Collaborative Research: Homological Invariants in Crystallography

建议： DMS-0204823、DMS-0204845主要研究者：Benji Fisher、大卫·拉布森 A B S T R A C T Fisher和Rabson正在对二维和三维的拟物质的对称性类型进行分类，并研究这些对称性的物理后果。 这种分类概括了世纪开始的关于晶体空间群的工作。 晶体学的傅立叶空间方法，其优点是避免了更高维度的空间群。 因此，分类从已知的正交群O（2）和O（3）的有限子群列表开始。 他们介绍了技术的groupcosomology，其中一些已长期用于“直接空间”，以傅立叶空间制定。 这些技术导致boththeoretical简化和有效的计算方法。 这些思想的一个重要应用是根据群同调描述某些auge不变量。 同调类的两种最简单的类型与已知的物理现象有关：衍射图样中的系统衍射和电子带的交叉（“带粘滞”）。 下一个最简单的类型首先出现在五阶四倍频调制晶体中，并且应该与一些类似的现象有关。 瓷砖模型的产生，以及想法也扩展到磁性和颜色组。晶体学的基础和通知很多物理，化学和地质学。目前的研究已经应用到最近的实验在液晶（与流行的液晶显示器上的手表和其他电子设备），等离子体，和调制晶体，高度对称的系统，不能描述的经典理论的晶体。 自从1784年法国修道院院长勒内·茹斯特·豪伊推导出晶体的微观结构以来，人们一直认为晶体不可能具有二十面体（或足球）的对称性。 200年后，这种被称为“准晶体”的材料被发现了，并且对其性质进行了大量的研究。 准晶具有奇特的电子和物理性质，已经在电外科手术刀片上作为高质量的不粘涂层得到应用。 本研究的目的是对晶体的对称性类型进行分类，并研究与这些对称性相关的物理性质，该项目由数学科学部和材料研究部联合资助。

Proposal: DMS-0204823, DMS-0204845Principal Investigators: Benji Fisher, David Rabson A B S T R A C TFisher and Rabson are classifying the symmetry types of quasicrystalsin two and three dimensions and studying the physical consequences ofthese symmetries. The classification generalizes the work begun inthe nineteenth century on space groups of crystals. The investigatorsfollow the Fourier-space approach to crystallography, which has theadvantage of avoiding space groups in higher dimensions. Thus theclassification starts from the known list of finite subgroups of theorthogonal groups O(2) and O(3). They introduce techniques of groupcohomology, some of which have long been used in "direct space," tothe Fourier-space formulation. These techniques lead to boththeoretical simplifications and efficient computational methods. Oneimportant application of these ideas is the description of certaingauge invariants in terms of group homology. The two simplest typesof homology class are connected with known physical phenomena:systematic extinctions in diffraction patterns and crossing ofelectronic bands ("band sticking"). The next simplest type firstoccurs in a rank-five tetragonal modulated crystal and should beconnected with some similar phenomenon. Tiling models are produced,and the ideas are also extended to magnetic and color groups.Crystallography underlies and informs much of physics, chemistry, andgeology. The present research has applications to recent experimentsin liquid crystals (related to the popular LCD displays onwristwatches and other electronic equipment), plasmas, and modulatedcrystals, highly symmetric systems that cannot be described by theclassical theory of crystals. Ever since 1784, when the French abbotRene Just Hauy deduced the microscopic structure of crystals, it hadbeen believed impossible for a crystal to have the symmetries of anicosahedron (or a soccer ball). Precisely 200 years later, suchmaterials, called "quasicrystals," were discovered, and much researchhas ensued into their properties. Quasicrystals possess strangeelectronic and physical properties and have already found applicationas high-quality, non-stick coatings on electrosurgical blades. Thepresent research aims to classify the symmetry types of crystals andto study physical properties associated with these symmetries.This project is funded jointly by the Division of Mathematical Sciencesand the Division of Materials Research.