Research

Summary: This paper proves sharp eigenfunction bounds on the torus for very large p, resolving the discrete restriction conjecture of Bourgain in that range. The discrete restriction conjecture asks how large eigenfunctions of the Laplacian on the torus can be in Lp if they are L2 normalized. The torus is a special manifold because the eigenfunction can be written down explicitly in terms of exponential sums, which invites number theory into the problem in an atypical way for eigenfunction estimates. Because of this, the numerology and techniques differ greatly for this problem compared to others. I refined the circle method approach by Bourgain-Demeter to get the result which, oddly enough, means the paper has little harmonic analysis. An analytic number theorist may be able to get more out of the exponential sum estimates, but proving sharp bounds gives far less wiggle room than proving estimates with a loss. This does also resolve the question of the additive energy of higher dimensional spheres. Some results with a log loss are proven as well in a better range of p. The numerology of this problem is heavily tied to the count of lattice point on spheres.

Summary: This paper is a combination, of sorts, or my previous results on sharp spectral projection estimates for the torus, which is good at upgrading results with epsilon loss to the sharp result, and work of Germain-Myerson, which is good for establishing the result with epsilon loss. We focus on the three dimensional torus as this is where the geometry of numbers arguments are the most tractable. We are able to get sharp results for large windows for all p and establish some sharp results for small windows and large p, which is the most difficult region to tackle. This paper is also semi-expository. We include every approach to the problem we are aware of, and it is our hope that this will be a useful reference for those who are interested in spectral projection estimates on the torus. For each approach, we have reached a natural barrier that, at least in my opinion, would require a new idea to get significantly beyond, although some optimizations may be possible with the current framework. 

Summary: This paper proves sharp estimates for spectral projectors on the torus at the critical exponent, improving a result of Germain-Myerson and resolving their conjecture in some region, specifically the region covered by the decoupling theorem of Bourgain-Demeter. The proof involves adapting an estimate valid on general manifolds to the torus setting to compliment decoupling. This result is interesting to me because it allows one to remove an epsilon loss that comes from an application of decoupling using structural information about the exponential sums being considered. It is an ongoing interest of mine whether this can be replicated in other settings.

Summary: This paper is a result of my undergraduate honors thesis done under Hans Christianson. It had previously been proved by Christianson that Neumann data equidistributed on the sides of triangular domains. This points to a quantum ergodicity like structure to these eigenfunctions. This paper does two things. Firstly, it shows that equidistribution does not hold on subsets of the sides in general. Secondly, it proves some theoretical results on equidistribution on average. Some interesting numerical structures arose when computing equidistribution results we could not fully explain. It would be interesting to return to this either from a computational or theoretical perspective.