First talk: Problems in the Theory of f-vectors of Simplicial Complexes (Wednesday, March 10, 10:30-12:00)
We first briefly survey results about classes of f-vectors of simplicial
complexes. In particular, we are interested in the g-conjecture
which is a conjectural classification of the set of f-vectors
of spimplicial complexes triangulating a sphere or more generally
Gorenstein* simplicial complexes. We exhibit two directions
of recent research:
1) f-vectors of barycentric subdivisions
2) f-vectors of Buchsbaum* simplicial complexes
which provide either special cases of the g-conjecture or lead
to the more general setting of triangulates orientable manifolds.
(this are joint works with Francesco Brenti and Christos Athanasiadis)
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).
Information
Time and Date:
Wednesday, March 10, 2010 - 10:30-12:00
Thursday, March 11, 2010 - 10:00-11:00
Saturday, March 13, 2010 - 10:30-12:00