Open problems in geometry of numbers, and the geometry of quantum phases

数几何和量子相几何中的未决问题

• 批准号: 5355-2007
• 负责人: Erdahl, Robert
• 金额: $ 0.73万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=364786
• 关键词: Open; problems; geometry; numbers; quantum

The first part of my program is to investigate fundamental questions in the theory of lattices and geometry of numbers.  I have been able to introduce new methods into these classical theories, which in two instances resulted in answering question posed by Voronoi nearly 100 years ago.  I have also established links between some classical results in geometry of numbers and a variety of modern mathematical topics.  The long-range goal is to understand more clearly the geometry of lattices in higher dimensional spaces, such as 6, 8, or even 20.  The second part of my program is to understand how phases are formed in solids.  Such phases change the material properties of solids in dramatic ways; examples are the super-conducting state, or the anti-ferromagnetic state.   I am developing new methods to calculate the properties of these quantum phases so that I can achieve a deeper understanding of how phases are formed.

我的第一部分是研究格理论和数论中的基本问题，我已经能够将新的方法引入这些经典理论，这在两个实例中导致回答问题提出的Voronoi近100年前。我还建立了一些经典的结果之间的联系，在几何的数字和各种现代数学主题。长期-我的目标是更清楚地理解更高维空间中的晶格几何，如6，8，甚至20。我的计划的第二部分是理解相是如何在固体中形成的。这些相以戏剧性的方式改变固体的材料性质;例子是超导状态，或反铁磁状态。 我正在开发新的方法来计算这些量子相的性质，以便我可以更深入地了解相是如何形成的。