Mathematical Problems from Materials Science and Finance

这个项目的前两个重点是数学，物理和材料科学之间的界面主题，即(a)模拟晶体表面在增韧温度以下的演变，以及(b)设计新的方案来“掩盖”电磁测量的空间区域。第三个重点在于数学和金融之间的边界，即(c)检查异质信念或“非理性投资者”对市场的影响。关于(a): 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。关于(b)：私家侦探继续其最近关于基于变量变化的隐身方案的工作；特别地，他探索了“近斗篷”的设计，在有限频率下使用正则变量而不是奇异变量的变化来获得。关注(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其他科学领域产生重大影响的主题。推力(a) -涉及晶体表面的演变-解决了材料科学的核心主题，对电子器件的设计和制造非常重要。此外，它的目标是连接不同长度和时间尺度的模型——这在现代科学的许多领域都是一个反复出现的问题。对这个例子的成功处理建立了一个范例，在其他情况下也可能有用。推力(b)——涉及“隐形”空间区域的方案——解决了当前光学领域的一个前沿问题。如果提出的隐形方案可以在实践中实现，它们将提供一种使物体难以“看到”的手段，例如使用雷达。但PI的重点不在于设备设计；相反，它是关于其他研究者最近提出的具体“隐形”方案的基本正确性。推力(c)——涉及异质信徒或“非理性投资者”——解决了现代经济学的一个核心问题。现实世界的市场是复杂的，但它们的许多关键特征可以用简单的模型来理解。最近的“互联网泡沫”提醒我们，即使在最有效的市场中，一项资产的市场价格也可能大大超过其内在价值。PI研究了用投资者信念的异质性来解释这类行为的模型。

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.