MATH 481
Measure theory and Hilbert spaces Spring 2023
Division III Quantitative/Formal Reasoning
This is not the current course catalog

How large is the unit square? One might measure the number of individual points in the square (uncountably infinite), the area of the square (1), or the dimension of the square (2). But what about for more complicated sets, e.g., the set of all rational points in the unit square? What’s the area of this set? What’s the dimension? In this course we’ll come up with precise ways to measure size—length, area, volume, dimension, etc.—that apply to a broad array of sets. Along the way we’ll encounter Lebesgue measure and Lebesgue integration, Hausdorff measure and fractals, space-filling curves and the Banach-Tarski paradox. We will also investigate Hilbert spaces, mathematical objects that combine the tidiness of linear algebra with the power of analysis and are fundamental to the study of differential equations, functional analysis, harmonic analysis, and ergodic theory, and also apply to fields like quantum mechanics and machine learning. This material provides excellent preparation for graduate studies in mathematics, statistics and economics.