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

Math Alive

Ayelet Heimowitz

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

Jiequn Han

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

Gregory W. Hammett

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.

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.

GEO 441 / APC 441 No Pass/D/Fail

Computational Geophysics

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 588 / APC 588 */AUD

Topics in Numerical Analysis: Optimization On Smooth Manifolds

Nicolas Boumal

This course covers current topics in numerical analysis. Specific topic information is provided when the course is offered. For the first topic we study theory and algorithms for optimization on smooth manifolds. The course mixes mathematical analysis and coding, aiming for methods that are actually practical and that we truly understand. No prior background in Riemannian geometry or optimization is assumed. Students should be comfortable with linear algebra and multivariable calculus.

MSE 515 / APC 515 / CHM 559 Graded A-F, P/D/F, Audit

Random Heterogeneous Materials

Salvatore Torquato

Composites, porous media, foams, colloids, geological media, and biological media are all examples of heterogeneous materials. The relationship between the macroscopic (transport, mechanical, electromagnetic, and chemical) properties and material microstructure is formulated. Topics include statistical characterization of the microstructure; percolation theory; fractals; sphere packings; Monte Carlo techniques; image analysis; homogenization theory; cluster and perturbation expansions; variational bounding techniques; topology optimization methods; and cross-property relations. Biological and cosmological applications are discussed.