What is Special About Polynomials? (Perspectives from Coding theory and Differential Geometry)
olynomials are a special class of functions. They are useful in many branches of mathematics, often in problems which don't mention polynomials. We discuss two examples: polynomials in error-correcting codes and polynomials in geometric inequalities. The special properties of polynomials that play a role in these stories are connected with the recent applications of polynomials in combinatorics. The ideas from coding theory helped inspire the proofs in Lecture 1, and the ideas in geometry helped inspire the proofs in Lecture 3.