Seminar Schedule (Fall 2025)

November 4th: Muze Ren (UZH)

Title:  Reduced coaction Lie algebra and double shuffle Lie algebra

Abstract:  In the study of intersections of curves on surfaces, the Goldman-Turaev Lie bialgebra and its noncommutative analogue, the necklace Lie bialgebra play a fundamental role. The reduced coaction was introduced as a refinement of those cobrackets. I will introduce a Lie algebra consists of skew-symmetric solutions to a noncommutative differential equation involving reduced coaction.  The double shuffle Lie algebra is a number theoretic object introduced by Racinet to study the relations in multiple zeta values and multiple polylogarithms. I will discuss the relation between those two Lie algebras, one in surface topology, one in number theory.

The talk is based on a joint work with M. Howarth, see also the preprints arxiv.2504.17416, arxiv.2509.20275

November 18th: Ben-Michael Kohli (Université de Genève)

Title: Alexander and Jones-type properties of the Links-Gould invariant of links

Abstract: The Links-Gould polynomial is a two-variable link invariant derived from the quantum supergroup Uq(sl(2|1)). As such, LG has a “hybrid” flavor between the Jones and Alexander polynomials. Thus one can wonder what properties of the Jones polynomial and what properties of the Alexander polynomial are inherited by LG. Several recent results have shown that the Links-Gould invariant shares some of the Alexander polynomial’s most geometric features—a surprising fact for a quantum invariant. In this talk, we will review these properties, then focus on proving that the Links-Gould polynomial and its colored counterparts provide lower bounds for the 3-genus of a knots that are quite precise. In particular, the 2-colored Links-Gould polynomial provides a genus bound that is sharp for all 58 million prime knots with up to 18 crossings.

November 25th: Gwenael Massuyeau (Institut de Mathématiques de Bourgogne)

Title: The twist group and the Lie algebra of tree diagrams with beads

Abstract:  Let V be a 3-dimensional handlebody. The twist group of V is the subgroup of the handlebody group (i.e., the mapping class group of V) that acts trivially on the fundamental group of V. The natural action of the handlebody group on (a Malcev-like completion of) the fundamental group of ∂V defines an embedding of the twist group into a Lie algebra of ``special derivations'', which can also be described as a Lie algebra of ``tree diagrams with beads''. At the graded level, we obtain analogues of the Johnson homomorphisms, whose images are constrained by analogues of Morita's traces (or ``divergence cocycles'' as in the Kashiwara-Vergne problem), providing new insights into the lower central series of the twist group. This is based on joint work with Kazuo Habiro, and ongoing work with Kazuo Habiro and Mai Katada.

December 2nd: Liam Rogel (University of Kaiserslautern-Landau)

Title:  Computing Idempotents and Dimensions in the Asymptotic Hecke Category

Abstract:  The diagrammatic Hecke category, introduced by Elias and Williamson, provides a categorical framework for studying Soergel bimodules using diagrams. We present results from recent joint work with Ben Elias and Dani Tubbenhauer on idempotents, traces, and dimensions in Hecke categories.

We begin by motivating general string-diagram notation for monoidal categories. We then define the diagrammatic Hecke category of Soergel bimodules and illustrate it with numerous examples. The main contribution of the paper is an algorithmic construction of clasp idempotents. Using these new idempotents, we define the asymptotic Hecke category and compute the dimensions of objects within it.

December 9th: Lukas Woike (Institut de Mathématiques de Bourgogne)

Title: Differential graded modular functors

Abstract: In my talk, I will present the construction of a chain complex valued modular functor from a non-semisimple modular category featuring homotopy coherent mapping class group representations and gluing laws using homotopy coends. This is a homotopy coherent version of Lyubashenko's well-known construction. Afterwards, I will cover in more detail the space of differential graded conformal blocks for the torus and its E_2 multiplicative structure generalizing the well-known Verlinde algebra. This is based on work with C. Schweigert from a few years ago.