The first two thrusts of this project are topics at the interface between mathematics, physics, and materials science, namely (a) modeling the evolution of a crystal surface below the roughening temperature, and (b) designing new schemes for "cloaking" regions of space from electromagnetic measurements. A third thrust lies at the boundary between mathematics and finance, namely (c) examining the impact on markets of heterogeneous beliefs or "irrational investors." Concerning (a): the PI seeks a unification of two widely-used but apparently distinct approaches -- one based on step dynamics, the other using a fourth-order partial differential equation. Concerning (b): the PI continues his recent work on a change-of-variable-based cloaking scheme; in particular he explores the design of "near-cloaks," obtained using regular rather than singular changes of variables, at finite frequency. Concerning (c): the PI explores how the heterogeneity of investors' beliefs can lead in some settings to speculation and market "bubbles."
The project addresses topics where mathematics can have great impact on other areas of science. Thrust (a) -- involving the evolution of a crystal surface -- addresses a core topic in materials science, of great importance for the design and manufacture of electronic devices. Moreover, its goal is the linkage of models with different length and time scales -- a recurring issue in many areas of modern science. Successful treatment of this example establishes a paradigm that could also be useful in other settings. Thrust (b) -- involving schemes for "cloaking" regions of space -- addresses a current frontier in optics. If the proposed cloaking schemes can be realized in practice, they will provide a means for making objects difficult to "see" using, for example, radar. But the PI's focus is not on device design; rather it is on the fundamental correctness of a specific "cloaking" scheme recently proposed by other investigators. Thrust (c) -- involving of heterogeneous beliefs or "irrational investors" -- addresses a central problem of modern economics. Real-world markets are complicated, but many of their key features can be understood using simple models. The recent "internet bubble" is a reminder that, even in the most efficient markets, the market price of an asset can greatly exceed its intrinsic value. The PI explores models in which this type of behavior is explained by the heterogeneity of investors' beliefs.
OVERVIEW: This project's primary thrusts were topics at the interface between mathematics, physics, and materials science. Early areas of focus included (1) the evolution of crystal surfaces, and (2) analyzing schemes for "cloaking" regions of space from electromagnetic measurements. Over time the PI identified and pursued additional scientific opportunities, including (3) wrinkling patterns in thin elastic sheets, and (4) membrane metamaterials. The project also had a secondary thrust related to finance, specifically (5) how the heterogeneity of investors' beliefs can lead to speculation. (1) CONCERNING THE EVOLUTION OF CRYSTAL SURFACES: The conventional approach uses a differential equation for the height of the surface as a function of space and time. The PI's work with Hala Al Hajj Shehadeh and Jonathan Weare (Physica D 240, 2011, 1771-1784) showed that other approaches can also be fruitful. One uses a discrete model for the motion of steps on a crystal surface, looking for a similarity solution that represents its stable large-time asymptotics. A second approach involves a differential equation for the slope as a function of height and time. These are related: the step motion law is a finite-difference scheme for the differential equation. (2) CONCERNING CLOAKING: The PI's work with Daniel Onofrei, Michael Vogelius, and Michael Weinstein (Comm Pure Appl Math 63, 2010, 973-1016) addressed a cloaking scheme proposed by Pendry, Schurig, and Smith in 2006, identifying a previously-unappreciated weakness. This work showed that while the scheme works for most inclusions, certain "cloak-busting" inclusions generate resonances and are therefore not cloaked at all. It also showed how this difficulty can be overcome, by using a lossy layer at the inner edge of the cloak. (3) CONCERNING WRINKLING PATTERNS IN THIN ELASTIC SHEETS: Stress-induced patterns in thin sheets are very familiar: our skin wrinkles and our clothes wrinkle; leaves, flowers, and hanging drapes have folds. When such patterns are observed, it is often unclear whether their geometry is due to elastic energy minimization, or due instead to the detailed history of growth or loading. To address this question, it is natural to explore the character of the elastic-energy-minimizing patterns. This topic fits within the framework of energy-driven pattern formation, a current frontier in the calculus of variations. A central theme is the identification of energy scaling laws, which show how the minimal energy depends on the thickness of the sheet and other physical parameters. A key step is the development of geometry-independent lower bounds, showing that a particular scaling law is optimal in the sense that no pattern can do better. The PI completed several projects in this area. One, with Peter Bella (J Nonlin Sci 24, 2014, 1147-1176) concerned "metric-driven wrinkling," a model for the intricate folding patterns sometimes seen in leaves and flowers. Another, with Jacob Bedrossian (Comm Pure Appl Math, in press) concerned the blister patterns sometimes seen when a thin film on an elastic substrate experiences compression due to thermal mismatch. A third, with Hoai-Minh Nguyen (J Nonlin Sci 23, 2013, 343-362) explained the "herringbone" pattern sometimes seen in compressed thin films on compliant substrates. And a fourth, with Peter Bella (Comm Pure Appl Math 67, 2014, 693-747), examined the wrinkling of an annular sheet with tensile loads at both boundaries. By focusing on energy scaling laws and geometry-independent lower bounds, the PI's work has helped explain "why" elastic energy minimization leads to the development of wrinkling patterns. (4) CONCERNING MEMBRANE METAMATERIALS: This term comes from the recent physics literature, which has explored the design of structured, flexible walls that can reflect sonic vibration. The physics literature showed that such walls exist, but it offered only a relatively crude explanation of how and why they work. Jens Jorgensen's PhD thesis ("Zero transmission in acoustic membranes", NYU, 2013) provided the first clear explanation of the mechanism. (5) CONCERNING THE CONSEQUENCES OF HETEROGENEOUS BELIEFS: Asset price pubbles occur even in the most efficient financial markets. A classic 1978 paper by Harrison and Kreps showed how the interaction of investors with different beliefs can lead to speculative behavior. The PI's work with Xi Chen (Fin Stoch 15, 2011, 221-241) provided a simple, transparent example of their mechanism. TRAINING YOUNG SCIENTISTS: A significant part of the project's budget was devoted to support for PhD students. The projects summarized above include thesis work by PhD students Hala Al Hajj Shehadeh (now an Assistant Professor at James Madison University), Peter Bella (now a postdoc at the Max Planck Institute for Mathematics in the Sciences), Jens Jorgensen (now doing research and development at 3Shape, a small high-tech firm), and Xi Chen (now in the finance industry at Field Street Capital Management).
