Applied and Computational Mathematics

Taken concurrently with EGR/MAT/PHY 191. An integrated course that covers the material of PHY 103 and MAT 201 with the emphasis on applications to engineering. Math topics include: vector calculus; partial derivatives and matrices; line integrals; simple differential equations; surface and volume integrals; and Green's, Stokes's, and divergence theorems. One lecture, two preceptorials.

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.

Draws problems from the sciences and engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics. This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upper class undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.

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. Prerequisites: Multivariable calculus and linear algebra.

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements.

An introduction to weak numerical methods used in computational geophysics. Finite- and spectral-elements, representation of fields, quadrature, assembly, local versus global meshes, domain decomposition, time marching and stability, parallel implementation and message-passing, and load-balancing. Parameter estimation and "imaging" using data assimilation techniques and related "adjoint" methods. Labs provide experience in meshing complicated surfaces and volumes as well as solving partial differential equations relevant to geophysics. Prerequisites: MAT 201; partial differential equations and basic programming skills. Two 90-minute lectures.

An introduction to lossless data compression algorithms, modulation/demodulation of digital data, error correcting codes, channel capacity, lossy compression of analog and digital sources. Three hours of lectures. Prerequisites: 301, ORF 309.