This talk will describe recent work on mathematical methods for signal recovery in high noise. The first part of the talk will explain the connection between the Wiener filter, singular value shrinkage, and Stein's method for covariance estimation, and review optimal shrinkage in the spiked covariance model. We will then present extensions to heteroscedastic noise and linearly-corrupted observations. Time permitting, we will also give an overview of the related class of orbit recovery problems. William Leeb is an Assistant Professor in the School of Mathematics at the University of Minnesota, Twin Cities. He earned a B.S. in Mathematics from the University of Chicago in 2010, a Ph.D in Mathematics from Yale University in 2015, and was a postdoc in the Program in Applied and Computational Mathematics at Princeton University until 2018, when he joined Minnesota. His research interests are inapplied and computational harmonic analysis, statistical signal processing, and machine learning.