Second talk:
The Betti Polynomial of Powers of an Ideal(Thursday, March 11, 10:00-11:00)

For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of $\beta_i(S/I^k)=\dim_K\Tor_i^S(S/\mm,S/I^k)$ as k goes to infinity. By Kodiyalam's result it is known that $\beta_i(S/I^k)$ is a polynomial for large $k$. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials have all highest possible degree. (this is joint work with Juergen Herzog)

Third talk:
Random to Random Shuffles and Commuting Families of Matrices (Saturday, March 13, 10:30-12:00)

We describe a family of matrices with rows and columns indexed by permutations. The entries generalize the inversion statistics on the symmetric group. These matrices are not only related to the inversion statistics but also are scaled versions of the transition matrices of Markov chains generalizing the random to random shuffle and can be factored into projections on a polytope generalizing the linear ordering polytope. Using techniques from enumerative combinatorics and respresentation theory We show some of the beautiful properties of these matrices. In particular, we study the algebra generated by the matrices, which can be seen as a subalgebra of the group algebra of the symmetric group. Finally we describe a generalization of the matrices within the symmetric group and for general finite Coxeter groups. (this is joint work with Franco Saliola and Vic Reiner).