All talks will take place in the HMI Seminar Room at the Hamilton Mathematics Institute
I will show how polylogarithms arise in Feynman diagrams, and discuss how physicists make use of their algebraic properties (symbol, coaction) with a view to confronting more complicated iterated integrals.
The linearized Kashiwara–Vergne and linearized double-shuffle Lie algebras, denoted respectively by lkv and ls, are two conjecturally isomorphic Lie algebras arising in the study of Grothendieck–Teichmüller theory and multiple zeta values. Their graded components can be realized as spaces of polynomials simultaneously annihilated by families of elements of the group algebra ℚ[GL_d(ℤ)], acting through the linearization of the symmetric powers of the standard representation of GL_d(ℤ).
This perspective suggests studying the conjectural equality lkv = ls by comparing the corresponding ideals of ℚ[GL_d(ℤ)] in each depth d. I will present an algorithmic approach to this comparison, which is effective in small depth, together with some first results obtained using this method. I will also explain a short argument showing that the equality lkv = ls follows from a conjecture of Racinet concerning the redundancy of the linearized double-shuffle equations.
To a genus h Riemann surface Σ and an integer n ≥ 1, one can attach a flat connection 𝒥_dhs^Σ over the trivial bundle with Lie algebra 𝔱ₕₙ on Cfₙ(Σ), in which 𝔱ₕₙ is the graded Lie algebra of the fundamental group of Cfₙ(Σ) (D’Hoker, Hidding, Schlotterer).
To a fibration E → B of complex manifolds, whose fibers are genus h Riemann surfaces, one attaches a fibration Cfₙ(E) → B, where Cfₙ(E) is the relative configuration space. It is equipped with a vertical flat connection 𝒥_dhs, inherited from the DHS construction.
An extension of 𝒥_dhs is a flat connection 𝒥 on Cfₙ(E) valued in der(𝔱ₕₙ), whose vertical restriction is 𝒥_dhs. Similarly, a (1,0)-extension of 𝒥_dhs is a section of an extension of the vertical cotangent bundle by the (1,0)-cotangent bundle of the base with the same properties; an extension of 𝒥_dhs then gives rise to a (1,0)-extension.
We show how to construct a (1,0)-extension of 𝒥_dhs explicitly (joint with E. D’Hoker, O. Schlotterer, F. Zerbini).
Drinfeld’s pentagon equation is central to the theory of associators and the Grothendieck–Teichmüller group. Hirose and Sato’s confluence relations, by contrast, arise from iterated integrals on a punctured projective line as two punctures merge. I will explain how these seemingly different constructions meet: for commutator group-like series, the pentagon equation is equivalent to the confluence relations. I will explain the proof in detail.
The Grothendieck–Teichmüller Lie algebra 𝔤𝔯𝔱₁ is a pro-nilpotent Lie algebra which admits several seemingly different but equivalent descriptions, depending on the context in which it arises.
One of them was given by Drinfeld in 1990: he defined 𝔤𝔯𝔱₁ as the set of Lie series in two variables satisfying four equations. It has since been shown that only one is necessary, the so-called pentagon equation, which takes values in the Drinfeld–Kohno Lie algebra 𝔱₄.
In this talk, we consider a relaxed version of this definition: given a Lie ideal I ⊆ 𝔱₄, we consider Lie series which satisfy the other three equations exactly, and the pentagon equation modulo I.
We will start by reviewing the closure properties of the first three equations under the Ihara bracket and product. Then, we will show that under certain conditions on I, the pentagon equation modulo I is multiplicative.
