We will talk about peak and descent polynomials, their roots, coefficients in a "special" binomial basis, as well as some q-analogues. Peak and descent polynomials "count" peaks and descents of permutations. Given a permutation pi=pi_1 pi_2 ... pi_n in S_n, we say an index i is a peak of pi if pi_{i-1} < pi_i > pi_{i+1}. Similarly, i is a descent of pi if pi_{i} > pi_{i+1}. I'll mention a generalization of these notions to other Coxeter groups, as well as some results regarding peak and descent subalgebras of the group algebra. 

Joint work with P. Harris, E. Insko, M. Omar, and B. Sagan.