Time : 2021/5/31, 6/2, 6/3, 6/7, 6/9 , 6/10 16:00~18:00
Location : Math Bldg #402 & Online (https://cgp.ibs.re.kr/activities/conferences/336)
Speaker : Anderson Vera (Pohang University of Science and Technology)
Title : The Le-Murakami-Ohtsuki invariant and Johnson-Morita theory
Abstract : After the discovery of the Jones polynomial for knots and its physical interpretation by Witten a new kind of invariants for links and 3-manifolds were defined, the so-called "quantum invariants". The first quantum invariants for 3-manifolds were defined by Reshetikhin and Turaev using quantum groups and their representations. There is an alternative approach developed by Le-Murakami-Ohtsuki which gave rise to the LMO invariant. The LMO invariant was extended to cobordisms between surfaces with one boundary component by Cheptea, Habiro and Massuyeau, giving rise to the so-called LMO functor. The LMO invariant and the LMO functor are very strong invariants taking values in some spaces of graphs known as Jacobi diagrams. The LMO invariant and LMO functor are universal among quantum invariants, in particular, all Reshetikhin-Turaev invariants can be recovered from them.
The construction of the LMO invariant and LMO functor is quite sophisticated, so an important effort is made into finding relationships with classical invariants.
In this series of lectures, we give a survey of the construction of the LMO invariant and the LMO functor and we focus on the relationships with some algebraic invariants defined on the mapping class group of a surface with one boundary component. We will start by reviewing some combinatorial presentations of 3-manifolds (surgery presentations) and then we will see the spaces involved in the definition of the Kontsevich integral, LMO invariant and LMO functor as well as the precise definition of these invariants. Finally, we will study some notions about the mapping class group and the Johnson-Morita theory to conclude with the explicit relationships between the LMO functor and several kinds of Johnson-type homomorphisms.