Daniil Rudenko | Goncharov depth conjecture and volumes of orthoschemes (in Russian)
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2021-01-28

Seminar on A1-topology, motives and K-theory, January 28, 2021
Daniil Rudenko (University of Chicago)
Goncharov depth conjecture and volumes of orthoschemes

Goncharov conjectured that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. In the first part of the talk I will explain how this conjecture fits into the general scheme of conjectures about mixed Tate motives. In the second part of the talk I will explain an idea behind the proof of the Goncharov conjecture. The proof is based on an explicit formula, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Surprisingly, almost the same formula gives a volume of a hyperbolic orthoscheme generalising the formula of Lobachevsky in dimension 3 to an arbitrary dimension. 

The talk is based on the preprint https://arxiv.org/abs/2012.05599v1