# Schedule CRM workshop July 2019

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Week 1: Introductions
Day 9-10:30 11-12:30 14:30-16
Tue 2 Dror Bar-Natan Dror Bar-Natan Dror Bar-Natan
Title Expansions Lie algebras Invariants
Wed 3 Anton Alekseev Florian Naef -
Title Goldman brackets, Turaev cobrackets and formality Formality of string topology in dimensions $$\geq 2$$ and Kashiwara-Vergne -
Thu 4 Roland van der Veen Roland van der Veen -
Title Universal invariants and quasi-triangular Hopf algebras Polynomial time knot invariants -
Fr 5 Leila Schneps Leila Schneps -
Title The mould theory approach to multiple zetas The mould theory approach to Kashiwara-Vergne and elliptic multizetas -
Week 2: Conference
Day 9-10:30 11-12:30 14:30-16
Mon 8 Gwenael Massuyeau Adam Sikora Tetsuya Ito
Title Generalized Dehn twists on surfaces and homology cylinders Quantization and Toric Degenerations of Character Varieties of Surfaces Garside theory and braid group representations
Tue 9 Dylan Thurston Ulf Kuehn Delphine Moussard
Title Sutured Manifolds and Hopf algebras Lie-algebras associated to multiple q-zeta values Finite type invariants of knots in homology 3- spheres
Wed 10 François Costantino Marcy Robertson GROUP PHOTO AFTER THE TALK -
Title Stated skein algebras of surfaces Weak modular operads as a model for the Teichmueller tower -
Thu 11 Pavol Severa Yusuke Kuno Sakie Suzuki RECEPTION AT 6th FLOOR LOUNGE AT 4:15pm
Title Quantization of Poisson Hopf algebras Symplectic/special expansions for surfaces Factorizations of the universal R matrix and the universal quantum invariant for framed 3-manifolds
Fri 12 Zsuzsanna Dancso Travis Ens Thang Le
Title Topological approaches to Kashiwara-Vergne Braidors: A Simplified Theory of Drinfel'd's Associators? On Witten's finiteness conjecture
Week 3: Workshop
Day 9-10:30 11-12:30 14:30-16
Mon 15
Title Coffee break at 10:30
Tue 16 Dror Bar-Natan
Title Coffee break at 10:30 Over-then-Under-tangles and the Drinfeld Double
Wed 17 Gwenael Massuyeau
Title Coffee break at 10:30 Turaev's loop operations on surfaces and their formal descriptions
Thu 18 Nariya Kawazumi
Title Coffee break at 10:30 Gate double derivatives
Fri 19 Dror Bar-Natan
Title Everything around $$\mathfrak{sl}^\epsilon_{2+}$$, is DoPeGDO. So what? Perhaps more after coffee break
Week 4: Workshop
Day 9-10:30 11-12:30 14:30-16
Mon 22 Jun Murakami
Title Coffee break at 10:30 Quantized SL(2) representations of knot groups
Tue 23 Brent Pym
Title Coffee break at 10:30 Multiple zeta values in deformation quantization
Wed 24 -
Title Coffee break at 10:30 - -
Thu 25 Hidekazu Furusho
Title Coffee break at 10:30 The associator relation and the confluence relations
Fri 26 Benjamin Enriquez
Title On double shuffle relations for MZVs
Mon 29
Title
Tue 30
Title
Fri 26
Title

## Abstracts

Anton Alekseev
Title: Goldman brackets, Turaev cobrackets and formality
Abstract: We will recall the notions of Goldman brackets and Turaev cobrackets on the vector space spanned by homotopy classes of free loops on an oriented 2-manifold. We will then set up the formality question for the Goldman-Turaev Lie bialgebra. This talk is an introduction to the talks by Yusuke Kuno and Florian Naef.

Dror Bar-Natan
Title: Expansions, Lie algebras and Invariants
Abstract: This will be an introduction/survey of several of the main topics of this workshop. Much of these lectures could have been given twenty years ago and nothing will be newer than as of five years ago.

Dror Bar-Natan 2
Title: OU and DD
Abstract: I will be giving an improptu lecture on Over-then-Under (OU) tangles, the Drinfel'd Double (DD), and quantization of Lie bialgebras. If you attend, at the end of the talk it should be clear to you why the Drinfel'd Double construction is natural and appealing from a knot-theory perspective, yet in what sense I still don't understand it.

