I'm giving a talk today on my recent and forthcoming work in collaboration with Theresa Anderson, Ayla Gafni, Robert Lemke Oliver, George Shakan, Frank Thorne, Jiuya Wang, and Ruixiang Zhang. The slides for my talk can be found here.
This talk includes some discussion of our paper to appear in IMRN (link to the arXiv version, which is mostly the same as what will be published). (See also my previous discussion on this paper). But I'll note that in this talk I lean towards a few ideas that did not make it into the paper, but which we are using in current work.
In particular, in our paper we don't need to use group actions or classify orbit sizes, but it turns out that this is a very strong idea! I'll note that in a very particular case, Thorne and Taniguchi have applied this type of orbit counting method in their paper "Orbital exponential sums for prehomogeneous vector spaces" to gain extremely strong, specific understanding of Fourier transform for their application.
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