In 1980, D. Johnson established a natural bijection between spin structures and Z/2-quadratic forms on surfaces. In the case where the surface has non-empty boundary, we generalize it to any commutative ring K with unit, where the quadratic form is the constant term of the framed based Turaev cobracket with coefficients in the ring K. Any solution of the positive genus Kashwara-Vergne problem provides the formality of the cobracket, while it induces an Earle cocycle of the mapping class group of the surface, which induces a unique K-framing through Furuta’s observation, if the genus is positive. We prove these K-framings coincide with each other if the boundary of the surface is connected. We will discuss how to modify this coincidence in the case where the boundary is not connected. The talk is based on the preprint arXiv:2604.27531
A central approach in the study of many generalisations of motivic multiple zeta values (MZVs) is to pass to the depth graded setting. Relations are simplified, but no linear information is lost. This is not the case for classical motivic MZVs. Instead we pick up additional relations coming from modular forms, somehow measuring the difference between the depth and coradical filtrations.
In a coradically graded world, there are no such surprise relations, yet ghosts of them can still be seen. We will discuss how these ghosts can be made visible via a surprising relationship between totally odd MZVs and coradically graded MZVs, giving an explicit way to transfer relations between the two settings and suggesting some possible approaches to various conjectures surrounding the structure of totally odd MZVs
We introduce the groupoid of parenthesized mixed braids and give a finite presentation for it, which generalizes the classical coherence theorem for braided monoidal categories. Our motivation comes from an explicit embedding of the set of Drinfeld associators into the set of solutions to the Kashiwara-Vergne equation, established by Alekseev, Enriquez and Torossian around 2010. Our goal is to revisit this embedding using the notion of emergent braids, a concept recently coined by Bar-Natan. We report some partial results in this direction. This talk is based on a joint work with Dror Bar-Natan, Zsuzanna Dancso, Tamara Hogan and Damian Lin.
Interestingly, Vogel's dream for a ``Universal Lie (super) algebra'' never came true: there are chord diagrams that are invisible to the span of all Lie superalgebras and therefore Vassiliev invariants nobody really understands. However, thanks to a result of Hinich and Vaintrob in 2000, we can characterise the algebra of chord diagrams as the universal enveloping algebra of a certain Lie algebra object in the PROP for Casimir Lie algebras. We make a similar characterisation for chord diagrams of welded knots (arrow diagrams).
Bar-Natan helped show that the graded Grothendieck-Teichmüller group (GRT) can be understood as the automorphism group of the operad of parenthesized chord diagrams (PaCD). On the other hand, Alekseev, Kawazumi, Kuno, and Naef showed that the graded Kashiwara-Vergne group (KRV) is essentially the automorphism group of the associated graded Goldman-Turaev (G-T) Lie bialgebra of a twice-punctured disk. In this talk, we show that the Alekseev-Enriquez-Torossian injection of GRT into KRV is the result of a natural construction that yields the G-T Lie bialgebra on the disk from the operad PaCD.
Multiple zeta values (MZVs) are real numbers that generalize Riemann zeta functions at integer values. MZVs have three distinguished set of relations, namely the stuffle relation, shuffle relation and regularization relations. G. Racinet introduced the double shuffle Lie algebra (dmr_0) and use it to study formally those three set of relations. The symmetric Kashiwara-Vergne Lie algebra (krv^{sym}_2) was introduced by A.~Alekseev and C.~Torossian in their study of Kashiwara--Vergne conjecture in Lie theory.
In this talk, I want to present a self contained (as much as I can) proof of the injection from dmr_0 into the krv^{sym}_2. The talk is based on the recent preprint arXiv:2607.28163 [math.QA] and ealier work of L.~Schneps and B.~Enriquez--H.~Furusho.
I will give a clear overview of the proof strategy and the main ideas, pointing out which parts are straightforward and where the technical parts are.
Time permitting, I will take the last 15 minutes to state the three recent results that were going to be the original subject of my lectures.
Kashiwara-Vergne problem is somewhat trivial in the case of Lie algebras with an invariant pairing, but it also gets related to symplectic and Poisson geometry, in particular of moduli spaces of flat connections. That triviality comes essentially from the fact that the Liouville volume form is preserved by Hamiltonian vector fields. Things change in the case of Lie superalgebras with an odd invariant pairing, since there is no canonical volume form on odd symplectic supermanifolds. Instead of that we have the Batalin-Vilkovisky operator acting on half-densities. I will review how this BV operator interacts with the KV problem.