Dror Bar-Natan 3
Title: Everything around $$\mathfrak{sl}^\epsilon_{2+}$$, is DoPeGDO. So what?
Abstract: I'll explain what "everything around" means: including classical and quantum Hopf algebra structure, ribbon element, associator, J, Cartan involution, dequantizator, and more, and all of their compositions. What DoPeGDO means: the category of Docile Perturbed Gaussians. And what $$\mathfrak{sl}^\epsilon_{2+}$$ means: the solvable approximation to the semi-simple Lie algebra $$\mathfrak{sl}_2$$. Knot theorists should rejoice because all this leads to very powerful and well-behaved poly-time-computable knot invariants. Quantum algebraists should rejoice because it's a realistic playground for testing complicated equations and theories. This is joint work with Roland van der Veen and continues work by Rozansky and Overbay.
A handout of a talk with the same title can be found here.

François Costantino
Title: Stated skein algebras of surfaces
Abstract: (Joint work with Thang Le) After providing the definition of stated skein algebras and surfaces and discussing their relations with standard skein algebras, I will state a result detailing their algebraic behavior under topological operations. This, together with the identification of the algebra of the bigon with $$O_q(sl_2)$$, will allow us to discuss an interesting functor into the category of $$U_q(sl_2)$$ bimodules and their tensor products. If time permits I will detail how this fits into a framework of  non symmetric operads in the sense of Markl.

Zsuzsanna Dancso
Title: Topological approaches to Kashiwara-Vergne
Abstract: I will give an overview of a problem in topology equivalent to the KV problem, concerning finite type invariants of a class of 4-dimensional tangles (joint with Dror Bar-Natan). I will point out some key differences from the Alekseev-Kawazumi-Kuno-Naef topological approach, described in previous talks, and mention a - so far not entirely successful - attempt at re-phrasing AKKN in more knot-theoretic terms. The true goal would be to clarify the mysterious, but likely existing, topological connection between these two very different-looking constructions.

Benjamin Enriquez
Title: On double shuffle relations for MZVs
Abstract: By a work of Furusho (2011), associator relations between multiple zeta values imply double shuffle relations (this result was also announded in a unfinished preprint by Deligne and Terasoma in 2005). The talk presents a new proof of this result. It turns out to be a consequence of the construction of a coproduct over a module over the underlying algebra of the harmonic coproduct, of "Betti" counterparts of both the harmonic coproduct and its module version, and of the statement that an arbitrary associator relates both coproducts with their Betti counterparts. In their turn, these statements rely on a geometric interpretation in terms of moduli spaces of the harmonic coproduct (following Deligne and Terasoma) and of its module version as well as of their Betti counterparts, and on the property of associators of relating braid groups with their infinitesimal counterparts. (Joint w. H. Furusho.)

Travis Ens
Title: Braidors: A Simplified Theory of Drinfel'd's Associators?
Abstract: I will present some incomplete work on a potentially equivalent theory to Drinfel'd's theory of associators, but which is simpler in some respects and in addition is more natural from a topological perspective. The fundamental idea is to replace braids in a disk by braids in an annulus. I will present some results which have been proven and several unproven conjectures which are backed by by computational evidence.

Hidekazu Furusho
Title: The associator relation and the confluence relations
Abstract: I will talk about the confluence relation, a relation among multiple zeta values, introduced by Hirose and Sato. I will explain that it is equivalent to Drinfeld’s associator relation.

Tetsuya Ito
Title: Garside theory and braid group representations
Abstract: A Garside structure is a combinatorial structure of groups that gives rise to a normal form solving the word and conjugacy problem. In this talk I discuss relations of braid group representations (of course, that is closely related to associator and knot invariants, the main theme of this Workshop) and Garside structures. Schematically speaking, I explain that we are able to see the normal form of braids directly' from the image of braid representations. This seems to suggest rich combinatorial structures on braid group representations.

Nariya Kawazumi
Title: Gate double derivatives
Abstract: Recently Turaev introduced the notion of a gate derivative on the group ring of the fundamental group of an oriented surface. Its double version gives a topological interpretation of a double divergence which connects the homotopy intersection form and the Turaev cobracket. We will explain the defintion of a gate double derivative and some of its properties. This is a joint work in progress with Anton Alekseev, Yusuke Kuno and Florian Naef.

