Schedule and Program

* The schedule may vary. 

Mini-course A Lecture 1:  A combinatorial approach to knotted surfaces in 4-space 

Speaker: Jean-Baptiste Meilhan (Grenoble-Alpes University)

Abstract: The aim of this mini-course is to introduce the notion of cut-diagrams, and present some of its applications to the study of surfaces in 4-space up to concordance and link-homotopy. We shall begin in Lecture 1 with reviewing the classical notion of broken surface diagrams and Roseman moves for studying knotted surfaces in 4-space, which leads to the theory of cut-diagrams and cut-moves. The latter is a diagrammatic generalization of surfaces in 4-space, which we will use in Lecture 2 to construct a family of concordance invariants. Lastly, Lecture 3 will show how cut-diagrams can be used to define link-homotopy invariants of link maps, generalizing works of P. Kirk. Several concrete topological application will be given in both Lectures 2 and 3.

No prior knowledge on 4-dimensional topology is needed for these lectures, which are based on a series of joint works with B. Audoux and A. Yasuhara. 

Coffee break

Mini-course B Lecture 1:  Introduction to the Kauffman bracket skein algebra 

Speaker: Helen Wong (Claremont McKenna College)

Abstract: We will give an introduction to the Kauffman bracket skein algebra of a surface and discuss its relationship with hyperbolic geometry.  The skein algebra is a generalization of the Jones polynomial for links that plays a key role in the skein-theoretic version of the topological quantum field theory associated with the Witten-Reshetikhin-Turaev invariant for 3-manifolds.  Although little is generally known about the topological or geometric content of the Jones polynomial and Witten-Reshetikhin-Turaev invariants, the relationship between the skein algebra and hyperbolic geometry is more established. In particular, the skein algebra is a quantization of the SL\_2(C) character variety, which contains a copy of the Teichmuller space of the surface.  In this series of lectures, we will discuss recent progress about the representation theory of the skein algebra and the geometric ramifications. If time permits, we will also discuss generalizations of the skein algebra,  as well as open conjectures related to the skein algebra.  

Short break

Talk 1:  Candidate for the contact first Kirby move 

Speaker: Prerak Deep  (IISER Bhopal) (TBC)

Abstract:  This talk is based on my article titled, "On a potential contact analogue of Kirby move of type 1". I will start the talk with giving necessary conditions for a contact surgery diagram to be a contact analogue of the first Kirby move. Then we will prove that contact positive integral surgery on Legendrian unknot of specific type has one presentation satisfying the necessary conditions. Thus, we get a collection of contact surgery diagrams as potential candidate for the contact first Kirby move. 

Lunch

Talk 2:  The covolumes of the mapping class group actions on many higher Teichmuller spaces are infinite

Speaker: Hongtaek Jung (Seoul National University)

Abstract:  For many higher Teichmuller spaces, we show that the quotient spaces of these spaces by the mapping class group action have infinite Atiyah-Bott-Goldman volume. Our result covers G-Hitchin components for G=PSL(n+1,R), PSO(n+1,n) and PSp(2n,R) with n>1 and the space of Sp(2n,R)-maximal representations with n>1. 

Talk 3:  Unbounded sl(3)-laminations and their shear coordinates

Speaker: Shunsuke Kano (Tohoku University)

Abstract: Title: Unbounded sl(3)-laminations and their shear coordinates

Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the moduli space $\mathcal{X}_{PGL_3,\Sigma}$ of framed $PGL_3$-local systems on a marked surface $\Sigma$ based on the Kuperberg's $\mathfrak{sl}_3$-webs.

We introduce their tropical cluster coordinates as an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any ideal triangulation.

We also describe tropical points of Goncharov--Shen's moduli space $\mathcal{P}_{PGL_3,\Sigma}$.

Then we give a tropical analogue of gluing morphisms among the moduli spaces as a geometric gluing procedure of $\mathfrak{sl}_3$-laminations with ``shearings''.

