On Random Polynomials and Counting Number Fields: Fourier Analysis Meets Arithmetic Statistics

Behaviors of objects of algebraic interest, such as polynomials, elliptic curves and number fields -- many of which are still unknown -- fall into the field of arithmetic statistics..  By bringing in Fourier analysis to interplay with a wide variety of number theoretic techniques, we were recently able to improve our counting tools.  Today I'll discuss predicting random polynomials and determining how many number fields of a given degree there are, both using these techniques in different ways.