Ulf Kuehn
Title: Lie-algebras associated to multiple q-zeta values
Abstract: The Lie Algebras associated with multiple zeta values are conjecturally related to Lie algebras of the Grothendieck Teichmueller groups and the Kashiwara Vergne problem. In this talk I present Lie Algebras that are associated with multiple q-zeta values and indicate how they complement the multiple zeta picture.

Yusuke Kuno
Title: Symplectic/special expansions for surfaces
Abstract: The 1-formality of a free group of finite rank is captured by the notion of group-like expansions. In this talk, we consider the situation where the free group under consideration is given as the fundamental group of a compact oriented surface with boundary. Then it is natural to ask group-like expansions to respect the topology of the surface, and the notion of special/symplectic expansions arises. I will present several examples of such expansions and results related to the Goldman bracket and the Tuarev cobracket.

Thang Le
Title: On Witten's finiteness conjecture
Abstract: Witten conjectured that over the field of rational functions in the Kauffman variable $$A$$, the skein module of any closed oriented 3-manifold is finite-dimensional. We will show that the conjecture holds true if the 3-manifold is a Dehn filling of a compact manifold with toroidal boundary satisfying certain $$q$$-holonomic condition. In particular, the conjecture holds for Dehn fillings of any 2-bridge knots or links, torus knots, and $$(-2,3,2n+1)$$-pretzel knots.

Gwenael Massuyeau
Title: Generalized Dehn twists on surfaces and homology cylinders
Abstract: (Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate'' the action of a Dehn twist on the (Malcev completion of) the fundamental group of S;  another possibility is to view C as a curve on the top boundary of the cylinder $$S \times [0,1]$$, to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a homology cylinder. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a symplectic expansion of the fundamental group of S.

Gwenael Massuyeau 2
Title: Turaev's loop operations on surfaces and their formal descriptions
Abstract: Turaev introduced in 1978 two operations on the fundamental group of a surface with boundary. The first operation measures the intersection of two loops on a surface, while the second operation measures the self-intersection of a single loop. In this "informal" talk, we will survey these loop operations before addressing the problem of their "formal" descriptions. We will see that a "formality" isomorphism for the self-intersection operation of a punctured disk arises from any Drinfeld associator. The proof is based on some 3-dimensional formulas for Turaev's loop operations.

Delphine Moussard
Title: Finite type invariants of knots in homology 3- spheres
Abstract: For null-homologous knots in rational homology 3- spheres, there are two equivariant invariants obtained by universal constructions a la Kontsevich, one due to Kricker and defined as a lift of the Kontsevich integral, and the other constructed by Lescop by means of integrals in configuration spaces. In order to explicit their universality properties and to compare them, we study a theory of finite type invariants of null-homologous knots in rational homology 3-spheres. We give a partial combinatorial description of the space of finite type invariants, graded by the degree. This description is complete for knots with a trivial Alexander polynomial, providing explicit universality properties for the Kricker lift and the Lescop equivariant invariant and proving the equivalence of these two invariants for such knots.

Jun Murakami
Title: Quantized SL(2) representations of knot groups
Abstract: For a braided Hopf algebra $$A$$ with braided commutativity, we introduce the space of $$A$$ representations of a knot $$K$$ as a generalization of the $$G$$ representation space of $$K$$ defined for a group $$G$$. By rebuilding the $$G$$ representation space from the view point of Hopf algebras, it is extended to any braided Hopf algebra with braided commutativity. Applying this theory to $$\mathrm{BSL}(2)$$ which is the braided quantum $$\mathrm{SL}(2)$$ introduced by S. Majid, we get the space of $$\mathrm{BSL}(2)$$ representations. It is a non-commutative algebraic scheme which provides quantized $$\mathrm{SL}(2)$$ representations of $$K$$.

Florian Naef
Title: Formality of string topology in dimensions $$\geq 2$$ and Kashiwara-Vergne
Abstract: String topology studies operations on the (circle-equivariant) homology of the free loop space of a manifold. In particular, there is a bracket and cobracket operation, defined in the surface case by Goldman and Turaev, respectively, and in the general case by Chas-Sullivan and Goresky-Hingston, respectively. In the surface case these structures are connected to the Kashiwara-Vergne problem, namely solutions to the Kashiwara-Vergne problem gives "nice" algebraic descriptions of these operations. We will generalize this to general closed manifolds, that is, we give an algebraic description that depends on evaluating certain Feynman diagrams up to loop order 1.