We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra by Ishibashi--Yuasa, which conjecturally leads to a quantum duality map.

This is a joint work with Tsukasa Ishibashi.

Coffee break

Talk 4:  Combinatorics of the product between a dendroidal set and a simplicial set

Speaker: Eric Dolores Cuenca (Pusan National University)

Abstract:  A dendroidal set is a contravariant functor from a category, whose objects are trees, to the category of sets. When we identify linear trees with simplices dendroidal sets generalize simplicial sets. In this talk we describe the combinatorics of the product of a dendroidal sets and a simplicial set. Our work brings us a step closer to understand the tensor product of dendroidal sets, which, via a Quillen equivalence between dendroidal sets and topological operads, induces the derived tensor product of topological operads.

Breakfast/Coffee

Mini-course C Lecture 1:  K-symplectic geometry

Speaker: Vladimir Fock (University of Strasbourg)

Abstract: For any three-dimensional manifold with boundary one can associate the symplectic space of flat connections on its boundary and a Lagrangian subvariety of flat connections extendible to the three-manifold. Many constructions in three-dimensional topology uses this construction.

The aim of the mini-course is to introduce a refinement of the notions of symplectic structure and of Lagrangian subvariety using the algebraic K-theory (which is not supposed to be known). This structure is on one side related to geometry, in particular to Dehn invariant and hyperbolic volume of polyhedra, and on the other hand to number theory such as Gauss reciprocity. Finally we will describe what this structure give more fine number-theoretic structure of invariants of knots and three-manifolds.

Coffee break

Mini-course A Lecture 2:  A combinatorial approach to knotted surfaces in 4-space 

Speaker: Jean-Baptiste Meilhan (Grenoble-Alpes University)

Abstract: The aim of this mini-course is to introduce the notion of cut-diagrams, and present some of its applications to the study of surfaces in 4-space up to concordance and link-homotopy. We shall begin in Lecture 1 with reviewing the classical notion of broken surface diagrams and Roseman moves for studying knotted surfaces in 4-space, which leads to the theory of cut-diagrams and cut-moves. The latter is a diagrammatic generalization of surfaces in 4-space, which we will use in Lecture 2 to construct a family of concordance invariants. Lastly, Lecture 3 will show how cut-diagrams can be used to define link-homotopy invariants of link maps, generalizing works of P. Kirk. Several concrete topological application will be given in both Lectures 2 and 3.

No prior knowledge on 4-dimensional topology is needed for these lectures, which are based on a series of joint works with B. Audoux and A. Yasuhara. 

Short break

Talk 5:  Parameterization and non-triviality of ribbon torus knots 

Speaker: Tumpa Mahato (Indian Institute of Science Education and Research Pune)

Abstract:  In this talk, we will discuss a method to provide a parameterization for a special class of knotted surfaces, called ribbon torus knots using elementary functions. We will also discuss if the parameterized ribbon torus knot is non-trivial. This uses the connection of ribbon torus knots with welded knots given by S. Satoh’s Tube map. We will explore the non-triviality of welded knots by studying a welded knot invariant, called welded unknotting number and utilize those results to examine the non-triviality of ribbon torus knots. 

Lunch

Mini-course B Lecture 2:  Introduction to the Kauffman bracket skein algebra 

Speaker: Helen Wong (Claremont McKenna College)

Abstract: We will give an introduction to the Kauffman bracket skein algebra of a surface and discuss its relationship with hyperbolic geometry.  The skein algebra is a generalization of the Jones polynomial for links that plays a key role in the skein-theoretic version of the topological quantum field theory associated with the Witten-Reshetikhin-Turaev invariant for 3-manifolds.  Although little is generally known about the topological or geometric content of the Jones polynomial and Witten-Reshetikhin-Turaev invariants, the relationship between the skein algebra and hyperbolic geometry is more established. In particular, the skein algebra is a quantization of the SL\_2(C) character variety, which contains a copy of the Teichmuller space of the surface.  In this series of lectures, we will discuss recent progress about the representation theory of the skein algebra and the geometric ramifications. If time permits, we will also discuss generalizations of the skein algebra,  as well as open conjectures related to the skein algebra.

