Dirac operators in representation theory

描述自然界中出现的对称性的数学结构是一个简单的代数对象，称为群。例如，可以考虑平面中圆的对称性，即，将圆映射到其自身的平面的长度保持变换。任何角度的旋转都保持圆不变。但是这组对称性具有某种内在的附加结构，例如，执行一个旋转之后执行另一个旋转给出了同一组中的另一个旋转，并且每个旋转具有反向旋转。这个集合与附加结构一起被称为平面上的特殊正交群，它是李群的一个例子。李群，以挪威数学家Sophus Lie的名字命名，是系统中固有的连续对称性的数学对象。这个建议福尔斯落在李群的表示领域。表示是李群可以表现自己的方式，例如，人们可以把特殊正交群看作是平面的旋转变换，而不是把它看作是一个抽象的对象。李群表示的研究有着悠久而辉煌的历史，并在数论和理论物理学中产生了变革性的影响。本项目的主要思想是导入和推广一个美丽的数学结构，称为狄拉克算子。狄拉克算子起源于物理学中的保罗·狄拉克在量子力学中的著名工作，随后，通过阿蒂亚和辛格的开创性工作在数学（几何）中找到了家，并在李群的表示论中找到了帕塔萨拉西，阿蒂亚-施密德，科斯坦特和许多其他人的工作。新的代数方法的理论是由沃根约15年前推出的狄拉克上同调，这开辟了新的和令人兴奋的角度研究数学。目前的项目将扩展狄拉克理论的代数设置和应用的狄拉克算子和狄拉克上同调的技术的p-adic李群和相关的代数结构（Hecke代数）的表示的世界与应用到现代数论和数学物理领域。

The mathematical structure that describes the symmetries that appear in nature is a simple algebraic object called a group. For example, one can consider the symmetries of a circle in the plane, i.e., length-preserving transformations of the plane that map the circle to itself. Rotations by any angle preserve the circle. But this set of symmetries has some intrinsic additional structure, e.g., performing one rotation followed by another gives another rotation in the same set and every rotation has an inverse rotation. This set, together with the additional structure, is called the special orthogonal group in the plane, and it is an example of a Lie group. Lie groups, named after the Norwegian mathematician Sophus Lie, are mathematical objects underlying the continuous symmetries inherent in a system. This proposals falls in the area of representations of Lie groups. Representations are ways in which Lie groups can manifest themselves, e.g., rather than regarding the special orthogonal group as an abstract object, one can think of its `representation' as transformations of the plane given by rotations. The study of representations of Lie groups has a long and illustrious history and has had transformative impact in number theory and theoretical physics. The main idea of the present project is to import and generalize a beautiful mathematical construction, called the Dirac operator. The Dirac operator originated in physics by the famous work of Paul Dirac in quantum mechanics, and subsequently, found a home in mathematics (geometry) by the seminal work of Atiyah and Singer, and in the representation theory of Lie groups in the work of Parthasarathy, Atiyah-Schmid, Kostant, and many others. The new algebraic approach to the theory was initiated by Vogan about 15 years ago with the introduction of Dirac cohomology and this has opened new and exciting perspectives of research in mathematics. The current project will extend the Dirac theory to an algebraic setting and apply the techniques of the Dirac operator and Dirac cohomology to the world of representations of p-adic Lie groups and of related algebraic structures (Hecke algebras) with applications to modern number theory and areas of mathematical physics.