# Events

# BTW4 Workshop

I am one of the organizers of the workshop BTW4: Heegaard Floer Homology, which will take place from February 12 to February 16, 2024, at the Station Biologique in Besse (France).

# BTW3 Workshop

I was one of the organizers of the workshop BTW3: Morse and Floer Theories, which took place from January 30 to February 3, 2023, at CIRM in Luminy, Marseille (France).

# BTW2 Workshop

I was one of the organizers of the workshop BTW2: Sliceness, Exotic Pairs, and Quantum Invariants, which took place from April 10 to April 15, 2022, at the Centre Paul-Langevin in Aussois (France).

# Zurich Topology Seminar

I was the organizer of the Topology Seminar of the University of Zurich between 2022 and 2023.

## Past Talks

May 25, 2023 - 11am - University of Zurich, room Y27-H-26

Speaker: Stefan Mihajlović (Alfréd Rényi Institute of Mathematics)

Title: Removing double points of surfaces in 4-manifolds via multi-tubing

Abstract: In joint work with Marco Marengon we present a simple but flexible method to simultaneously remove multiple double points of immersed surfaces in 4-manifolds. One consequence is that in an appropriate sense many knots bound disks in 4-manifolds, and in particular, we significantly improve the best known bound in the K3 surface. Namely, for a small 4-ball in K3, we show that all unknotting number 21 knots on its boundary 3-sphere will bound disks in the interior of K3. In work in progress in a different direction I aim to improve upper bounds on the minimal genus function in some 4-manifolds, an important smooth invariant generally hard to pinpoint. Finally, I speculate about connections to a third well-known question asking how big or small a self-intersection of a sphere can be in a fixed 4-manifold.

May 11, 2023 - 11am - University of Zurich, room Y27-H-26

Speaker: Bruno Roso (University College London)

Title: Mod p Borel Floer Cohomology

Abstract: In this talk, I consider a G-equivariant Borel Floer theory for 3-manifolds equipped with a G-action and consider what such a theory implies about the non-equivariant Floer theories of the manifold Y and its quotient. In particular, by defining a G-equivariant contact class, I shall demonstrate a relationship between the contact topology of a mod p L-space Y and of any manifold p-fold covered by Y.

January 26, 2023 - 2pm - University of Zurich, room Y27-H-28

Speaker: Volodymyr Lyubashenko (University of Zurich)

Title: Remarks on "Relating quantum character varieties and skein modules" by Julien Korinman and Jun Murakami

Abstract: I will describe the main results of the paper mentioned in title, as well as few results from "A note on quantum fundamental groups and quantum representation varieties for 3-manifolds" by Kazuo Habiro, "The half-twist for U_q(g) representations" by Noah Snyder and Peter Tingley, "The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds" by Wade Bloomquist and Thang Lê and "On the structure of modular categories' by Michael Müger.

December 15, 2022 - 10:30am - University of Zurich, room Y27-H-26

Speaker: Lukas Lewark (University of Regensburg)

Title: Rasmussen invariants of Whitehead doubles and other satellites

Abstract: Take a knot P in a solid torus; then tie this solid torus into a knot K. The result is the so-called satellite knot P(K) with pattern P and companion K. Satellite knots are an important source of examples. For many knot invariants y, the value y(P(K)) is determined by y(P(unknot)) and y(K). Not so for the Rasmussen invariants! The Rasmussen invariants are integer-valued concordance homomorphisms coming from Khovanov homology. We will see that for certain satellites, namely twisted Whitehead doubles, the values of Rasmussen invariants are controlled by another, new knot invariant, which is itself a concordance homomorphism. This talk is based on joint work with Claudius Zibrowius. The proofs of the main results rely on the multicurve description for Khovanov homology of four-ended tangles, about which Claudius spoke in this seminar on October 6.

December 8, 2022 - 10:30am - University of Zurich, room Y27-G-28

Speaker: Rhea Palak Bakshi (ETH Zurich)

Title: On Torsion in Skein Modules and Framings of Links in 3-manifolds

Abstract: Skein modules are invariants of 3-manifolds which were introduced by Józef H. Przytycki in 1987 as generalisations of the Jones, HOMFLYPT, and Kauffman bracket polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Over time, skein modules have evolved into one of the most important objects in knot theory and quantum topology, having strong ties with many fields of mathematics such as algebraic geometry, hyperbolic geometry, and the Witten-Reshetikhin-Turaev 3-manifold invariants, to name a few. One avenue in the study of skein modules is determining whether they reflect the geometry or topology of the manifold, for example, whether the module detects the presence of incompressible or non-separating surfaces in the manifold. Interestingly enough this presence manifests itself in the form of torsion in the skein module. In this talk we will discuss various skein modules which detect the presence of non-separating surfaces. We will focus on the framing skein module and show that it detects the presence of non-separating 2-spheres in a 3-manifold by way of torsion.

