This project centers on a new approach to the grand challenge of understanding, predicting, and eventually controlling the behavior of waves in complex and disordered media. Building on advances in both rigorous mathematics and cutting-edge physical experiments, the PIs are revealing hidden structures which arise from the cacophony of disordered data describing the materials and the geometry which guide wave behavior in real-world systems, with the ultimate aim of obtaining a new understanding of matter.
The investigators made progress in many directions this year.
- One of the main achievements of the year is a new approach to the structure of the phase space of quantum states in disordered systems. The approach relies on an intricate merger of the localization landscape tool set pioneered in the present project with the formalism of the Weyl transform and Wigner function. The first fruits of this approach have been an immediate and successful achievement of the first prediction of the spectral function in cold atoms, both in the quantum and the classical regime, without any adjustable parameters, and the prediction of absorption curves in three-dimensional semiconductors with alloy disorder. We are still in the process of unveiling the full potential and rich underlying mathematical structure of this exciting new development and comparing these theoretical results with the experimental measurements of the ultra-cold atoms team.
- Simultaneously, the PIs have established the Landscape Law (proved last year in the continuum setting) for discrete tight-binding Hamiltonians. Through numerical simulations they have shown that, with the appropriate and predictable choice of scaling constants, the Landscape Law provides a highly accurate approximation to the integrated density of states in disordered systems in both the lower and upper portions of the spectrum.
- The year 2020 also brought the first validation of the landscape approach for random matrices. It is astonishing, even to the PIs, that the concept of the landscape, born in the realm of PDEs, proved to be useful to address localization of the eigenfunctions in the general setting of M-matrices, opening a door to open problems from a completely new area of mathematics.
- Another focal point of our work is the impact of boundary structure on solutions, and the related issue of regularity of free boundaries. In 2020, the PIs achieved a long sought-after result in this direction, the absolute continuity of the elliptic measure associated to a suitable degenerate PDE on all rectifiable sets, with boundaries of arbitrary, possibly lower, dimension.
- Also within the geometric component of the project, the PIs have discovered what appears to be a new extrinsic geometry of level surfaces of harmonic functions, parallel to the classical geometry of hypersurfaces. We expect this to lead to new, natural qualitative and quantitative estimates of level surfaces of harmonic functions and eigenfunctions.
- Exploring further the interplay between spatial localization and geometry, the PIs established a deep connection between crystalline measures (distributions which, together with their Fourier transform, are supported on a locally finite set), quantum graphs, and quasicrystals.
- Disordered semiconductors are an essential test bench for our theories. This year, the PIs made important progress in the understanding and the computation of light interaction in these materials. On the computational side, the PIs developed an original landscape-based overlapping domain decomposition approach to compute absorption curves of a 1-dimensional analogue of InGaN semiconductors. Their new method required the calculation of about 1,000 times fewer inner products compared to the gold standard computational technique, while still capturing the essential absorption peaks.
- Using full 3D landscape-based device simulations, the PIs identified a new mechanism of sequential carrier injection in active multiple quantum wells tied to increased energy barriers to electron transport. This mechanism lies at the origin of the puzzling lack of efficiency of green LEDs, the notorious green gap problem. Following this discovery, they invented new structures that circumvent the barriers by lateral injection into the quantum wells. Fabrication is underway.
- Finally, the PIs began the study of a new class of materials, perovskites, which bear tremendous potential for solar cell applications. The perovskites are characterized not only by static disorder, which is addressed well by the landscape theory, but also by dynamical disorder. This brings new challenges requiring new theoretical tools.
The Third Pillar of Science
Professor of Mathematics, School of Mathematics, University of Minnesota
In the late 20th century, science underwent a revolution as computational science emerged as the third mode of scientific inquiry alongside experiment and theory. Computer simulation of physical reality has played an equally transformative role in virtually all areas of technology, affecting many aspects of modern life. We now depend on simulation to design, predict and optimize natural and engineered systems of all sorts, ranging from mechanical to chemical to electronic and scales ranging from atomic to terrestrial to cosmological. Mathematical algorithms have been crucial to these advances, even more so than advances in computer technology. In this talk, Douglas Arnold will discuss some of the key ideas that have emerged and the ongoing challenges facing computational mathematics in simulating the physical world.
More information is available at the lecture’s page.
- Talk Slides