As much for my own future reference as for anything, here’s a summary of some things I’d like to learn, maybe not this season, but soon.
First on the docket, I’d like to learn Vasy’s method. This is a technique for meromorphically continuing the resolvent of the Schrödinger operator on an asymptotically hyperbolic manifold — that is, a manifold which, near its boundary, looks like the Poincaré model of hyperbolic space does near its boundary. A priori the definition of the the resolvent only makes sense on a small open subset of the complex plane, and one hopes to show that the definition of the resolvent makes sense on the entire plane, except possibly a discrete set of poles.
On a somewhat similar note, I’d like to learn the Atiyah-Singer index theorem. This theorem equates the Fredholm index of an elliptic pseudodifferential operator on a line bundle to its “topological index”, which is a rational number defined in terms of the cohomology of . This is largely motivated by my quest to understand the sense in which cohomology counts solutions to PDE, c.f. my recent post on the genera of Riemann surfaces. I previously tried to learn the heat-kernel proof of Atiyah-Singer shortly after I first learned about pseudodifferential operators but got nowhere. This time, I will armed with the knowledge of the Riemann-Roch theorem, which may make all the difference.
Unlike the previous two requests, which are both PDE-analytic in nature, I think that my knowledge of complex analysis has prepared me to learn the proof that there are twenty-seven lines on a cubic surface in . This would entirely be for fun, and I may blog about it, so as to tell the story of a hapless analyst faffing around hopelessly in deep algebra.
Finally, I would like to fix up and publicize the Sage code that is mentioned by my paper on computation of Kac-Moody root multiplicities with Joshua Lin and Peter Connick. I suspect that this will require learning some nontrivial representation theory and complexity theory, though in its current form the algorithm is essentially a consequence of elementary facts about quadratic forms over .