Mathematical Problems from Materials Science and Finance

这个项目的前两个主题是数学、物理和材料科学之间的交界点，即(A)对低于强化温度的晶体表面的演化进行建模，以及(B)设计新的方案来通过电磁测量来“隐形”空间区域。第三个重点在于数学和金融学之间的分界线，即(C)考察不同信仰或“非理性投资者”对市场的影响。关于(A)项：PI寻求两种广泛使用但明显不同的方法的统一--一种基于阶跃动力学，另一种使用四阶偏微分方程式。关于(B)：PI继续他最近关于基于变量变化的隐形方案的工作；特别是他探索了在有限频率下利用变量的规则变化而不是奇异变化来获得的“近隐形”的设计。关于(C)：PI探讨了投资者信念的异质性如何在某些情况下导致投机和市场“泡沫”。该项目涉及数学可以对科学的其他领域产生影响的主题。推力(A)--涉及晶体表面的演化--解决了材料科学中的一个核心主题，对电子设备的设计和制造非常重要。此外，它的目标是将不同长度和时间尺度的模型联系起来--这是现代科学的许多领域中反复出现的问题。对这个例子的成功处理建立了一个范例，在其他环境中也可能有用。推力(B)--涉及“隐形”空间区域的计划--解决了当前的前沿光学问题。如果拟议的隐身方案能够在实践中实现，它们将提供一种手段，使物体难以使用，例如，雷达。但PI的关注点并不是OnDevice的设计，而是其他研究人员最近提出的一种特定的“隐形”方案的基本正确性。主旨(C)--涉及不同信仰或“非理性投资者”--解决了现代经济学的一个中心问题。现实世界的市场是复杂的，但它们的许多关键特征可以通过简单的模型来理解。最近的“互联网泡沫”提醒人们，即使在效率最高的市场，一项资产的市场价格也可能大大超过其内在价值。PI探索了用投资者信念的异质性来解释这种行为类型的模型。

Kohn0807347 The first two thrusts of this project are topics at theinterface between mathematics, physics, and materials science,namely (a) modeling the evolution of a crystal surface below theroughening temperature, and (b) designing new schemes for"cloaking" regions of space from electromagnetic measurements. Athird thrust lies at the boundary between mathematics andfinance, namely (c) examining the impact on markets ofheterogeneous beliefs or "irrational investors." Concerning (a):the PI seeks a unification of two widely-used but apparentlydistinct approaches -- one based on step dynamics, the otherusing a fourth-order partial differential equation. Concerning(b): the PI continues his recent work on achange-of-variable-based cloaking scheme; in particular heexplores the design of "near-cloaks," obtained using regularrather than singular changes of variables, at finite frequency. Concerning (c): the PI explores how the heterogeneity ofinvestors' beliefs can lead in some settings to speculation andmarket "bubbles." The project addresses topics where mathematics can havegreat impact on other areas of science. Thrust (a) -- involvingthe evolution of a crystal surface -- addresses a core topic inmaterials science, of great importance for the design andmanufacture of electronic devices. Moreover, its goal is thelinkage of models with different length and time scales -- arecurring issue in many areas of modern science. Successfultreatment of this example establishes a paradigm that could alsobe useful in other settings. Thrust (b) -- involving schemes for"cloaking" regions of space -- addresses a current frontier inoptics. If the proposed cloaking schemes can be realized inpractice, they will provide a means for making objects difficultto "see" using, for example, radar. But the PI's focus is not ondevice design; rather it is on the fundamental correctness of aspecific "cloaking" scheme recently proposed by otherinvestigators. Thrust (c) -- involving of heterogeneous beliefsor "irrational investors" -- addresses a central problem ofmodern economics. Real-world markets are complicated, but manyof their key features can be understood using simple models. Therecent "internet bubble" is a reminder that, even in the mostefficient markets, the market price of an asset can greatlyexceed its intrinsic value. The PI explores models in which thistype of behavior is explained by the heterogeneity of investors'beliefs.