Here are some suggestions for the short project. The idea is to write
an expository account, at most 5 pages in length, explaining a
certain topic in a clear and accurate language. If you have other topics
of interest in mind, feel free to talk to me about them. Once you choose
your topic, contact me so I can give you advice and provide you with
references that you might need.
- Generalized 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 derivation of the fundamental solution of the
1-dimensional heat equation. This would be a nice exercise demonstrating
how the Gaussian (normal) distribution shows up as the solution of the
heat equation on the real line.
- 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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MATH 328