Introduction #

Analysis is the mathematical language of convergence and approximation. Its ideas are ubiquitous in modern mathematics appearing in fields as diverse as machine learning, probability, dynamical systems, differential equations, as well as geometry and topology, and number theory. At its heart is the following question:

Given a mathematical object such as a function or a matrix, it is possible to approximate it using simpler objects? And can we answer important questions about the original object by studying just the simpler ones?

Rather than studying convergence individually in each of the many contexts within which it appears in mathematics, we unify the course around the notion of a metric space; in this way, our work can be applied universally. Two courses naturally follow: Math 2805: Mathematical principles of machine learning and Math 3603: Advanced analysis. Both explore important applications of this class.

Course description: Building on the theoretical underpinnings of calculus, develops the rudiments of mathematical analysis. Concepts such as limits and convergence from calculus are made rigorous and extended to other contexts, such as spaces of functions. Specific topics include metric spaces, point-set topology, sequences and series, continuity, differentiability, the theory of Riemann integration, and functional approximation and convergence.