What I want to learn, Spring 2021

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 L to its “topological index”, which is a rational number defined in terms of the cohomology of L. 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 \mathbb P^3. 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 \mathbb Z.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s