Mathematics Colloquia and Seminars

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How 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/q² - 1/q³ + ... = q / (q+1). In joint work with J. Ellenberg and B. 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 give a general picture of this representation stability for sequences of S_n-representations, and 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, and vice versa.