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The major focus of the research that my advisor has me doing centers around the idea of asymptotic behavior of partial-spectral-projections on compact manifolds. In a few sentences, here is the context for the research:

  • $(M,g)$ is a compact Riemannian manifold without boundary, and $-\Delta_g$ is the (positive) Laplace-Beltrami operator of the metric $g$.

  • The operator $\sqrt{-\Delta_g}$ is defined in the usual way its collection of $L^2$-normalized eigenfunctions is denoted by $\{e_j(x)\}_{j=0}^{\infty}$, with eigenvalues $0 = \lambda_0 < \lambda_1 \leq \lambda_2 \leq \cdots \to \infty$.

  • For a fixed $\lambda > 0$, we then define the partial-projection operator $$ \sum_{j=0}^{\infty}\langle f,e_j\rangle e_j(x) = f(x) \mapsto \sum_{\{j \,:\, \lambda_j \in [\lambda, \lambda+1)\}} \langle f,e_j \rangle e_j(x) $$ as the projection of $f(x)$ onto the direct sum of eigenspaces which have eigenvalues in the unit-interval $[\lambda, \lambda+1)$.

  • We then denote the Schwartz kernel of the corresponding integral operator as $K(x,y;\lambda)$, where $$ f(x) \mapsto \int_{M} K(x,y;\lambda)f(y) \,dV_g(y) $$ agrees with the partial-sum definition above.

  • The goal of our research is to then analyze the big-oh behavior of this Schwartz-kernel as $\lambda \to \infty$. Usually this is formulated as $$ \sup_{x,y \in M}\big| K(x,y;\lambda) - F(x,y;\lambda) \big| = O(\lambda^{n-1}), $$ where the term $F(x,y;\lambda)$ comes from some parametrix approximation or something.

At this point I'm a little embarrassed to say admit that while I can do the mathematical research needed, I am unsure as to why people actually care about such a specific kind of linear operator?

I understand that the Weyl law is an old result in functional and harmonic analysis, but sadly I'm not sure why this specific problem is useful in the larger field of research. I've tried asking this of my advisor before, but he has not offered me all that much in the way of an answer. Also, while reading through the literature of similar problems to my own, I find many references to the myriad of results and slightly-different hypotheses, but still an answer of WHY? eludes me.

Specifically, why does everyone also study these partial-projections onto unit-length interval? What would be different if we projected only an interval of length 2? Or length $L$? Or onto a compact set of some fixed, finite measure?

Any insight into these kinds of problems, and their important to the mathematical body at-large would be much appreciated. Thank you in advanced, as always.

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  • $\begingroup$ If you dilate $M$ by a factor $c$, don't the eigenvalues of $\sqrt{-\Delta}$ scale by a factor of $c^{-1}$, or something like that? That could explain why you're specifically looking at unit intervals, because going to arbitrary finite lengths really isn't any more general. $\endgroup$ – Nik Weaver Jun 2 at 4:12
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    $\begingroup$ But to the broader point, I don't think you should expect a problem given to you by your advisor to have great, fundamental importance. Problems of great importance that can be solved by a typical grad student are quite rare. Some people explode out of the gates, but for most of us it takes many years of experience to get to the point where we can come up with important problems whose solutions are within reach. $\endgroup$ – Nik Weaver Jun 2 at 4:16
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The unit length hypothesis at here is not important, and very crude estimates are available using Sobolev embedding only. The main issue is that studying the spectrum on the manifold itself is not enough for recovering underlying topological/geometric information of the manifold. This is a subtle topic even for 2 dimensional surfaces, where a lot of work has been done.

For very recent work, check some papers by Sogge and Xi:

https://arxiv.org/abs/1711.04707

I would suggest that instead of working through the detailed estimates (on a sphere, on a torus, on negatively curved manifold, etc), think about some other ways to understand the spectrum of the Laplacian on the manifold. For example, a compact Riemann surface of genus $g\ge 2$ can be realized as the quotient of the upper half plane $\mathcal{H}/\Gamma$. There is a lot of interesting work that can be done to understand the relationship between the group action and the spectrum. The interplay between the algebraic nature of the surfaces and flexibility of the analysis tools made the subject really interesting.

A survey paper by Sanark may be a good start:

http://web.math.princeton.edu/facultypapers/sarnak/baltimore.pdf

For 3-manifolds this becomes deep and is related to heat kernels in geometric analysis. The subject is related to Ricci flow and there are plenty written up online already.

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  • $\begingroup$ Thank you for taking the time to post these insights; they are rather interesting. However I'm not sure if I understand some of the bigger implications. Are you saying that studying the spectrum of the manifold (as a whole) is insufficient? But that by looking at the spectrum's distribution amongst higher frequencies (spread out among compact intervals) we can glean more important topological/geometric information about the manifold? $\endgroup$ – Patch Jun 2 at 17:28
  • $\begingroup$ And if this indeed the case, then can you explain briefly/broadly why know this kind of information about the spectrum tells you that kind of information? $\endgroup$ – Patch Jun 2 at 17:30
  • $\begingroup$ @Patch: I did not say "by looking at the spectrum's distribution amongst higher frequencies (spread out among compact intervals) we can glean more important topological/geometric information about the manifold? " anywhere in the answer. You put word into my mouth. I suggest you talk to your advisor. I would be direct: I do not think this is an interesting thesis topic, and I suggest you work on other things. Hope this is clear enough. $\endgroup$ – Bombyx mori Jun 2 at 20:54
  • $\begingroup$ Sorry, I never mean to put words in your mouth. Since I was asking why people would be interesting in these kinds of estimates, I was thinking you were implying something more. It's one thing to say knowing the spectrum alone isn't enough, but that didn't tell me why projecting onto these restricted Eigenspaces would be of interest. Thank you, again, for your time. $\endgroup$ – Patch Jun 2 at 21:22

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