Gravity and Regulators of Number Fields

Abstract: We are going to describe two applications of a new tool - higher dimensional iterated integrals. One of the applications is in number theory and the other - in quantum physics.

The number theoretic application is about Borel regulator of a number field: We express the values of the Dedekind zeta function at the positive in terms of multiple polylogarithms. Zagier has conjectured that single-valued polylogarithms are enough.

The second application is a new approach to quantum gravity. A starting point for the gravity that we consider is general relativity in terms of a connection, using spinors. (Some people who have worked in this direction are sir Penrose, Ashtekar, Gambini, Lano, Fedosin, Agop, Buzea and Ciobanu, Mashhoon, Gronwald, and Lichtenegger, Clark and Tucker. There are many relations between the two application. For example: the loop expansion in this approach to quantum gravity is done in terms of Borel regulators of number fields.