ABSTRACT
Normal forms: a historical overview One of the most common things to do in mathematics is solving equa-
tions. Since the introduction of the coordinate method for geometric problems by Descartes, everyone knows that combining algebra and geometry leads to mutual benefits: geometric problems may be approached in a uniform way through solving algebraic equations, and algebraic equations may become easier to deal with if one tries to think in terms of geometric properties of solutions to those equations. On the other hand, in the 20th century it became common to view geometric objects via algebras of functions on those objects. Once this viewpoint is taken, it becomes absolutely crucial to be able to work with algebras in an effective way. That does not necessarily have to mean using computer algebra systems; a computation using pen and paper, or blackboard and chalk, also needs to represent elements of algebras in a concrete way in order to write them down, decide whether two elements are equal to each other, etc. This naturally leads to hunting for “normal forms”, some canonical ways to represent elements. In this book, we mainly use that philosophy to study algebraic objects that are somewhat more abstract than polynomial equations that Descartes would have used: noncommutative algebras, twisted associative algebras, and operads.
