on their application to bioelectronic systems. The course is divided into two parts. The first part covers the fundamentals of cleanroom-based fabrication, including thin-film deposition, lithographic and alternative patterning techniques, etching processes, and self-assembly strategies. Students also become familiar with key characterization methods such as ellipsometry, profilometry, and advanced microscopy. The second part of the course focuses on applying these fabrication strategies to the development of bioelectronic devices. Lectures explore how materials and device architectures can be designed to interface with neural and peripheral systems, with applications in electrophysiology, electrical stimulation, optogenetics, and chemical neuromodulation. In addition to lectures, students engage in literature-based presentations and discussions. The course culminates in the preparation of a perspective article on a selected area of bioelectronics, giving students the opportunity to synthesize technical knowledge, critically assess research directions, and articulate their own vision for the future of the field.
T 1:30pm-3:25pm

ECE 9631b, Network Algorithms and Stochastic Optimization  Leandros Tassiulas

This course focuses on resource allocation models as well as associated algorithms and design and optimization methodologies that capture the intricacies of complex networking systems in communications computing as well as transportation, manufacturing, and energy systems. Max-weight scheduling, back-pressure routing, wireless opportunistic scheduling, time-varying topology network control, and energy-efficient management are sample topics to be considered, in addition to Lyapunov stability and optimization, stochastic ordering, and notions of fairness in network resource consumption.
MW 11:35am-12:50pm

ECE 9909a or b, Special Investigations  Staff

Faculty-supervised individual projects with emphasis on research, laboratory, or theory. Students must define the scope of the proposed project with the faculty member who has agreed to act as supervisor, and submit a brief abstract to the director of graduate studies for approval.
HTBA

ENAS 5000a / MSCI 5000a, Mathematical Methods I  Owen Miller

A beginning, graduate-level introduction to ordinary and partial differential equations, vector analysis, linear algebra, and complex functions. Laplace transform, series expansion, Fourier transform, and matrix methods are given particular attention. Applications to problems frequently encountered in engineering practice are stressed throughout.
TTh 9am-10:15am

ENAS 5080b and ENAS 5190b, Responsible Conduct of Research  Staff

Required of first-year students. Presentation and discussion of topics and best practices relevant to responsible conduct of research including academic fraud and misconduct, conflict of interest and conflict of commitment, data acquisition and human subjects, use and care of animals, publication practices and responsible authorship, mentor/trainee responsibilities and peer review, and collaborative science.  0 Course cr per term
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ENAS 5390a / INP 5110a, Small Objects  Timothy Newton, Nathan Burnell, and Alyse Guild

This course is offered to graduate and undergraduate students who wish to pursue their own special talents, follow their passions, and expand possibilities and creative impulses to create a small object of their own design. The course is cross-listed with architecture, neuroscience, and engineering & applied science and intentionally brings together students with different backgrounds and experiences. The course explores the ideation, design processes, and fabrication of a functioning prototype. Potential areas of exploration include, but are not limited to: jewelry, furniture, experimental scientific instruments, electronic devices, architectural objects, lighting, cutlery, packaging, and musical instruments. Proposal submissions are due by August 18. See course syllabus for course and proposal details. Selection for the course is through application only.
TF 9:20am-11:15am

ENAS 5410b / CB&B 5230b / MB&B 5230b / MSCI 5410b / PHYS 5230b, Biological Physics  Yimin Luo

This course has three aims: (1) to introduce students to the physics of biological systems, (2) to introduce students to the basics of scientific computing, and (3) to familiarize students with characterization methods and analysis tools. We focus on studies of a broad range of biophysical phenomena including diffusion, polymer statistics, entropic forces, membranes, and cell motion using computational tools and methods. We provide intensive tutorials for Matlab including basic syntax, arrays, functions, plotting, and importing and exporting data.
TTh 2:35pm-3:50pm

ENAS 5620b / CB&B 5620b / INP 7562b / MB&B 5620b / MCDB 5620b / PHYS 5620b, Modeling Biological Systems II  Thierry Emonet and Harry McNamara