Coffee break

Talk 6:  On knots in $S_{g} \times S^{1}$ and its invariant. 

Speaker: Seongjeong Kim (Jilin university)

Abstract:  In knot theory not only classical knots, which are embedded circles in S^{3} up to isotopy, but also knots in other 3-manifolds are interesting for mathematicians. In particular, virtual knots, which are knots in thickened surface $S_{g} \times [0,1]$ with an orientable surface $S_{g}$ of genus $g$, are studied and they provide interesting properties. 

In this talk, we will talk about knots in $S_{g} \times S^{1}$ where $S_{g}$ is an oriented surface of genus $g$. We introduce basic notions and properties for them. In particular, for knots in $S_{g} \times S^{1}$ one of important information is “how many times a half of a crossing turns around $S^{1}$”, and we call it winding parity of a crossing. We extend this notion more generally and introduce a topological model. In the end we apply it to classify knots in $S_{g}\times S^{1}$ with small number of crossings. 

Breakfast/Coffee

Mini-course B Lecture 3:  Introduction to the Kauffman bracket skein algebra 

Speaker: Helen Wong (Claremont McKenna College)

Abstract: We will give an introduction to the Kauffman bracket skein algebra of a surface and discuss its relationship with hyperbolic geometry.  The skein algebra is a generalization of the Jones polynomial for links that plays a key role in the skein-theoretic version of the topological quantum field theory associated with the Witten-Reshetikhin-Turaev invariant for 3-manifolds.  Although little is generally known about the topological or geometric content of the Jones polynomial and Witten-Reshetikhin-Turaev invariants, the relationship between the skein algebra and hyperbolic geometry is more established. In particular, the skein algebra is a quantization of the SL\_2(C) character variety, which contains a copy of the Teichmuller space of the surface.  In this series of lectures, we will discuss recent progress about the representation theory of the skein algebra and the geometric ramifications. If time permits, we will also discuss generalizations of the skein algebra,  as well as open conjectures related to the skein algebra.

Coffee break

Mini-course C Lecture 2:  K-symplectic geometry 

Speaker: Vladimir Fock (University of Strasbourg)

Abstract: For any three-dimensional manifold with boundary one can associate the symplectic space of flat connections on its boundary and a Lagrangian subvariety of flat connections extendible to the three-manifold. Many constructions in three-dimensional topology uses this construction.

The aim of the mini-course is to introduce a refinement of the notions of symplectic structure and of Lagrangian subvariety using the algebraic K-theory (which is not supposed to be known). This structure is on one side related to geometry, in particular to Dehn invariant and hyperbolic volume of polyhedra, and on the other hand to number theory such as Gauss reciprocity. Finally we will describe what this structure give more fine number-theoretic structure of invariants of knots and three-manifolds.

Short break

Talk 7:  (Co)Homology of symmetric quandles over homogeneous Beck modules 

Speaker: Deepanshi Saraf (Indian Institute of Science Education and Research (IISER) Mohali)

Abstract: A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this talk, I will intro- duce the category of symmetric quandle modules and will see that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, this establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the symmetric quandle (co)homology developed by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)]. Our constructions also apply to symmetric racks. This is a joint work with Biswadeep Karmakar and Dr. Mahender Singh.

