Olivetti Club

Itamar OliveiraCornell University
Unexpected applications of polynomials

Tuesday, September 7, 2021 - 4:35pm
Malott 406

Erdős asked: given $n$ points in the plane, what is the minimum number of distinct distances between them? Szemerédi and Trotter asked: given finite collections $\mathcal{P}$ and $\mathcal{L}$ of points and lines, respectively, in the plane, how many "incidences" happen (i.e. pairs $(p,\ell)$ with $p\in \mathcal{P}$, $\ell\in\mathcal{L}$ and $p\in\ell$)? Kakeya and Besicovitch asked: How "big" is a zero-measure set in $\mathbb{R}^{n}$ that contains a unit segment in every direction?

These are a few examples of completely unrelated problems in distinct areas that we will go over in the talk. After Dvir's breakthrough in the finite field Kakeya problem, it was noticed that polynomial methods were applicable to study other questions in discrete combinatorics, incidence geometry, geometric measure theory and (more recently) harmonic analysis. This is still a very active topic and new applications are frequently found, so maybe you'll figure out how to finish your thesis with it as well. No analysis background is necessary, the only prerequisite you have to master is the abstract concept of sandwich.