Mathematics

The course is to introduces the basic concepts of Homological Algebra including modules, diagrams, functors, homology of complexes, tensor products, group homology, and cohomology. 

This course introduces numerical methods, which are now an essential component in a diverse range of disciplines. Topics include creation and manipulation of arrays, solutions of linear systems, Gaussian elimination with partial pivoting, numerical differentiation and integration, introductory numerical differential equations, root finding methods, including bisection and fixed-point iteration, Newton's method in one and higher dimensions, and functional minimization in multiple dimensions. Within these topics students are introduced to variables and functions, floating point arithmetic, flow control, container types, plotting, and symbolic expressions.

This course examines the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. The course starts with the foundations of calculus and the real numbers system. It goes on to study the limiting behavior of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.