If you are interested in developing quantum algorithms, there is a nice up-to-date survey by Childs and van Dam called Quantum algorithms for algebraic problems, available at http://arxiv.org/abs/0812.0380. For PDE's, you might look into hyperbolic polynomials and hyperbolicity cones. The latter are characterized algebraically, there are lots of open questions about them, and their study is also practially relevant to engineering applications like convex optimization (in case practical applications, or perhaps hot funding areas, are what you're looking for).
|
1 | [made Community Wiki] | ||
|
|
||||
