# Mathematics Colloquia and Seminars

### Abelianization and its $q$-deformation
I will describe joint work in progress with Fei Yan, concerning a new geometric construction which quantizes the “abelianization” map between moduli spaces of flat $GL(N)$-connections on a surface $C$ and flat $GL(1)$-connections on spectral covers of $C$. The physical context of the construction is the study of framed BPS states in $N=2$ supersymmetric field theories. The construction extends and reinterprets previous work of Bonahon-Wong, Gabella, and Galakhov-Longhi-Moore. A byproduct is a new scheme for computing specializations of the HOMFLY polynomials of links.