4 problems

Let $\mathcal{O}_2=k[x_1,x_2]$ and let $\mathcal{I}\triangleleft\mathcal{O}_2$ be a finite-codimensional, integrally closed ideal. Let $\mathcal{O}$ denote the ambient module in th…

Let $k[x_1,x_2]$ be the polynomial ring in two variables, and let $\mathcal{I}\triangleleft k[x_1,x_2]$ be a finite-codimensional, integrally closed ideal. Write…

Let $G$ be a plumbing tree, let $Y(G)$ be its associated plumbed 3-manifold, and let $\boldsymbol{\mathbf{\mathit{HF}}}^-(Y(G))$ denote the completed Heegaard Floer homology. Write…

Let $G$ be a negative definite plumbing tree, let $Y_G$ be its corresponding 3-manifold, and let $\mathbb{HF}^-(G)$ denote the lattice homology associated to $G$. Némethi's conject…