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

我的计划的第一部分是研究格论和数几何中的基本问题。我已经能够将新的方法引入这些经典理论中，在两个例子中，我回答了近100年前沃罗诺伊提出的问题。我还在数字几何中的一些经典结果与各种现代数学主题之间建立了联系。我的长期目标是更清楚地理解更高维空间中的格几何，如6，8，我的计划的第二部分是了解固体中的相是如何形成的。这些相以戏剧性的方式改变了固体的材料性质；例如超导态，或反铁磁态。我正在开发新的方法来计算这些量子相的性质，以便我可以对相是如何形成的有更深的理解。

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.