# Grothendieck-Teichmüller, Kashiwara-Vergne and MZVs

**Place: **Room 1-05, UNIGE Section of Mathematics, rue du Conseil-Général 7-9, Geneva

#### Monday August 12

**10:00 - 11:00 Iva Halacheva (Northeastern) **

*Kashiwara-Vergne solutions degree by degree *

Solutions to the Kashiwara-Vergne equations (KV solutions) in Lie theory have been shown to play an important role in topology, describing finite-type invariants for a family of knotted objects known as welded foams. The KV solutions come from the degree-completed free Lie algebra on 2 generators, which has a grading, so a natural question that arises is whether, given a solution up to degree n it can be extended to degree n+1. We show this is indeed the case and leads to further interesting properties of the Kashiwara-Vergne groups acting on this set of solutions. This is joint work with Zs. Dancso, M. Robertson, and G. Laplante-Anfossi.

**11:30 - 12:30 Guillaume Laplante-Anfossi (Melbourne) **

*Kashiwara—Vergne operads *

Gluing genus zero surfaces along boundaries endows their mapping class groups with the structure of an operad. A deep theorem of Boavida de Brito, Horel and Robertson from 2017 identifies the homotopy automorphisms of this operad with the Grothendieck—Teichmüller group, a mysterious profinite group containing the absolute Galois group of the rational numbers.

Intersecting loops on genus zero surfaces defines a Lie bialgebra structure on their fundamental groups, called the Goldman—Turaev Lie bialgebra. Around the same time, Alekseev, Kawazumi, Kuno and Naef defined group homomorphisms from the Grothendieck—Teichmüller group to the group formed by some special tangential automorphisms of the Lie bialgebra associated with any genus zero surface.

Are these two results related? I will describe ongoing joint work with Zsuzsanna Dancso, Iva Halacheva and Marcy Robertson, where we show that the tangential automorphisms known as Kashiwara—Vergne solutions, as well as their two symmetry groups, form operads. I will also mention what we know so far about their precise relationship to the Grothendieck—Teichmüller group.

**15:00 - 16:00 Marcy Robertson (Melbourne) **

*Duality and GT actions on Tangles *

I will describe how one uses cyclic structure on the operad of parenthesized braids to determine an action of the Grothendieck-Teichm\”uller group, $GT$, on a category of “parenthesized tangles”. This action extends to a $GT$-action on a categorical models for virtual and welded tangles which, at least conjecturally, should extend to an action of the Kashiwara-Vergne symmetry groups. This talk contains joint work with Chandan Singh.

#### Tuesday August 13

**10:00 - 11:00 Dror Bar-Natan (Toronto) **

*Secondary Operations, Emergent Knots, and the Goldman-Turaev Lie Algebra *

I will explain what are "secondary operations" (that's a technical term, not an insult) and explain how and why I like to see the Goldman bracket and the Turaev co-bracket as secondary operations on spaces of "emergent knots" ("emergent" is in fact a mild insult). Link

**11:30 - 12:30 Yusuke Kuno (Tsuda), by Zoom **

*Emergent version of Drinfeld's associator equations*

This is based on an ongoing joint work with Dror Bar-Natan. Following the idea presented in "Tangles in a pole dance studio" by Bar-Natan, Dancso, Hogan, Liu and Scherich, we consider a simplified version of the defining equations for Drinfeld's associators. These equations take place in the quotient of the Drinfeld-Kohno algebra by a certain ideal, which we call the "emergent quotient". We further discuss the relationship of these equations to the Kashiwara-Vergne equations.

**15:00 - 16:00 Rodrigo Navarro Betancourt (Dublin) **

*Another proof GRT injects into KRV *

The graded Grothendieck-Teichmüller group GRT is fundamentally present in several branches of mathematics. With regards to Lie theory, Alekseev and Torossian showed GRT injects into KRV, a graded group acting freely and transitively on solutions of the Kashiwara-Vergne conjecture. In this talk, we revisit this result and offer a different proof of the injection of GRT into KRV. We will present GRT as the group of automorphisms of the operad of parenthesized chord diagrams, and exploit a new characterization of KRV as the isotropy group of a fixed relative Lie cohomology class.

#### Wednesday August 14

*Geneva working group *

**(Anton Alekseev, Francis Brown, Megan Howarth, Florian Naef, Muze Ren, Pavol Severa) **

**10:00 - 12:30 **

**Anton Alekseev (Geneva) **

*On the relation between double shuffle and Kashiwara-Vergne: introduction *

**Pavol Severa (Geneva) **

*Furusho's polylogs made explicit *

**Muze Ren (Geneva) **

*Polylogarithm characterization of double shuffle regularization relations *

**Florian Naef (Dublin) **

*Putting things together and conclusions *

#### Thursday August 15

**10:00 - 11:00 Anton Alekseev (Geneva) **

*Tetrahedral sums of Hermitian matrices and related problems *

The Horn problem is a Linear Algebra question asking to determine the range of eigenvalues of the sum (a+b) of two Hermitian matrices with given spectra. The solution was conjectured by Horn, and it is given by a set of linear inequalities on eigenvalues. The proof of the conjecture is due to Klyachko and Knutson-Tao. It is interesting that exactly the same set of inequalities describes singular values of matrix products, maximal multipaths in concatenation of planar networks, and non-vanishing of Littlewood-Richardson coefficients for representations of GL(N).

In this talk, we consider the multiple Horn problem which is asking to determine the range of eigenvalues of (a+b), (b+c) and (a+b+c) for a, b and c with given spectra. Now the four different problems described above no longer have the same solution. We will present some results for the additive, multiplicative, and maximal multipaths problems. It turns out that under some further assumptions the maximal multipaths problem is related to the octahedron recurrence from the theory of crystals.

Based on joint works in progress with A. Berenstein, M. Christandl, T. Fraser, A. Gurenkova and Y. Li.

**11:30 - 12:30 Zsuzsanna Dancso (Sydney) **

*Knots, Graphs and Lattices *

In a 2011 breakthrough, Greene used the "Tait graph" construction for knots, a lattice-valued invariant of graphs, and the Discrete Torelli Theorem to prove that the Heegaard-Floer homology of the double branched cover is a complete mutation invariant of alternating knots. We generalise this construction to knots on surfaces, show that the resulting mutation invariant is well-defined but not complete, and propose a stronger invariant. I'll briefly explain the computational methods used - which are interesting in their own right - and end with a list of open questions. Based on joint work with Hans Boden, Damian Lin and Tilda Wilkinson-Finch.

**15:00 - 17:00 Dror Bar-Natan (Toronto) **

*Knot Invariants from Finite Dimensional Integration *

For the purpose of today, an "I-Type Knot Invariant" is a knot invariant computed from a knot diagram by integrating the exponential of a pertubed Gaussian Lagrangian which is a sum over the features of that diagram (crossings, edges, faces) of locally defined quantities, over a product of finite dimensional spaces associated to those same features. Q. Are there any such things? A. Yes. Q. Are they any good? A. They are the strongest we know per CPU cycle, and are excellent in other ways too. Q. Didn't Witten do that back in 1988 with path integrals? A. No. His constructions are infinite dimensional and far from rigorous. Q. But integrals belong in analysis! A. Ours only use squeaky-clean algebra. URL: http://drorbn.net/ge24.

#### Friday August 16

*Free discussions*