Brent Pym
Title: Multiple zeta values in deformation quantization
Abstract: In 1997, Kontsevich gave a universal solution to the "deformation quantization" problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions. His formula is a Feynman expansion, involving volume integrals on the moduli space of marked holomorphic disks. I will describe recent joint work with Banks and Panzer, in which we develop a theory of integration on these moduli spaces via suitable sheaves of polylogarithms, and use it to prove that Kontsevich's integrals evaluate to integer-linear combinations of multiple zeta values, yielding an algorithm for their exact symbolic calculation.

Marcy Robertson
Title: Weak modular operads as a model for the Teichmueller tower
Abstract: The Grothendieck-Teichmueller group is an explicitly defined (profinite) group introduced by Drinfeld which is closely related to (and contains) the absolute Galois group. The idea was based on Grothendieck's suggestion that one should study the absolute Galois group by relating it to its action on the Teichmueller tower of fundamental groupiods of the moduli stacks of genus g curves with n marked points.  In this talk, we give an reimagining of the Teichmueller tower in terms of a profinite completion of an algebraic object called a `Segal modular operad.'' Using this reinterpretation, we show that the homotopy automorphisms of this model for the Teichm\"uller tower is isomorphic to a subgroup of the (profinite) Grothendieck-Teichmüller group introduced by Hatcher, Lochak and Schneps. We then show a non-trivial action of the absolute Galois group on our tower -- giving evidence that this is a good model for the Teichmueller tower.  This talk will be aimed a general audience and will not assume any previous knowledge of the Grothendieck-Teichmueller group or operads.

Pavol Severa
Title: Quantization of Poisson Hopf algebras
Abstract: I will describe a method for quantization of Poisson Hopf algebras (in arbitrary $$\mathbb Q$$-linear symmetric monoidal categories) which is compatible with tensor products. The main idea comes from nerves of groups: they are symmetric simplicial sets. Nerves of Hopf algebras then turn out to be braided rather than symmetric and nerves of Poisson Hopf algebras to be infinitesimally braided. The problem is thus solved via the standard machinery of Drinfeld associators. A little bit can be said also if groups are replaced with groupoids and Hopf algebras with Hopf algebroids. Joint work with Jan Pulmann.

Title: Quantization and Toric Degenerations of Character Varieties of Surfaces
Abstract: We explore the properties of character varieties of surfaces and their knot theory inspired quantizations, called skein algebras, by applying the theory of pseudo- Anosov diffeomorphisms and of measured foliations. In particular, we prove that every sufficiently generic measured foliation of a surface defines a toric degeneration of the character variety and a quantum toric degeneration of the skein algebra.

Sakie Suzuki
Title: Factorizations of the universal R matrix and the universal quantum invariant for framed 3-manifolds
Abstract: Take the Drinfeld double $$D(B)$$ of the Borel subalgebra $$B$$ of the quantized enveloping algebra of sl2. We consider two embeddings of $$D(B)$$ as an algebra, into a double of Heisenberg double and into a quantum torus algebra. With both embeddings, each image of the universal R matrix has a factorization into a product of four elements each satisfying a pentagon relation. This setting leads us to the Kashaev invariant of links and to quantum Teichmüller theory. In this talk I will explain these situations and show our trials to unify these studies in a view point of the universal S tensor and framed 3-manifolds.  This talk includes a joint work with Y. Terashima.

Dylan Thurston
Title: Sutured Manifolds and Hopf algebras
Abstract: Kuperberg showed how Hopf algebras can give invariants of (framed) 3-manifolds. We look at possible extensions to open expressions, with free variables; these appear to be related to marked sutured 3-manifolds. In particular, there is a geometric interpretation of the Drinfel'd double construction. Some of this was done by Rohit Thomas, and portions are joint work with Dror Bar-Natan.

Roland van der Veen
Title: Quasi-triangular Hopf algebras
Abstract: The first half is a survey on quasi-triangular Hopf algebras and the corresponding universal quantum knot invariants. In fact, the universal invariants provide a convenient graphical calculus to discuss the algebra. The Drinfeld double construction and relations to Poisson Lie groups and their quantization will also be mentioned. In the second half we present some techniques for efficient computation in Hopf algebras of PBW type. As an application we discuss how this leads to a wealth of strong knot invariants that are still computable in polynomial time. Our main example is related to the n-loop approximations to the universal sl2 invariant. Joint work with Dror Bar-Natan.