The volume and Chern-Simons invariant in topology and number theory.

根据Mostow刚性，双曲3流形的体积不变量和chen - simons不变量是拓扑不变量。它们通常被认为是复值不变量复体积的实部和虚部。复体积可以更一般地定义为3流形群的抛物表示成简单复李群的不变量，例如SL（n，C）。PI和他的合作者将研究这种表示，并扩展以前仅为SL（2，C）所知的几个结果。特别地，我们将给出复体积的具体公式。新的想法是抛物线表示在诺伊曼的扩展布洛赫群中有一个基本类。利用PI的结果，扩展Bloch群可以定义在一个数域上，并且与代数k群同构。这可能会使人们对复数体积在数论中的作用有新的认识。体积不变量和chen - simons不变量是有趣而重要的不变量，它们出现在数学的各个领域，包括数学物理、双曲几何和数论。该提案引入了通过一个被称为扩展Bloch群的对象来计算这些不变量的新技术。扩展Bloch群与出现在数学的另一个分支代数k理论中的群是同构的。与k理论的关系可能会导致对体积和chen - simons不变量在数论中的作用的更好理解，这到目前为止是完全神秘的。

By Mostow rigidity, the volume and Chern-Simons invariant of a hyperbolic 3-manifold are topological invariants. They are often regarded as the real and imaginary part of a complex valued invariant known as the complex volume. The complex volume can be defined more generally as an invariant of a parabolic representation of a 3-manifold group into a simple complex Lie group, e.g. SL(n,C). The PI and his collaborators will study such representations and extend several results previously only known for SL(2,C). In particular, we will give a concrete formula for the complex volume. The new idea is that a parabolic representation has a fundamental class in Neumann's extended Bloch group. By a result of the PI, the extended Bloch group can be defined over a number field, and is isomorphic to an algebraic K-group. This may shed new light on the role of the complex volume in number theory.The volume and Chern-Simons invariant are interesting and important invariants that appear in a wide range of areas of mathematics including mathematical physics, hyperbolic geometry and number theory. The proposal introduces new techniques for computing these invariants via an object known as the extended Bloch group. The extended Bloch group is isomorphic to a group appearing in a different branch of mathematics known as algebraic K-theory. The relationship with K-theory may lead to an improved understanding of the role of the volume and Chern-Simons invariant in number theory, which has so far been completely mysterious.