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.

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.

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.

Phase-plane methods and single-degree-of-freedom nonlinear oscillators; invariant manifolds, local and global analysis, structural stability and bifurcation, center manifolds, and normal forms; averaging and perturbation methods, forced oscillations, homoclinic orbits, and chaos; Melnikov's method, the Smale horseshoe, symbolic dynamics, and strange attractors.