Projects

To pass the course each student is required to carry out one of the projects which will be described starting in the second week. Since I may require these to be revised before they are acceptable I would suggest that you start rather early.

Here are some possible projects that I am thinking about:-

1. Radon-Nikodym theorem.
2. Kuiper's theorem: The group of unitary operators on a (separable infinite dimensional) Hilbert space is contractible.
3. Seeley's extension theorem.
4. Gibb's phenomenon.
5. Surjectivity of any non-trivial constant coefficient differential operator, $P:\mathcal{S}'(\mathbb{R}^n)\longrightarrow \mathcal{S}'(\mathbb{R}^n).$
6. Every elliptic differential operator with constant coefficients is surjective as a map on $\mathcal{C}^{\infty}(U),$ for any open set $U\subset\mathbb{R}^n.$
7. Lidskii's theorem on trace class operators on $L^2(\mathbb{R}^n).$