Sophus Lie Days

     at Cornell University

            April 27–28, 2009

Monday, April 27, 2009

Pavel Etingof (MIT), Calogero-Moser systems and representation theory

Calogero-Moser systems are integrable systems (classical and quantum) which are attached to finite Coxeter groups. Attempts to prove their integrability lead to the discovery of Dunkl operators (certain remarkable pairwise commuting differential-reflection operators), and ultimately, the fundamental algebraic structure behind them - the Cherednik algebra. I will explain the proof of commutativity of Dunkl operators and integrability of Calogero-Moser systems, and then discuss representation theory of Cherednik algebras.

David Vogan (MIT), Inflatable mathematics: Schubert varieties and combinatorics

One of the most powerful tools in mathematics is the Gaussian elimination method for solving systems of linear equations. This tool shows how to reduce the number of dimensions in a system of equations, until finally you arrive at a lower triangular system that can be solved almost by inspection. The "inflatable" in the title refers to the fact that a big system of equations can be thought of as inflated from a small one.

The small tool makes it possible to understand the geometry of some important spaces called Schubert varieties, by regarding them as "inflated" from lower-dimensional Schubert varieties. I'll explain the connection and some of the combinatorial ideas used to describe the geometry. Finally, I'll describe a (VERY) large computer calculation about Schubert varieties, done in 2006 by a group of mathematicians including Dan Barbasch.

Tuesday, April 28, 2009

David Vogan (MIT), Calculating signatures of invariant Hermitian forms

A basic problem in representation theory is describing all the irreducible unitary representations of a group. These are the irreducible representations preserving a positive definite Hermitian form on the underlying vector space. For simple Lie groups, Knapp (almost thirty years ago) described all the irreducible representations preserving a possibly indefinite Hermitian form. To identify the unitary representations, it would suffice to calculate the signature of any such invariant form.

I will describe recent work of Jeff Adams' research group "Atlas of Lie groups and representations" on an algorithm for solving this problem.  

All talks are accessible to advanced undergraduates and beginning graduate students. If you are interested in attending, please send a note to liedays@math.cornell.edu. If you need help with your travel expenses, let us know.

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