Mathematical Problems from Materials Science and Finance

科恩0807347 该项目的前两个重点是数学、物理学和材料科学之间的接口，即（a）模拟晶体表面在粗糙化温度以下的演化，（B）设计新的方案，从电磁测量中“掩盖”空间区域。 第三个重点在于数学和金融之间的边界，即（c）研究异质信念或“非理性投资者”对市场的影响。“关于（a）：PI寻求统一两种广泛使用但明显不同的方法-一种基于阶跃动力学，另一种使用四阶偏微分方程。 关于（B）：PI继续着他最近在基于变量变化的隐身方案上的工作，特别是他探索了“近隐身”的设计，在有限频率下使用变量的常规变化而不是奇异变化来获得。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。" 该项目的主题是数学可以对其他科学领域产生巨大影响。 推力（a）--涉及晶体表面的演化--涉及材料科学的一个核心课题，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--这是现代科学许多领域反复出现的问题。 对这个例子的恰当处理建立了一个范例，在其他环境中也可能是有用的。 推力（B）--涉及“隐形”空间区域的方案--解决了当前光学领域的前沿问题。 如果所提出的隐形方案能够在实践中实现，它们将提供一种使物体难以“看到”的方法，例如雷达。 但PI的重点不是设备设计，而是最近由其他研究人员提出的一个特定的“隐身”方案的基本正确性。 推力（c）--涉及异质信念或“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，即使在最有效率的市场中，资产的市场价格也可能远远超过其内在价值。 PI探索的模型中，这类行为是由投资者信念的异质性来解释的。

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.