Images courtesy of Wikipedia: Riemann vs. Lebesgue integration (left); a Fourier series (middle); and blowups of a 1D Brownian motion trajectory (right)

### MATH 487 - Real Analysis II (Spring 2020): Measure Theory, Fourier Analysis, and Brownian Motion

#### UNDER CONSTRUCTION

**Instructor:** Joe P. Chen

**Contact:** McGregory 214, jpchen-at-(the obvious edu)

**Course philosophy:**
This is a second-semester course on mathematical analysis, covering two topics which are mutually connected:
**Convergence of (series of) functions**.
- A more robust theory of integration called
**Lebesgue integration.**

In treating these topics together, we will simultaneously learn about **function spaces**, in particular, the space of continuous functions, and \(L^p\) spaces (\(p\in [1,\infty]\)).

\(L^2\) spaces are important in their own right---they are Hilbert spaces---and allow us to characterize properties of **Fourier series**.

An additional motivation for this course comes from *probability* (coin tosses and random walks) and *partial differential equations* (Laplace’s equation, and its discrete analog, the gambler’s ruin problem; and the heat equation).

Towards the end of the course, we will use what we learned to construct the space-time scaling limit of random walks, called **Brownian motion**---it can be realized as a random Fourier series!

**Prerequisites:**
Strong performance in Real Analysis I (MATH 377). Some exposure to complex variables and/or basic probability is helpful, but not required.

Back to the Colgate math department