November 17, 2022 - 10:30am - University of Zurich, room Y27-H-26

Speaker: Danica Kosanović (ETH Zurich)

Title: 2-knots and knotted families of arcs

Abstract: Knowing when one can embed a surface into a 4-manifold is a question of fundamental importance for 4-manifold theory. It gives rise to a field of 2-knot theory, which is in a certain sense even harder than classical knot theory. However, in some situations – like in the setting when embedded surfaces (disks) have a common “light bulb” (in the boundary) – one can completely classify them up to isotopy. I will explain how in joint work with Peter Teichner we provide such a classification using some fairly general techniques from homotopy theory, leading to surprising applications of higher homotopy groups of spaces of embeddings.

November 3, 2022 - 10:30am - University of Zurich, room Y27-H-26

Speaker: Sakie Suzuki (Tokyo Tech)

Title: Quantum invariants of closed framed 3-manifolds based on ideal triangulations

Abstract: We construct a new type of quantum invariant of closed framed 3-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of 3-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner (2, 3) move of ideal triangulations and the non-involutiveness of the Hopf algebra reflects framings. This is a joint work with S. M. Mihalache and Y. Terashima.

October 6, 2022 - 10:30am - University of Zurich, room Y27-H-26

Speaker: Claudius Zibrowius (University of Regensburg)

Title: Classification of multicurve invariants

Abstract: The Fukaya category of the 4-punctured sphere plays a central role in understanding link Floer and Khovanov theory via decompositions along Conway spheres. Yet, the multicurve invariants for both theories seem to inhabit only a very small part of this Fukaya category. I will discuss what is known, what is conjectured, and what some of the consequences and applications are.

September 29, 2022 - 10:30am - University of Zurich, room Y27-H-26

Speaker: Quentin Faes (University of Burgundy)

Title: The Casson invariant and subgroups of the Mapping class group of surfaces

Abstract: In this talk, we will present some equivalence relations (defined by surgery) on the set of 3-manifolds. These relations can be characterized with the help of finite-type invariants, and in particular the Casson invariant. This is all strongly related to the study of some subgroups of the Mapping class group of surfaces, such as the Chillingworth subgroup or the Johnson kernel. Recently, some (homological) representations of these subgroups have been defined, which are interesting if one wants to clarify the relations between finite-type invariants, quantum invariants, and homological representations.

May 30, 2022 - 2pm - University of Zurich, room Y21-F-70

Speaker: Cristina Palmer-Anghel (University of Geneva)

Title: Coloured Jones and coloured Alexander invariants from two Lagrangians intersected in a symmetric power of a surface

Abstract: The coloured Jones polynomials and the coloured Alexander polynomials are quantum invariants that come from the representation theory of the quantum group Uq(sl(2)). We construct a unified topological model for these two sequences of invariants. More precisely we prove that the Nth coloured Jones and Nth coloured Alexander invariants are different specializations of a graded intersection between two explicit Lagrangians in a symmetric power of the punctured disc. In particular, the Jones polynomial and the Alexander polynomial are two specializations of the same graded intersection in a configuration space. Then, we show that the intersection before specialization is (up to a quotient) an explicit interpolation between the Jones polynomial and Alexander polynomial.

May 9, 2022 - 3pm - University of Zurich, room Y22-F-68

Speaker: Peter Feller (ETH Zurich)

Title: On the length of knots on a Heegaard surface of a 3-manifold

Abstract: In this talk we explore connections between the topology and the geometry of 3-manifolds. Concretely, we use Heegaard-splittings (topology) of a 3-manifold to describe hyperbolic structures (geometry) on it. More concretely, for a knot K that lies on a Heegaard surface F of a closed oriented connected 3-manifold M, we describe a sufficient condition for M to carry a hyperbolic structure. The sufficient condition is in terms of topological information about the triple (M,F,K). Furthermore, whenever our criterion applies, we provide bounds on the length of K. The upshot is that there is no Ricci-flow machine running in the background. Instead, the motor behind what we do is effective hyperbolic Dehn surgery ala Hodgson and Kerckhoff. This talk is based on work in progress with A. Sisto and G. Viaggi.

