6 problems

Let $X$ be a nilpotent complex algebraic variety such that $\pi_2(X)$ is a free abelian group of rank $r$. Assign to each holomorphic map $f:\mathbb{C}\mathbb{P}^1\to X$ a multiple…

Rational cohomology conjecture. The rational cohomology of $\mathscr{B}un_{G,X}^0$ is

The mod-$p^r$ hyperbolicity conjecture. Let $X$ be a finite simply-connected $CW$-complex. If $X$ is rationally hyperbolic, then it is mod-$p^r$ hyperbolic for all primes $p$ and p…

Let $X$ have dimension at most $n$, let $Y$ be simply connected, and let $f:X\to Y$ be nullhomotopic with Lipschitz constant at most $L$. Let … be a filtration of the indecomposabl…

Let $D$ be the exceptional divisor of a resolution of an isolated rational hypersurface singularity of dimension at least $3$, and let $\Delta=\Delta(D)$ be its dual complex. Ratio…

Directional isoperimetric conjecture. There is such a $(k+1)$-chain $y$ for which, for every $(k+1)$-tuple $I$,