Semiinvariants of Finite Reflection Groups

Let G be a finite group of complex n by n unitary matrices generated by reflections acting on C^n. Let R be the ring of invariant polynomials, and \chi be a multiplicative character of G. Let \Omega^\chi be the R-module of \chi-invariant differential forms. We define a multiplication in \Omega^\chi and show that under this multiplication \Omega^\chi has an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between \chi-invariant forms and logarithmic forms.Comment: Paper presented at 1999 Joint Meetings in San Antonio, special session on Geometry in Dynamics. Typo correcte

Stochastic modeling for the COMET-assay

We present a stochastic model for single cell gel electrophoresis (COMET-assay) data. Essential is the use of point process structures, renewal theory and reduction to intensity histograms for further data analysis

Poincare-Birkhoff-Witt Theorems

We sample some Poincare-Birkhoff-Witt theorems appearing in mathematics. Along the way, we compare modern techniques used to establish such results, for example, the Composition-Diamond Lemma, Groebner basis theory, and the homological approaches of Braverman and Gaitsgory and of Polishchuk and Positselski. We discuss several contexts for PBW theorems and their applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and symplectic reflection and related algebras.Comment: 30 pages; survey article to appear in Mathematical Sciences Research Institute Proceeding