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. compression, animation 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.

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.

This course gives a broad introduction to numerical algorithms used in scientific computing. It covers classical methods to solve Ordinary and Partial Differential Equations such as spectral, finite difference and finite volume methods. A brief introduction to finite element methods is given. Explicit and implicit time integration using various high-order methods are discussed. We review basic methods to solve linear and non-linear systems of equations. Issues related to the implementation of efficient algorithms on modern high-performance computing systems are discussed. Hyperbolic systems of conservations laws are covered in detail.

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

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.

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.