The backbone and virtue of Poincare-Birkhoff-Witt

In 1900, Poincare established a fundamental result in representation theory which was independently proved by Birkhoff and also by Witt in 1937. The three letters "PBW" are now sprinkled throughout combinatorics, representation theory, and noncommutative ring theory to advertise the sincerity behind an algebra's introduction: There is no less than meets the eye. We will discuss Poincare-Birkhoff-Witt properties with a view toward deformation theory, touching on quantum polynomial rings, Weyl algebras, universal enveloping algebras, skew group algebras, graded affine Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, and Koszul algebras as examples. We will sketch the use of Hochschild cohomology in securing PBW theorems, with special interest in groups acting on algebras and cases when the underlying field exhibits troublesome characteristic.