| February |
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(St. Olaf) Title: "Garnir modules, Hall-Littlewood polynomials, and the cohomology of Springer fibers" Abstract: One classical construction of the irreducible representations of the symmetric group is as spaces of polynomials in n variables. However, apparently no one has previously studied the ideal these polynomials generate. I will describe the representation theory of these ideals. It turns out that determining this representation theory involves not only algebra and combinatorics, but also algebraic geometry and topology. This is joint work with Mark Haiman |
| Week 2. |
(University of South Caroline, Aiken) Title: On Renner Monoids Abstract: If you are interested in Lie theory, then you are a close friend of the Weyl group. If you are interested in semigroup theory, you are a great friend of the symmetric inverse semigroup. In this talk, I would like to introduce a new friend to you, who is a relative of the symmetric inverse semigroup and is much younger than the Weyl group, but whose size is much bigger than that of the Weyl group. The name of this guy is the so-called Renner monoid. This talk will discuss properties of the Renner monoid and its connection with combinatorics. Traditional examples consisting of matrices will be used to demonstrate the concepts. |
| Week 3. |
(Brandeis University) Title: Symplectic and smooth four-manifolds via singular fibrations Abstract: We will survey various results on the existence and uniqueness of maps with prescribed singularities on four-manifolds, along with certain classes of singular fibrations, by means of which the smooth type of the underlying manifold can be prescribed combinatorially. |
| March |
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(UC Davis & Universite Paris Sud) Title: "Sorting monoids and algebras on Coxeter group" Abstract: The usual combinatorial model for the 0-Hecke algebra H_n(0) (in type A) is to consider the algebra (or monoid) generated by the bubble sort operators pi_1,...,pi_{n-1}, where pi_i acts on words of length n and sorts the letters in positions i and i+1. This construction generalizes naturally to any finite Coxeter groups. By combining several variants of those operators (sorting, antisorting, affine) we construct several monoids and algebras. Astonishingly, they are endowed with very rich structures which relate to the combinatorics of descents and of several partial orders (such as Bruhat and left-right weak orders). These structures can be explained by numerous connections with representation theory, and in particular with affine Hecke algebras, and symmetric functions. While the focus of this talk will be on the combinatorial nature of the problem, we will show how our research was driven by this algebraic background together with computer exploration of examples by mean of the MuPAD-Combinat and Sage-Combinat software. |
| April |
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