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### Combinatorial stability and representation stability

**Algebra & Discrete Mathematics**

Speaker: | Thomas Church, Stanford |

Location: | 2112 MSB |

Start time: | Mon, Jan 26 2015, 5:10PM |

How many roots does a random squarefree polynomialf(T) inF_{q}[T] have? On average, it's a bit less than one root per polynomial. The precise answer depends on the degree off(T), but as degf(T) goes to infinity, the expectation stabilizes and converges to 1 - 1/q+ 1/q^{2}- 1/q^{3}+ ... =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 ofS_{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.