This course covers advanced topics in computational biology. How do cells compute, how do they count and tell time, how do they oscillate and generate spatial patterns? Topics include time-dependent dynamics in regulatory, signal-transduction, and neuronal networks; fluctuations, growth, and form; mechanics of cell shape and motion; spatially heterogeneous processes; diffusion. This year, the course spends roughly half its time on mechanical systems at the cellular and tissue level, and half on models of neurons and neural systems in computational neuroscience. Prerequisite: a 2000-level biology course or permission of the instructor.
TTh 2:35pm-3:50pm

ENAS 5710a / MB&B 5910a / MCDB 5910a / PHYS 5710a, Integrated Workshop  Corey O'Hern

This required course for students in the PEB graduate program involves a series of modules, co-taught by faculty, in which students from different academic backgrounds and research skills collaborate on projects at the interface of physics, engineering, and biology. The modules cover a broad range of PEB research areas and skills. The course starts with an introduction to MATLAB, which is used throughout the course for analysis, simulations, and modeling.
TTh 9am-10:15am

MENG 5020b, Mechatronics Laboratory  Madhusudhan Venkadesan

Hands-on synthesis of control systems, electrical engineering, and mechanical engineering. Review of Laplace transforms, transfer functions, software tools for solving ODEs. Review of electronic components and introduction to electronic instrumentation. Introduction to sensors, mechanical power transmission elements, programming microcontrollers, and PID control.
F 10:30am-1:30pm, MW 1:05pm-2:20pm

MENG 5040b, Applied Numerical Methods for Algebraic Systems, Eigensystems, and Function Approximation  Beth Anne Bennett

The derivation, analysis, and implementation of various numerical methods. Topics include root-finding methods, numerical solution of systems of linear and nonlinear equations, eigenvalue/eigenvector approximation, polynomial-based interpolation, and numerical integration. Additional topics such as computational cost, error analysis, and convergence are studied in several contexts throughout the course.
MW 2:35pm-3:50pm

MENG 5050a or b / MSCI 5050, Computer-Aided Engineering  Staff

Aspects of computer-aided design and manufacture (CAD/CAM). The computer’s role in the mechanical design and manufacturing process; commercial tools for two- and three-dimensional drafting and assembly modeling; finite-element analysis software for modeling mechanical, thermal, and fluid systems.
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MENG 5359a, Neuromuscular Biomechanics  Madhusudhan Venkadesan

Mechanics and control of animal movement, including skeletal muscle mechanics, systems-level neural and sensory physiology, elements of feedback control, and optimal control. Deriving equations of motion for multibody mechanical systems that are actuated by muscles or muscle-like motors; incorporating sensory feedback; analyzing system properties such as stability and energetics.
MW 4pm-5:15pm

MENG 5441a, Nonlinear Dynamics  Bauyrzhan Primkulov

This course introduces nonlinear dynamics and chaos in dissipative systems, tailored broadly for undergraduate students in science and engineering. It focuses on simple dynamical models, the mathematical principles underlying their behaviors, their connection to natural phenomena, and techniques for data analysis and interpretation. Key topics include forced and parametric oscillators, phase space analysis, periodic, quasiperiodic, and aperiodic flows, sensitivity to initial conditions, and strange attractors such as the Lorenz attractor. The course also explores phenomena like period doubling, intermittency, and quasiperiodicity, emphasizing nonlinear processes describable by a limited number of time-evolving variables.
TTh 11:35am-12:50pm

MENG 6278a, Advanced Robotic Mechanisms  Aaron Dollar

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TTh 1:05pm-2:20pm

MENG 7469a, Aerodynamics  Staff

Review of fluid dynamics. Inviscid flows over airfoils; finite wing theory; viscous effects and boundary layer theory. Compressible aerodynamics: normal and oblique shock waves and expansion waves. Linearized compressible flows. Some basic knowledge of thermodynamics is expected.
TTh 7pm-8:15pm

MENG 8370b / BENG 5570b, Computational Mechanics  Martin Pfaller

This course integrates fundamental concepts from nonlinear continuum mechanics and finite element methods applied to solid and fluid mechanics, focusing on theoretical understanding and numerical techniques. Topics covered are fundamentals of tensor calculus, kinematics, balance equations, constitutive relationships, geometric and material nonlinearities, nonlinear solution strategies, stability, nonlinear dynamics, errors, convergence, and adaptivity. Applications in biomedical engineering are stressed throughout the course. Prerequisites: fundamentals in calculus, differential equations, and linear algebra.
TTh 9am-10:15am

