T-ERC_STG - Tremplin-ERC Starting

New Applications of Quantum Invariants to 3- and 4-dimensional Topology – NAQI-34T

Submission summary

The goal of the proposal is to develop New Applications of Quantum Invariants to 3- and 4-dimensional
Topology. Its main objectives will be divided into three parts:
First, applications of quantum representations to the theory of mapping class groups of surfaces, where we
propose to study the faithfulness of two families of linear representations of mapping class groups: the non
semi-simple quantum representations and the Heisenberg homological representations of configuration
spaces on surfaces. Both are serious candidates for being faithful linear representations of mapping class
groups, which would solve one of the biggest open problem on mapping class groups.
We will also study
Ivanov's question about finite index subgroups of mapping class groups of surfaces having finite
abelianization on the family of kernels of semi-simple quantum representations modulo some ideals.
Our second axis is to build up methods to relate quantum invariants to twisted homology on configuration
spaces, hopefully providing new insights on the geometric content of those invariants. We expect such a
program to give applications to the various conjectures relating quantum and classical invariants, and to
detection problems (e.g. do the colored Jones polynomials detect the unknot ?).
Finally, the third aspect addresses new 4-dimensional applications of quantum invariants. We plan to use the
theory of trisections to extract new invariants of 4-manifolds or of knotted surfaces in S^4 from the
established quantum invariants in dimension 3: colored Jones polynomials, WRT invariants...
We will focus particularly on one interesting lead that results from the recent proof of Witten's finiteness conjecture for
skein modules: we plan to investigate how to define a 3+1 TQFT whose values on 3-manifolds are the
Kauffman bracket skein modules. In particular, we hope to find effective methods for computing dimensions
of skein modules, giving a constructive proof of the finiteness conjecture.

Project coordinator

Monsieur Renaud Detcherry (Institut de Mathématiques de Bourgogne)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

IMB Institut de Mathématiques de Bourgogne

Help of the ANR 113,500 euros
Beginning and duration of the scientific project: May 2023 - 24 Months

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