The block filtration and motivic multiple zeta values

Abstract: Multiple zeta values are a class of transcendental numbers, going back to Euler in the 1700s and with ties to the Riemann zeta function. They are found arising naturally in many areas of mathematics and physics, from algebraic geometry to Feynman amplitudes. Unlike in the case of single zeta values, we know many algebraic relations satisfied by multiple zeta values: the double shuffle relations, the associator relations, the confluence relations. However it is unknown if any of these sets of relations are complete. Assuming Grothendieck's period conjecture, a complete set of algebraic relations are given by the motivic relations, arising from a connection to P^1 minus three points. However these relations are inexplicit.

In this talk, we introduce a new filtration, called the block filtration, on the space of multiple zeta values. By considering the associated graded, we describe several new families of motivic relations, that provide a complete description of relations in low block degree. A generalisation of these results would thus provide a complete description of relations among multiple zeta values and aid in settling several open problems about the motivic Galois group of mixed Tate motives.

number theory

Audience: researchers in the topic

( slides )

Warsaw Number Theory Seminar