Collaborative Research: Homological Methods in Crystallography

建议：DMS-0204823，DMS-0204845主要研究人员：Benji Fisher，David Rabson A B S，T R A C Tfiner和Rabson正在对准晶的二维和三维对称性类型进行分类，并研究这些对称性的物理后果。这一分类概括了19世纪开始的关于晶体空间群的工作。研究人员遵循傅立叶空间方法进行结晶学研究，这种方法的优点是避免了更高维度的空间群。因此，分类从正交群O(2)和O(3)的有限子群的已知列表开始。他们将群上同调的技术引入傅立叶空间公式，其中一些技术在“直接空间”中已被长期使用。这些技术导致了理论上的简化和高效的计算方法。这些思想的一个重要应用是用群同调来描述某些不变量。两种最简单的同源类与已知的物理现象有关：衍射图的系统消光和电子带的交叉(“带粘”)。第二种最简单的类型首先出现在五阶四方调制晶体中，应该与一些类似的现象有关。产生了瓷砖模型，这种想法也扩展到磁性和彩色群体。结晶学是物理、化学和地质学的基础，并提供了许多信息。目前的研究已应用于最近在液晶(与手表和其他电子设备上流行的LCD显示器有关)、等离子体和调制晶体中的实验，这些高度对称的系统不能用经典晶体理论来描述。自1784年法国阿伯特雷纳·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。整整200年后，这种被称为“准晶”的材料被发现，随后对它们的性质进行了大量研究。准晶材料具有奇特的电学和物理性质，已被应用于电刀上的高质量不粘涂层。这项研究旨在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 being funded jointly by the Division of Mathematical Sciencesand the Division of Materials Research.