A coherent-constructible correspondence for affine Springer fibers

Affine Springer fibers are moduli spaces whose geometry plays an important role in a variety of things - for example orbital integrals on reductive groups, singularities of the Hitchin fibration, and representations of double affine Hecke algebras. The physics of 3d mirror symmetry suggests a certain equivalence of categories of constructible and coherent sheaves on a partial resolution of the commuting variety (PRCV), and following the physical heuristics it is possible to distill a particular case of this equivalence to a mathematical construction of a (quasi-)coherent sheaf on the PRCV, starting from an affine Springer fiber. In the first 30 minutes, I will give an elementary introduction to affine Springer fibers and related geometry. In the second part of the talk I will introduce BFN-Coulomb branches and the commuting variety, as well as explain the construction and some of its consequences in detail.