Wave Localization and its Landscape 116



Douglas Arnold, University of Minnesota, USA


2020.04.11 09:30-10:30




Conference ID: 955-626-784
PIN Code: 667844



The puzzling phenomenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating media. Localization arises in many physical and mathematical systems and has many important implications and applications. It is closely associated with Philip Anderson (1923-2020) who received the Nobel prize in 1977 for his 1958 discovery of localization of the eigenfunctions of the Schrödinger equation with a random potential, a process which dramatically affects the electrical properties of materials disordered by impurities. Despite this long history, many aspects of localization remain mysterious even today. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.


Douglas N. Arnold is the McKnight Presidential Professor of Mathematics at the University of Minnesota. He is a research mathematician and educator specializing in computational mathematics. Prof. Arnold’s work centers on numerical analysis, especially as applied to partial differential equations arising in physics and engineering. Many of his numerical methods have become standard tools for studying elastic bodies, plate theory, fluid mechanics, and other applications. He may be best known as the founder of finite element exterior calculus, a paradigm for structure-preserving discretization which obtains stability numerical methods by preserving relevant geometric and topological structure. Recently Prof. Arnold is using numerical methods to understand and control the localization of waves, especially for applications in quantum mechanics.

Arnold’s numerous awards include a plenary lecture at the International Congress of Mathematicians in Beijing in 2002, a Guggenheim fellowship in 2008, foreign membership in the Norwegian Academy of Science and Letters in 2009, the SIAM Prize for Distinguished Service to the Profession in 2013, and the J. Tinsley Oden Medal from the US Association of Computational Mechanics in 2015. He delivered the Feng Kang Distinguished, Lecture at the Chinese Academy of Sciences in 2009, and the Erwin Schrodinger Lecture at Vienna’s Erwin Schrodinger Institute this past December. Prof. Arnold is a fellow of the American Mathematical Society, SIAM, and the American Association for the Advancement of Science. He has also held many scientific leadership roles including the Presidency of the Society for Industrial and Applied Mathematics SIAM in 2009 and 2010, and directorship of the Institute for Mathematics and its Applications (IMA) before that. Among Arnold’s priorities are efforts to increase public understanding of mathematics and its role in society. In 2007 he coauthored an award winning video, Möbius Transformations Revealed, which went viral on YouTube, garnering over two million views.