Marina Iliopoulou - Three Polynomial Methods for Point Counting, Lecture I

Explore three powerful polynomial methods for point counting in this lecture by Marina Iliopoulou. Dive into Dvir's polynomial method for solving the Kakeya problem in finite fields, Guth and Katz's polynomial partitioning technique for addressing the Erdös distinct distances problem in the plane, and the slice rank method developed by Croot, Lev, Pach, Ellenberg, and Gijswijt for demonstrating the small size of sets with no 3-term arithmetic progressions in finite-field settings. Learn about key concepts such as the Kakeya problem, Kakeya universe, vanishing quality, zero polynomials, and gain insights through proofs and sketches. Enhance your understanding of advanced mathematical techniques in this comprehensive 54-minute lecture from the Hausdorff Center for Mathematics.

Introduction
Notation
Kaiga problem
Kaiga universe
The polynomial method
Vanishing quality
Zero polynomials
Proof
Sketch
First method