I will discuss recent joint work with Ben Elias, in which we develop a combinatorial method for computing the triply graded homology (equivalently, the superpolynomial) of certain links, especially torus links. Conjecturally, this link homology has deep connections with Hilbert schemes and symmetric functions. For instance the superpolynomial of the $(n,nk)$ torus link is conjecturally an evaluation of $nabla^k (p_1)^n$. I will discuss some evidence for this conjecture, including recent work of Andy Wilson.

In this talk I will try to prove De Giorgi’s Theorem on the regularity of minimal Caccioppoli Sets(Theorem 8.4). He used a measure theoretic method closely related to properties of function of bounded variation, which is technical and powerful. I will focus on Chapter 5-8 in E. Giusti’s book. It is based on M. Miranda’s simplification of De Giorgi’s original proof.