Explore a mathematical lecture on sums in progressions to squarefree moduli among polynomials over finite fields. Delve into topics such as primes in progressions, Fourier analysis approaches, and bounds on sums in progressions. Examine the polynomial analogue and the anatomy of polynomials, including an introduction, theorem, and methods for counting polynomials with large values. Investigate the Poisson Dirichlet distribution control and progress towards the main theorem by studying arithmetic functions and the trivial character. Conclude with a general formulation of the main theorem and its applications in this 58-minute presentation from the Hausdorff Center for Mathematics.