The Geometry of Polynomials
Prerequisites: Strong undergraduate-level familiarity with probability, multivariable calculus, linear algebra, and a little analysis. Mathematically mature undergraduates are welcome. Grading: Homework (20%), participation (20%), quizzes (20%), and a final project (40%). The starting point of the field is to encode combinatorial objects as multivariate polynomials. Typically, we will encode discrete probability distributions. For example, the distribution over { 𝑒 , 𝑓 } {e,f} which selects { 𝑒 } {e} with probability 1 / 2 1/2 and { 𝑒 , 𝑓 } {e,f} with probability 1 / 2 1/2 is encoded as 𝑝 ( 𝑥 , 𝑦 ) = 1 2 𝑥 + 1 2 𝑥 𝑦 p(x,y)= 2 1 x+ 2 1 xy. We will then analyze the zero-free regions of these polynomials in 𝐶 𝑛 C n . These regions often have underlying geometry, hence the name of the course. We will study how this geometry can help us understand the original combinatorial object. We will discuss some important recent results that use ideas from this field, such