APC 199 / MAT 199 (QR) Graded A-F, P/D/F, Audit

Math Alive

Ian M. Griffiths

Adam W. Marcus

Mathematics has profoundly changed our world, from the way we communicate with each other and listen to music, to banking and computers. This course is designed for those without college mathematics who want to understand the mathematical concepts behind important modern applications. The course consists of individual modules, each focusing on a particular application (e.g., digital music, sending secure emails, and using statistics to explain, or hide, facts). The emphasis is on ideas, not on sophisticated mathematical techniques, but there will be substantial problem-set requirements. Students will learn by doing simple examples.

APC 350 / CEE 350 / MAT 322 (QR) Graded A-F, P/D/F, Audit

Introduction to Differential Equations

Staff

This course will introduce the basic theory, models and techniques for ordinary and partial differential equations. Emphasis will be placed on the connection with other disciplines of science and engineering. We will try to strike a balance between the theoretical (e.g. existence and uniqueness issues, qualitative properties) and the more practical issues such as analytical and numerical approximations.

APC 523 / AST 523 / MAE 507 Graded A-F, P/D/F, Audit

Numerical Algorithms for Scientific Computing

Michael E. Mueller

A broad introduction to numerical algorithms used in scientific computing. The course begins with a review of the basic principles of numerical analysis, including sources of error, stability, and convergence. The theory and implementation of techniques for linear and nonlinear systems of equations and ordinary and partial differential equations are covered in detail. Examples of the application of these methods to problems in engineering and the sciences permeate the course material. Issues related to the implementation of efficient algorithms on modern high-performance computing systems are discussed.

APC 537 / ELE 537 P/D/F Only

Information Theory and Machine Learning Seminar

Emmanuel A. Abbe

This advanced seminar discusses topics in information theory, coding theory and machine learning. The goal is to cover different approaches driven by both probabilistic and worst-case models, as well as information-theoretic and computational limits. Focus is put on compression and unsupervised learning. The class has lectures and students presentations.

AST 559 / APC 539 Graded A-F, P/D/F, Audit

Turbulence and Nonlinear Processes in Fluids and Plasmas

Gregory W. Hammett

A comprehensive introduction to the theory of nonlinear phenomena in fluids and plasmas, with emphasis on turbulence and transport. Experimental phenomenology; fundamental equations, including Navier-Stokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudo-spectral and particle-in-cell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and direct-interaction approximation; PDF methods and intermittency; variational techniques. Applications from neutral fluids, fusion plasmas, and astrophysics.

ELE 486 / APC 486 Graded A-F, P/D/F, Audit

Transmission and Compression of Information

Emmanuel A. Abbe

Our digital world relies heavily on our ability to extract, store, and transfer information. Consequently, much effort has been devoted to perform efficiently such tasks. This class covers the fundamental limits and algorithms for data compression and transmission, providing an introduction to information theory and coding theory. It also discusses connections with unsupervised machine learning problems, in particular with data clustering.

GEO 441 / APC 441 No Pass/D/Fail

Computational Geophysics

Dmitry Borisov

Herurisa Rusmanugroho

Jeroen Tromp

An introduction to weak numerical methods, in particular finite-element and spectral-element methods, used in computational geophysics. Basic surface & volume elements, representation of fields, quadrature, assembly, local versus global meshes, domain decomposition, time marching & stability, parallel implementation & message-passing, and load-balancing. In the context of parameter estimation and 'imaging', will explore data assimilation techniques and related adjoint methods. The course offers hands-on lab experience in meshing complicated surfaces & volumes as well as numerically solving partial differential equations relevant to geophysics

MAE 502 / APC 506 No Pass/D/Fail

Mathematical Methods of Engineering Analysis II

Clarence W. Rowley

Topics in complex analysis and functional analysis, with emphasis on applications in physics and engineering. Topics include power series, singularities, contour integration, Cauchy's theorems, and Fourier series; an introduction to measure theory and the Lebesgue integral; Hilbert spaces, linear operators, and adjoints; the spectral theorem, and its application to Sturm-Liouville problems.

MAT 522 / APC 522 Graded A-F, P/D/F, Audit

Introduction to PDE

Sergiu Klainerman

The course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, Fokker-Planck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the Navier-Stokes equations.

MAT 586 / APC 511 / MOL 511 / QCB 513 Graded A-F, P/D/F, Audit

Computational Methods in Cryo-Electron Microscopy

Amit Singer

This course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

MSE 515 / APC 515 Graded A-F, P/D/F, Audit

Random Heterogeneous Materials

Salvatore Torquato

Composites, porous media, foams, colloidal suspensions, geological media, polymer blends, and biological media are all examples of heterogeneous materials. Often the microstructure of such materials is random. The relationship between the macroscopic (transport, mechanical, electromagnetic, and chemical) properties and microstructure of random heterogeneous materials is formulated. Topics include statistical characterization of the microstructure via n-point distribution functions; percolation theory; fractal concepts; sphere packings; Monte Carlo simulation techniques; and image analysis of microstructures; homogenization theory; effective-