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Explore the mathematical foundations of random perturbations in Toeplitz matrices through this Members' Colloquium lecture delivered at the Institute for Advanced Study. Delve into the pioneering 1947 work of John Von Neumann and Herman Goldstine, who introduced the concept of modeling computing noise through probabilistic methods while developing IAS computing machines. Discover how their groundbreaking perspective on "probabilistic noise" continues to provide crucial insights into modern computational challenges, particularly in understanding the severe numerical errors that occur when computing eigenvalues of poorly conditioned matrices. Examine Toeplitz matrices, a fundamental class of matrices with widespread applications across mathematics, physics, and engineering, known for their extreme sensitivity to small perturbations. Learn how the century-old theoretical framework surrounding these matrices enables remarkably precise descriptions of their spectral behavior when subjected to random perturbations. Gain insights into contemporary research addressing the intersection of numerical analysis, random matrix theory, and computational mathematics, based on collaborative work with Mireille Capitaine and François Chapon.
