Abstract: The shuffle conjecture was a big open problem which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This conjecture was finally solved after 14 years by Carlsson and Mellit via the introduction of a new interesting algebra denoted $A_{q,t}$. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. Afterwards Carlsson, Mellit, and Gorsky showed this algebra and its representation could be realized using parabolic flag Hilbert schemes and in addition to containing the generators of the elliptic Hall algebra. Despite the various formulations of this algebra, computations within it are extremely complicated and non-intuitive.

In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of $A_{q,t}$ and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of $A_{q,t}$ and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. If time permits, I will also discuss a categorification of our construction as certain functors over the derived trace of the Soergel category.