Georgia Institute of Technology College of Sciences

A broad introduction to the local and global behavior of nonlinear dynamical systems arising from maps and ordinary differential equations.

The theory of curves, surfaces, and more generally, manifolds. Curvature, parallel transport, covariant differentiation, Gauss-Bonet theorem

Point set topology, topological spaces and metric spaces, continuity and compactness, homotopy and covering spaces

Method of characteristics for first and second order partial differential equations, conservation laws and shocks, classification of second order systems and applications.

Topics from complex function theory, including contour integration and conformal mapping

Differentiation of functions of one real variable, Riemann-Stieltjes integral, the derivative in R^n and integration in R^n

Real numbers, topology of Euclidean spaces, Cauchy sequences, completeness, continuity and compactness, uniform continuity, series of functions, Fourier series

Linear algebra in R^n, standard Euclidean inner product in R^n, general linear spaces, general inner product spaces, least squares, determinants, eigenvalues and eigenvectors, symmetric matrices.

Sampling distributions, Normal, t, chi-square and F distributions. Moment generating function methods, Bayesian estimation and introduction to hypothesis testing

Simple random walk and the theory of discrete time Markov chains