Group photo/Lunch

Talk 8:  Khovanov-instanton Floer theory and immersed cobordism maps

Speaker: Hayato Imori (KAIST)

Abstract: Khovanov homology theory and instanton Floer theory have provided powerful tools with functorial properties in knot theory. Kronheimer and Mrowka constructed a spectral sequence linking Khovanov homology and instanton Floer homology to demonstrate that Khovanov homology detects the unknot. Furthermore, Baldwin, Hedden, and Lobb showed that this spectral sequence is functorial with respect to embedded surface cobordisms. In this talk, we show that Kronheimer--Mrowka's spectral sequence is also functorial for immersed surface cobordisms. We also provide several topological applications using the functoriality of the spectral sequence for immersed surfaces. This talk is based on a joint work with Taketo Sano, Kouki Sato, and Masaki Taniguchi.

Talk 9:  Center of stated SL(n)-skein algebras 

Speaker: Hiroaki Karuo (Gakushuin University)

Abstract:  To understand the representation theory of non-commutative algebras, the Unicity theorem is helpful and implies the importance of understanding of their centers. Since there exists the quantum trace map, an embedding of the (reduced) stated SL(n)-skein algebra into the (extended) Fock--Goncharov algebra (a quantum torus), we can use some properties of quantum tori to understand the properties of the skein algebra. In the talk, I will give the center of the (reduced) stated SL(n)-skein algebra using that of the Fock--Goncharov algebra. Consequently, thanks to the Unicity theorem, we can access to the representation theory of (reduced) stated SL(n)-skein algebras potentially related to quantum moduli algebras and quantum cluster algebras.

Coffee break

Talk 10:  The moduli space of decorated G-local systems and skein algebras 

Speaker: Tsukasa Ishibashi (Tohoku University)

Abstract: The moduli space of decorated (twisted) G-local systems on a marked surface, originally introduced by Fock–Goncharov, is known to have a natural cluster K_2 structure. In particular, it admits a quantization via the framework of quantum cluster algebras, due to Berenstein—Zelevinsky and Goncharov—Shen.

In this talk, I will explain its connection to the skein algebras: we have two generating sets of the function ring of the moduli space - cluster variables and matrix coefficients of Wilson lines - and the quantum lift of their relations would lead to an isomorphism between two types of skein algebras.

This talk is based on several joint works with Hironori Oya, Linhui Shen, and with Wataru Yuasa.

Dinner

Breakfast/Coffee

Mini-course C Lecture 3:  K-symplectic geometry 

Speaker: Vladimir Fock (University of Strasbourg)

Abstract: For any three-dimensional manifold with boundary one can associate the symplectic space of flat connections on its boundary and a Lagrangian subvariety of flat connections extendible to the three-manifold. Many constructions in three-dimensional topology uses this construction.

The aim of the mini-course is to introduce a refinement of the notions of symplectic structure and of Lagrangian subvariety using the algebraic K-theory (which is not supposed to be known). This structure is on one side related to geometry, in particular to Dehn invariant and hyperbolic volume of polyhedra, and on the other hand to number theory such as Gauss reciprocity. Finally we will describe what this structure give more fine number-theoretic structure of invariants of knots and three-manifolds.

Coffee break

Mini-course A Lecture 3:  A combinatorial approach to knotted surfaces in 4-space 

Speaker: Jean-Baptiste Meilhan (Grenoble-Alpes University)

Abstract: The aim of this mini-course is to introduce the notion of cut-diagrams, and present some of its applications to the study of surfaces in 4-space up to concordance and link-homotopy. We shall begin in Lecture 1 with reviewing the classical notion of broken surface diagrams and Roseman moves for studying knotted surfaces in 4-space, which leads to the theory of cut-diagrams and cut-moves. The latter is a diagrammatic generalization of surfaces in 4-space, which we will use in Lecture 2 to construct a family of concordance invariants. Lastly, Lecture 3 will show how cut-diagrams can be used to define link-homotopy invariants of link maps, generalizing works of P. Kirk. Several concrete topological application will be given in both Lectures 2 and 3.

No prior knowledge on 4-dimensional topology is needed for these lectures, which are based on a series of joint works with B. Audoux and A. Yasuhara. 

Closing remarks

Lunch