1. Compare the cobordism level theories of Reshetikhin-Turaev and Le-Murakami-Ohtsuki invariants; Ohtsuki series
2. Study the U(n) weight system specification for the LMO functor
3. Study the representations of the Torelli groups and homology cylinders induced by the LMO functor, connections with Johnson homomorphisms and dynamics of surface diffeomorphisms.
4. Study Ricci surfaces.
5. Study the functoriality of the Lax equation.
6. Study resonance varieties.
7. Study a possible interpretation of LMO representations through configuration spaces, as well as related areas (Toeplitz quantization, skein theory, categorification).
8. Study the quantization of isomonodromy deformations equations in the case of higher order poles.
9. Study the minimum of dilation for some classes of pseodo-Anosov homeomorphisms of punctured spheres and surfaces
10. Explore research opportunities that could appear while studying the above problems.