I will discuss some of the major problems and results pertaining to Dehn surgery, with a highlight on the application of combinatorial methods and Heegaard Floer homology. In particular, I will report on progress on two guiding conjectures, the cabling conjecture and the Berge conjecture.

The main result presented in this talk is a plethystic formula for the
specialization at $t = 1/q$ of the $Q_{u,v}$ operators studied in [math
arKiv:1405.0316]. This discovery yields elementary and direct
derivations of several identities relating these operators at $t =1/q$
to the Rational Compositional Shuffle Conjecture of [math arKiv:
1404.4616]. In particular we are able to give a direct derivation of a
simple formula for the symmetric polynomial
$$Q_{km,kn}1|_{t=1/q} \ \mbox{(for all $m,n$ co-prime and $k \geq 1
$.)}$$

We also give an elementary proof that this polynomial is Schur positive.
Moreover, by combining our main result with the Rational Compositional
Shuffle Conjecture, we obtain a completely elementary derivation of the
identity expressing this polynomial in terms of Parking functions in the
$km \times km$ rectangle.

We investigate the role of eigenvector norms in spectral graph theory to various combinatorial problems including the densest subgraph problem, the Cheeger constant, among others. We introduce randomized spectral algorithms that produce guarantees which, in some cases, are better than the classical spectral techniques. In particular, we will give an alternative Cheeger “sweep” (graph partitioning) algorithm which provides a linear spectral bound for the Cheeger constant at the expense of an additional factor determined by eigenvector norms. Finally, we apply these ideas and techniques to problems and concepts unique to directed graphs.