MENG 8664b / MSCI 8664b, Forces on the Nanoscale  Udo Schwarz

Modern materials science often exploits the fact that atoms located at surfaces or in thin layers behave differently from bulk atoms to achieve new or greatly altered material properties. The course provides an in-depth discussion of intermolecular and surface forces, which determine the mechanical and chemical properties of surfaces. In the first part, we discuss the fundamental principles and concepts of forces between atoms and molecules. Part two generalizes these concepts to surface forces. Part three then gives a variety of examples. The course is of interest to students studying thin-film growth, surface coatings, mechanical and chemical properties of surfaces, soft matter including biomembranes, and colloidal suspensions.
TTh 11:35am-12:50pm

MENG 8672b / MSCI 8672b, Electronic and Optical Properties of Energy Materials  Diana Qiu

This course explores the electronic and optical properties of materials from the perspective of electronic and molecular structure with a special focus on the microscopic origin and design of properties of interest for energy harvesting, storage, and transport. The course begins by briefly introducing concepts in quantum mechanics, such as wave functions and the time-independent Schrödinger equations. Then, we explore electronic structure in the context of designing materials for energy harvesting and generation, such as photovoltaics, thermoelectrics, and piezoelectrics. We also study dynamical processes, such as hot electron relaxation, multi-exciton generation, charge transport, and energy transport at a phenomenological level. Finally, we overview common energy storage materials, with a focus on solid-state batteries and solar fuels.
TTh 9am-10:15am

MENG 8673a / MSCI 8673a, Introduction to Nanomaterials and Nanotechnology  Cong Su

Survey of nanomaterial synthesis methods and current nanotechnologies. Approaches to synthesizing nanomaterials; characterization techniques; device applications that involve nanoscale effects.
MW 1:05pm-2:20pm

MENG 8675b / MSCI 8675b, Thermodynamics, Kinetics, and Structure of Materials  Jan Schroers

This advanced-level course focuses on the thermodynamic and kinetic aspects of materials and how they define structure and properties. We first discuss thermodynamics relevant to materials. This includes thermodynamic laws, auxiliary functions to develop convenient equations of state to describe equilibrium, Gibbs Free Energy (G), experimental determination of G, model calculations of G such as ideal solutions and regular solutions, using G curves to construct equilibrium conditions, phase diagram constructions, reading of phase diagrams. We then focus on solidification which we develop from the phenomena of undercooling, nucleation and growth. Combining both allows us to predict microstructures formed during solidification far and close to equilibrium. We also discuss glass formation, the case when nucleation and growth can be suppressed, and the liquid freezes upon cooling into a glass.
MW 11:35am-12:50pm

MENG 8850a / APHY 7250a / MSCI 7250a, Advanced Synchrotron Techniques and Electron Spectroscopy of Materials  Charles Ahn

This course provides descriptions of advanced concepts in synchrotron X-ray and electron-based methodologies for studies of a wide range of materials at atomic and nano-scales. Topics include X-ray and electron interactions with matter, X-ray scattering and diffraction, X-ray spectroscopy and inelastic methods, time-resolved applications, X-ray imaging and microscopy, photo-electron spectroscopy, electron microscopy and spectroscopy, among others. Emphasis is on applying the fundamental knowledge of these advanced methodologies to real-world materials studies in a variety of scientific disciplines.
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MENG 9000a or b, Research Seminars in Mechanical Engineering & Materials Science  Madhusudhan Venkadesan

The purpose of this course is to introduce graduate students to state-of-the-art research in all areas of Mechanical Engineering & Materials Science (MEMS), as well as related disciplines, so that students understand the range of current research questions that are being addressed. An important goal is to encourage students to explore research topics beyond their particular field of study and develop the ability to contextualize their work in terms of larger research questions in MEMS. We therefore require that MEMS Ph.D. students enrolled in this course attend at least eight research seminars during the term: six must be part of the official MEMS seminar series, and two can be from any other relevant Yale graduate department/program seminar series. This course is graded Sat/Unsat with sign-in sheets used to monitor attendance. Required of first- and second-year MEMS Ph.D. students.  0 Course cr
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