Equivariant noetherianity: theory and applications

The infinite variable polynomial ring R=C[x1,x2,...] is often one's first example of a non-noetherian ring. Cohen proved that it is, however, noetherian up to the action of the infinite symmetric group. This has been used to prove uniformity results for finite-dimensional structures in algebraic geometry, statistics, and algebraic topology, and motivates the following question: what other algebraic and geometric properties of R exhibit finiteness up to equivariance? In this talk, we will give an overview of the field of equivariant commutative algebra and highlight some applications to algebraic statistics and tensors.