Abstract: “I will report on a joint work in progress with Pablo Boixeda
Alvarez, Michael McBreen and Zhiwei Yun where categories of microlocal 
sheaves on some affine Springer fibers are described in terms of the
Langlands dual group. In particular, in the slope 1 case we recover 
the regular block in the category of (graded) modules over the small
quantum groups. Assuming a general formalism connecting microlocal
sheaves to Fukaya categories, this yields new examples of homological
mirror symmetry, including the mirror dual to T^*(G/B).”