May 2, 2022 - 3pm - University of Zurich, room Y22-F-68

Speaker: Jules Martel (University of Burgundy)

Title: Non semi-simple quantum representations reconstructed from homology

Abstract: Twisted homologies of configuration spaces of a surface S are more or less naturally endowed with an action of the mapping class group Mod(S). When S is a punctured disk, the construction is due to Lawrence while Bigelow used homological intersection to obtain their faithfulness and thus the linearity of braid groups. We have added an action of the quantum group of sl2 on these homological modules and proved that they recover a quantum representation arising from a TQFT (non semi simple), theirself producing highly organized invariants: of knots, 3-manifolds, of Mod(S) for all S... Could we use these homologies to provide these so called quantum invariants with a topological flavour that is often missing? Indeed, these TQFTs are constructed from algebraic tools and their topological content is the subject of many conjectures. I will present this framework and how to recover homologically quantum representations of mapping class groups (j.w. M. De Renzi). I might say a word on how to deal with other quantum groups than sl2 (in progress j.w. S. Bigelow).

April 4, 2022 - 3pm - University of Zurich, room Y22-F-62

Speaker: Christian Blanchet (Paris Cité University)

Title: Action of the Mapping Class Group on Heisenberg homology of surface configurations

Abstract: Together with Awais Shaukat and Martin Palmer we defined Heisenberg homology of configurations in surfaces with one boundary component. We will present this construction and deduce representations of the Mapping Class Groups, which intertwine an action of quantum sl(2).

March 16, 2022 - 3pm - Zoom

Speaker: Kyle Hayden (Columbia University)

Title: Where are the complex curves in Khovanov homology?

Abstract: Since the advent of gauge theory, many modern tools exhibit a close connection with complex curves and a heightened sensitivity to objects from the complex realm. Surprisingly, this is true even for Khovanov homology, whose construction is combinatorial rather than geometric. I will discuss this in the context of joint work with Isaac Sundberg that uses Khovanov homology to study knotted surfaces in 4-space, especially (compact pieces of) complex curves in the 4-ball.

# Non-Semisimple TQFTs Work Group

The goal of this work group was to understand the family of non-semisimple TQFTs constructed by Blanchet, Costantino, Geer, and Patureau in Non-Semi-Simple TQFTs, Reidemeister Torsion and Kashaev's Invariants. Talks were held every other Tuesday at Paris Diderot University.

## Past Talks

November 28, 2017 - 9:30 am - 1016 - Sophie Germain

Speaker: Sakie Suzuki

Title: The universal quantum invariant and colored ideal triangulations

Abstract: The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the S-tensor) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of 3-manifolds up to colored moves. In this construction, a copy of the S-tensor is attached to each tetrahedron, and invariance under the colored Pachner (2,3) moves is shown by the pentagon relation of the S-tensor.

March 6, 2017 - 9:30 am - 1016 - Sophie Germain

Speaker: Daniel Lopez

Abstract: Modular categories are semisimple monoidal categories satisfying certain finiteness and nondegeneracy conditions. These categories turn to be a central object in low dimensional topology, as Reshetikhin and Turaev showed in the early 90s that one can obtain 3-manifold invariants from any modular category. The main goal of this talk (based on an article of L. Funar) is to give an infinite family of pairs of non-homeomorphic 3-manifolds, namely, certain torus bundles, which are not distinguished by their Reshetikhin-Turaev invariants for any modular category. We will show that any pair of matrices (A,B) in SL(2,Z), non-conjugate in GL(2,Z), but conjugate in any SL(2,Z/NZ) produces such an example.

January 5, 2017 - 2 pm - 1016 - Sophie Germain

Speaker: Daniel Lopez

Title: A surgery presentation of the cobordism category with an application to the classification of (2+1)-TQFTs

Abstract: In this talk, we will present a generators/relations presentation of the oriented cobordism category in terms of surgery, based on a yet unpublished work of András Juhász (see arXiv:1408.0668). We will use this to classify (2+1)-TQFTs in terms of a new algebraic structure, called J-algebras, which roughly speaking are graded Frobenius algebras with an involution and a splitting together with mapping class group representations satisfying certain relations.

December 6, 2016 - 9:30 am - 1016 - Sophie Germain

Speaker: Cristina Anghel

Title: Modified 3-manifold invariants from the Lie super-algebra sl(2|1)

Abstract: In this talk we will present 3-manifold invariants constructed from the Lie super algebra sl(2|1) at roots of unity. In 2011, N. Geer, B. Patureau and V. Turaev defined a construction that having as a starting point any pivotal category with additional properties, gives invariants for links in 3-manifolds. They used a modified quantum dimension on objects and the corresponding modified 6j-symbols in a state-sum setting, in order to obtain topological invariants. Following this line, we will present 3-manifold invariants constructed using the representation theory of the quantum enveloping algebra of the Lie super algebra sl(2|1) at roots of unity. We show that there exists a modified quantum dimension on the category of representations, and we use a purification of a certain subcategory by negligible morphisms in order to have the right algebraic structure that leads to the invariants. (This is a joint work with Nathan Geer.)

