Department of MathematicsHow many roots does a random squarefree polynomial f(T) in F_q[T] have? On average, it's a bit less than one root per polynomial. The precise answer depends on the degree of f(T), but as deg f(T) goes to infinity, the expectation stabilizes and converges to 1 - 1/q + 1/q^2 - 1/q^3 + ... = q / (q+1). Many other statistics such as the number of irreducible quadratic factors stabilize in a similar way,
In joint work with Ellenberg and Farb, we proved that the stabilization of this combinatorial formula is equivalent to a representation-theoretic stability in the cohomology of braid groups. I will describe how combinatorial stability for statistics of squarefree polynomials, of maximal tori in GL_n(F_q), and other natural geometric counting problems can be converted to questions of representation stability in topology for configuration spaces and flag varieties, and vice versa.