A multi-linear geometric estimate

This is my talk at Virginia Tech's analysis and mathematical physics seminar for the fall of 2021, which is about joint work with Alex Iosevich in which we generalize a geometric estimate on level sets of non-degenerate bilinear forms originally made in a 2008 paper of Hart and Iosevich to the case of multi-linear forms. I discuss applications to particular sets of points in the plane over a finite field which I whimsically named omphaloi after the mythical Greek center of the world as well as the relationship between this work and the sum-product conjecture.

Slides: https://aten.cool/documents/slides/aten_geometric_estimate.pdf
Preprint: https://arxiv.org/abs/2112.00810

The Hart and Iosevich paper: https://arxiv.org/abs/0705.4256
Alex Iosevich: https://alexiosevich.com/