November 8, 2016 - 11 am - 1016 - Sophie Germain

Speaker: Marco De Renzi

Title: Invariants de 3-variétés de Hennings-Kauffman-Radford

November 8, 2016 - 10:30 am - 1016 - Sophie Germain

Speaker: Christian Blanchet

Title: Algèbres de Hopf enrubannées de dimension finie, l'exemple de sl(2) quantique - deuxième partie

October 18, 2016 - 9:30 am - 1016 - Sophie Germain

Speaker: Léo Benard

Title: Algèbres de Hopf enrubannées, l'exemple de sl(2) quantique

Abstract: On commencera par définir, exemples à l'appui, les algèbres de Hopf enrubannées, puis on montrera qu'une certaine déformation de l'algèbre enveloppante de sl(2) en est une. L'exposé sera voué à être introductif, aucune connaissance particulière du sujet ne sera présupposée.

October 4, 2016 - 11 am - 1016 - Sophie Germain

Speaker: Daniel Lopez

Title: Link invariants from ribbon Hopf algebras

Abstract: In this talk, we will define ribbon Hopf algebras, which are Hopf algebras endowed with an extra element giving solutions to the Yang-Baxter equation. Following ideas of R. Lawrence and T. Ohtsuki, we use these Hopf algebras to construct link invariants.

March 29, 2016 - 11 am - 1016 - Sophie Germain

Speaker: Clinton Reece

Title: Turaev-Viro invariants for unrolled quantum sl(2)

January 12, 2016 - 11 am - 1016 - Sophie Germain

Speaker: Christian Blanchet

Title: Non-semisimple TQFTs: the graded BCGP functor

December 8, 2015 - 9:30 am - 2018 - Sophie Germain

Speaker: Ramanujan Santharoubane

Title: Finite dimensionality of BCGP TQFTs via skein calculus

November 24, 2015 - 9:30 am - 2018 - Sophie Germain

Speaker: Marco De Renzi

Title: Universal Construction and surgery axioms

Abstract: I will start the construction of the BCGP TQFTs. The first step will be the definition of the decorated cobordism category which will be used to carry out the Universal Construction. I will also introduce the modified surgery axioms which are satisfied by CGP invariants and which will be used in the following in order to figure out the properties of TQFT vector spaces. Every step will be compared to the classical case of simisimple WRT TQFTs.

November 3, 2015 - 9:30 am - 2018 - Sophie Germain

Speaker: Ramanujan Santharoubane

Title: Torsion de Reidemeister, polynôme d'Alexander et TQFT non semi-simple en r=2

Abstract: Nous verrons comment la spécialisation en r=2 des invariants de Costantino, Geer et Patureau donne : 1) Le polynôme d'Alexander-Conway pour des entrelacs. 2) La torsion de Reidemeister-Turaev pour les 3-variétés closes. Cet exposé sera basé sur l'article de C. Blanchet, F.Costantino, N. Geer et B. Patureau intitulé "Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants".

October 20, 2015 - 9:30 am - 2018 - Sophie Germain

Speaker: Marco De Renzi

Title: CGP invariants of closed 3-manifolds

Abstract: I will present some of the results contained in the paper Quantum Invariants of 3-Manifolds via Link Surgery Presentations and Non-Semi-Simple Categories by Costantino, Geer and Patureau. We will construct two families of invariants of closed 3-manifolds indexed by a natural parameter r>1. These invariants are built out of the non-semisimple category of representations of the unrolled quantum group U_q^H(sl_2) at a 2r-th root of unity we saw in the first talk. The secondary invariants conjecturally extend the original Reshetikhin-Turaev invariants for the small quantum group \tilde{U}_q(sl_2). The use of richer categories pays off as these non-semisimple invariants are strictly finer than the original semisimple ones: indeed they can be used to recover the classification of lens spaces, which Reshetikhin-Turaev invariants could not always distinguish.

Reference: A small survey I wrote on the subject can be found here.

October 6, 2015 - 9:30 am - 2018 - Sophie Germain

Speaker: Cristina Anghel

Title: The modified link invariants for representations of unrolled quantum sl(2)

September 22, 2015 - 2 pm - 1020 - Sophie Germain

Speaker: Nathan Geer

Title: Unrolled quantum sl(2)

Abstract: In this talk I will discuss a particular quantization of the Lie algebra sl(2). I will start by giving the definition of the algebra. We will see that the finite dimensional modules over this algebra from a ribbon category, where the simple modules are indexed by the complex numbers. This category is not semi-simple. However, I will prove that generically the tensor product of two simple modules is semi-simple.

Reference: Talk