Emmy Noether and the History of Mathematical Structuralism: Invariants and IdealsAudrey Yap (University of Victoria)
Audrey Yap (University of Victoria) will give a talk titled "Emmy Noether and the History of Mathematical Structuralism: Invariants and Ideals".Abstract: In Hermann Weyl's obituary of Emmy Noether, he identifies several different periods in her work in which she transitions through different methodological styles. Though she began her career studying under Paul Gordan, and working in an algorithmic, constructive style (her early work), she truly grew into her own as an algebraist, having been encouraged to study abstract algebra by Ernst Fischer. In the second period Weyl identifies, Noether worked on invariant theory, some of which comprised her Habilitation work, but then turned to the theory of ideals, which is arguably one of her most important mathematical contributions. We will consider how her structural approach to mathematics developed as her work on invariant theory transitioned away from the algorithmic Gordan style of calculating invariants. Though in many ways, her contributions to ideal theory are extensions of work that had already been done by others, most notably Dedekind, this structural approach and a greater range of mathematical tools allowed her to generalize Dedekind’s work to great effect. It is exactly her emphasis on generalization that embodies her pioneering approach to abstract algebra. Her student B.L van der Waerden writes that the maxim by which she always let herself be guided was that ``all relations between numbers,functions, and operations become clear, generalizable, and truly fruitful only when they are separated from their particular objects and reduced to general concepts.” This paper will show how Noether's emphasis on abstraction and generalization of frameworks and results contributed to the more abstract conception of structure used in contemporary mathematics.
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