Explore three powerful polynomial methods for point counting in this advanced mathematics lecture. Delve 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. Gain insights into problem-solving techniques, grid configurations, proof sketches, and refined estimates. Examine the effectiveness of the polynomial method, learn about the bisection method and topology in polynomial partitioning, and understand the Section Lemma. This comprehensive lecture covers advanced mathematical concepts and is ideal for those interested in combinatorics, finite field theory, and geometric problem-solving.