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There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like.
It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused.
I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch, Liam Watson and Ben Webster.[-]

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...[+]

57M25 ; 57M27

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The Reidemeister torsion may be viewed as a volume form on the character variety of a 3-manifold with boundary. I will explain a conjectural differential equation that this form should satisfy, motivated by the study of the asymptotical behaviour of quantum invariants.

53D50 ; 57M25 ; 57M27 ; 57R56

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A surface system for a link in $S^3$ is a collection of embedded Seifert surfaces for the components, that are allowed to intersect one another. When do two $n$–component links with the same pairwise linking numbers admit homeomorphic surface systems? It turns out this holds if and only if the link exteriors are bordant over the free abelian group $\mathbb{Z}_n$. In this talk we characterise these geometric conditions in terms of algebraic link invariants in two ways: first the triple linking numbers and then the fundamental groups of the links. This involves a detailed study of the indeterminacy of Milnor's triple linking numbers.
Based on joint work with Chris Davis, Matthias Nagel and Patrick Orson.[-]

A surface system for a link in $S^3$ is a collection of embedded Seifert surfaces for the components, that are allowed to intersect one another. When do two $n$–component links with the same pairwise linking numbers admit homeomorphic surface systems? It turns out this holds if and only if the link exteriors are bordant over the free abelian group $\mathbb{Z}_n$. In this talk we characterise these geometric conditions in terms of algebraic link ...[+]

57M25 ; 57M27 ; 57N70

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We say a manifold $M$ is exceptional if for any $n$ all degree $n$ covers of $M$ are homeomorphic. For example closed surfaces and all tori are exceptional. We classify exceptional 3-manifolds.
This is based on joint work with Junghwan Park, Bram Petri and Aru Ray.

57M27 ; 57M25 ; 57M50

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In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...[+]

57M27 ; 57M50 ; 57M25

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In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...[+]

57M27 ; 57M50 ; 57M25

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In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...[+]

57M27 ; 57M50 ; 57M25

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Heegaard-Floer theory is a 4-dimensional topological fi theory. It has been partially extended down to dimension 2: Lipshitz-Oszvath-Thurston constructed a differential algebra for a surface equipped with some extra structure. Douglas and Manolescu started building a partial extension down to dimension 1. I will discuss joint work with Andy Manion where we explain a gluing mechanism for surfaces. This is based on the construction of a monoidal 2-category of 2-representations of $gl(1|1)^+$.[-]

Heegaard-Floer theory is a 4-dimensional topological fi theory. It has been partially extended down to dimension 2: Lipshitz-Oszvath-Thurston constructed a differential algebra for a surface equipped with some extra structure. Douglas and Manolescu started building a partial extension down to dimension 1. I will discuss joint work with Andy Manion where we explain a gluing mechanism for surfaces. This is based on the construction of a monoidal ...[+]

57R58 ; 57M27 ; 17B37

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This talk will report on a work with S. Bettin (University of Genova) in which we obtained exact modularity relations for the q-Pochhammer symbol, which is a finite version of the Dedekind eta function. We will overview some of their useful aspects and applications, in particular to the value distribution of a certain knot invariants, the Kashaev invariants, constructed with q-Pochhammer symbols.

11B65 ; 57M27 ; 11F03 ; 60F05

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In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a 3-manifold and a choice of Spin-C structure, the q-series invariant turns out to be a character of a (logarithmic) vertex algebra that depends on the 3-manifold. [-]

In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a ...[+]

17B69 ; 57R56 ; 57M27 ; 58B32

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Documents 57M27 14 résultats

A hitchhiker's guide to Khovanov homology - Part I - Turner, Paul (Auteur de la conférence) | CIRM H Nouveau

Differential equation for the Reidemeister torsion - Marché, Julien (Auteur de la conférence) | CIRM H Nouveau

Surface systems for links​ - Powell, Mark (Auteur de la conférence) | CIRM H Nouveau

Exceptional 3-manifolds​ - Friedl, Stefan (Auteur de la conférence) | CIRM H Nouveau

Structure of hyperbolic manifolds - Lecture 1 - Purcell, Jessica (Auteur de la conférence) | CIRM H Nouveau

Structure of hyperbolic manifolds - Lecture 2 - Purcell, Jessica (Auteur de la conférence) | CIRM H Nouveau

Structure of hyperbolic manifolds - Lecture 3 - Purcell, Jessica (Auteur de la conférence) | CIRM H Nouveau

Higher representations of $gl(1|1)^+$ - Rouquier, Raphaël (Auteur de la conférence) | CIRM H Nouveau

Modularity of the q-Pochhammer symbol and application - Drappeau, Sary (Auteur de la conférence) | CIRM H Nouveau

Surprising VOA structures from Quantum Topology - Gukov, Sergei (Auteur de la conférence) | CIRM H Nouveau

Surface systems for links - Powell, Mark (Auteur de la conférence) | CIRM H Nouveau

Exceptional 3-manifolds - Friedl, Stefan (Auteur de la conférence) | CIRM H Nouveau