5 problems

Conjecture. For every smooth quasi-projective variety $X$, … The lower bound $\operatorname{Rdim}D^b(X)\geq\dim X$ is known quite generally. The difficult direction is to construct…

Let $X$ be a scheme, and let $\operatorname{D}^{b}_{\operatorname{coh}}(X)$ denote the bounded derived category of coherent sheaves on $X$. The Rouquier dimension of a triangulated…

Bondal–Van den Bergh's conjecture. The category $\operatorname{perf}(X)$ is strongly generated if and only if there exists an affine open cover of $X$ whose coordinate rings all ha…

Let $M$ be a smooth compact manifold, let $R$ be a commutative ring, and equip $\mathcal W(T^*M;R)$ with its canonical grading and orientation data. Let $H^*(M;R)$ be the cohomolog…

Let $M$ be a smooth manifold, and let $\operatorname{Cuplength}(M)$ denote the cuplength of $H^*(M;k)$ for any field $k$. Let $\mathcal C(M)$ denote the wrapped Fukaya category ass…