Lusztig’s theory of canonical bases reveals a remarkably rigid and positive algebraic structure on quantum groups and their modules. In symmetric types, it is known that the structure constants for multiplication in the negative part U −, as well as for the action of Chevalley generators Ei and Fi on a single simple module, all belong to N[v, v−1 ].

Lusztig conjectured that this strong positivity holds for the multiplication within the modified quantum group and the action on the tensor product of modules. In this talk, I will present recent joint work with Jiepeng Fang towards this conjecture.

 A key innovation in our approach is the ”thickening philosophy”, an algebraic technique inspired by geometric ideas from total positivity, building on my earlier work with Huanchen Bao. This method embeds a suitable approximation of the tensor product into the negative part U˜ − of a larger quantum group, constructed via a framed quiver. This allows us to inherit the desired positivity directly from the well-established positivity of the canonical basis of U˜ −. This approach demonstrates how the large Kac-Moody groups can provide a powerful framework for elucidating the structure and representations of quantum groups even for the finite and affine types.

A volume conjecture relates a certain asymptotical growth of a given quantum topological invariant of a hyperbolic 3-manifold to the hyperbolic volume of this manifold.

In this talk I will mention several of these volume conjectures, their common points and differences, notably those associated to the Baseilhac-Benedetti invariants and to the Andersen-Kashaev TQFT.

A general strategy to prove a volume conjecture is to use the combinatorial properties of a given triangulation of the manifold to simplify the expression of the quantum invariant, and hopefully to successfully apply the saddle point method in the desired asymptotics.

I will use the figure-eight knot complement as a recurring example, as it is the simplest member of two infinite families, the hyperbolic twist knots and the once-punctured torus bundles over the circle.

No prerequisite in quantum topology or hyperbolic geometry will be needed.

(This talk will cover joint works with François Guéritaud, Stéphane Baseilhac and Ka Ho Wong)

TBA