Emmy Noether and the Methods of Rationality

Emmy Noether and the Methods of Rationality

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Daniel Kriz, Princeton University
Fine Hall 110

Suppose you were to encounter a finite group G in the wild - you might ask yourself, is G a Galois group over a given field F? In 1917, Emmy Noether set forth an ambitious program to answer this question by asking another related question: if one lets G act linearly on an affine space over F, is the quotient variety always rational? Noether's problem, while still widely open and not having an affirmative answer in general, has inspired much work in, and revealed surprising connections to, areas such as number theory, birational geometry and representation theory. I will discuss some of what is known about Noether's problem as well as some of its relations to the above subjects. In particular, I will exposit recent work of Sophie Kriz, who answers Noether's problem for certain G and F, including when G is the 3-Sylow subgroup of the group of positions of the corners of a Rubik's cube.