We consider the construction of refined Chern–Simons torus knot invariants by M. Aganagic
and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's …

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m, n) torus
knot from the unique finite-dimensional simple representation of the rational DAHA of type A …

The “classical” parking functions, counted by the Cayley number (n+ 1) n− 1, carry a natural
permutation representation of the symmetric group S n in which the number of orbits is the …

For any finite path $ v $ on the square grid consisting of north and east unit steps, starting at
(0, 0), we construct a poset Tam $(v) $ that consists of all the paths weakly above $ v $ with …

We observe that for a and b relatively prime, the" abacus construction" identifies the set of
simultaneous (a, b)-core partitions with lattice points in a rational simplex. Furthermore …

We consider two motivic generating functions defined on a variety, and reveal their tight
connection. They essentially count torsion-free bundles and zero-dimensional sheaves. On …

We explore the link between combinatorics and probability generated by the question “What
does a random parking function look like?” This gives rise to novel probabilistic …

For any finite path v on the square lattice consisting of north and east unit steps, we construct
a poset Tam (v) that consists of all the paths lying weakly above v with the same endpoints …

Given a strictly increasing sequence $\mathbf {t} $ with entries from $[n]:=\{1,\ldots, n\} $, a
parking completion is a sequence $\mathbf {c} $ with $|\mathbf {t}|+|\mathbf {c}|= n $ and …

These notes cover the lectures of the first named author at 2021 IHES Summer School on
“Enumerative Geometry, Physics and Representation Theory” with additional details and …