6 problems

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ and $A=mathbbm{k}[t]$. Let $\operatorname{D^b_{DM}}(S)$ and $\operatorname{D^b_{DM}}(A)$ denote the relevant bounded derived categories of differe…

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ and $A=mathbbm{k}[t]$. Let $mathbbm{B}$ be the vector space of Betti vectors, and let $\mathcal{E}$ be a vector bundle on $mathbbm{P}^{n-1}$ with…

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ be a polynomial ring over a field $mathbbm{k}$. A degree sequence is the sequence of generating degrees of a pure resolution, and folding such a r…

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ be a polynomial ring over a field $mathbbm{k}$. For an integer $a$, let $BS_{\text{mod}}(S)$ be the rational cone of Betti tables of graded finite…

Small pure resolution conjecture. For any pair of partitions $\lambda^{(0)}\subsetneq\lambda^{(1)}$, such a small pure resolution exists, equivariantly for both $\mathbf{GL}(V)$ an…

Let ${\bf d}$ be a degree sequence, and let $\pi({\bf d})$ denote the smallest integer diagram on the ray $t\pi({\bf d})$ for $t>0$. Such integer diagrams are $m\pi({\bf d})$ for…