Here are a few suggested topics for the short project. The idea is to write an expository account, at most 5 pages in length, explaining a topic clearly and accurately. If you want to write about a topic other than the ones suggested below, feel free to contact me about them. But bear in mind they must be intimately connected to the ideas and techniques developed in this course. Once you have chosen your topic, I'll give you advice and provide you with references that you might need.

• More general aspects of Fourier series: L²[a,b] as a Hilbert space, Orthogonal families in L²[a,b], Bessel's inequality and Parseval's equality, convergence in mean, completeness of the trigonometric system.

• Probabilistic interpretation of the fundamental solution of the 1-dimensional heat equation. This would be a nice exercise demonstrating how the Gaussian (normal) distribution occurs as a solution of the heat equation on the real line.

• Heat flow (diffusion) in a semi-infinite solid via Fourier integrals; application in approximating the "age of the earth" a la Lord Kelvin.

• Harmonic functions from the complex-variable point of view: The Cauchy-Riemann equations, the relation between holomorphic and harmonic functions in dimension 2, harmonic conjugates.

• The Laplace transform and its basic properties. Examples of how it can be used to solve PDE's.

• Dirac's delta function, the convolution integral, fundamental solutions of linear PDE's and their role in finding more general solutions.

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