TBA

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.

I will present some finite stratified homotopy types constructed from smooth data which are not the homotopy types of conically smooth manifolds, disproving a 2015 conjecture of Ayala, Francis and Rozenblyum. The main tool is a new combinatorial and fully faithful exit path infinity-category construction. Time permitting, I will discuss a future direction for such conjectures.