Explore advanced algorithms for heavy-tailed statistics in this 20-minute conference talk from the Association for Computing Machinery (ACM). Delve into high probability estimation techniques, Gaussian covariance estimation, and linear regression methods. Learn about covariance estimation and linear regression rates under weak assumptions, and examine key SOS (Sum of Squares) assumptions. Investigate the Median of Means framework, including one-dimensional cases and high-dimensional tournament estimators. Analyze SOS relaxation techniques, covering optimization problems, concentration steps, and expectation steps. Discuss Matrix Bernstein inequalities and their limitations, as well as evidence of hardness for covariance estimation. Conclude by exploring low-degree tests for detection in heavy-tailed statistical scenarios.