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

通过Mostow刚性，双曲三维流形的体积和Chern-Simons不变量是拓扑不变量。它们通常被认为是复值不变量（称为复体积）的真实的和虚部。复体积可以更一般地定义为一个3-流形群到一个简单复李群的抛物线表示的不变量，例如SL（n，C）。PI和他的合作者将研究这种表示，并扩展以前只知道SL（2，C）的几个结果。特别地，我们将给出复体积的具体公式。新的想法是抛物表示在诺依曼扩展布洛赫群中有一个基本类。利用PI的一个结果，推广的Bloch群可以定义在数域上，并且同构于代数K-群。复体积和Chern-Simons不变量是一个有趣而又重要的不变量，它们广泛地出现在数学物理、双曲几何和数论等数学领域。该提案引入了新的技术来计算这些不变量通过一个对象被称为扩展布洛赫群。扩展的布洛赫群同构于出现在数学的另一个分支中的群，称为代数K-理论。与K理论的关系可能会导致对体积和Chern-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.