**Abstract:** Group actions have implicitly played a role in applications since at least the work of Fourier on solutions of the heat equation. Namely, one can generate the classical Fourier basis as an action of the group \$\mathbb{T}\$ on \$L^2([0,1])\$. Hundreds of years later in the 20th century, the two most important transforms in applied harmonic analysis arose, the wavelet transform and the short-time Fourier transform, which were created using projective unitary representations of the affine group and (a group related to the) the Weyl-Heisenberg group, respectively. Even in the modern era of data-driven approaches, group symmetries still play an important role, from generating transformations like diffusion wavelets and the scattering transform to characterizing overtrained networks in neural collapse. The talk will start as a colloquium-style talk on applications of group symmetry and finish with some results on the relationship between group symmetry and optimality.

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