Abstract for colloquium talk at UT Arlington:

The classical Poincare-Birkhoff-Witt (PBW) Theorem sheds light on the structure of Lie algebras: These are, by definition, nonassociative rings, and the PBW Theorem states that nonetheless, a Lie algebra embeds into an associative ring, namely its universal enveloping algebra, that behaves in many ways like a polynomial ring (and this can be made precise). Many other rings share this advantageous property. In particular, they have PBW bases, which greatly facilitate their study. In this talk, we will first recall Lie algebras and the classical PBW Theorem. Then we will mention some more recent appearances of PBW-type theorems in the contexts of quantum groups, symplectic reflection algebras, graded Hecke algebras, and generalizations.