Georgia Institute of Technology College of Sciences

An introduction to proofs in advanced mathematics, intended as a transition to upper division courses including MATH 4107, 4150 and 4317. Fundamentals of mathematical abstraction including sets, logic, equivalence relations, and functions. Thorough development of the basic proof techniques: direct, contrapositive, existence, contradiction, and induction. Introduction to proofs in analysis and algebra.

Fourier series, Fourier integrals, boundary value problems for partial differential equations, eigenvalue problems

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.