8 problems

Let $\Gamma_{n,s}$ be the graph automorphism group with parameters $n$ and $s$, and let $2n+s-3$ be its virtual cohomological dimension. Below-vcd vanishing conjecture. All classes…

Stabilization conjecture. For any dominant coweight $\lambda\in\Lambda^\vee$ and dominant $x,y\in W$, there exists an integer $N_0=N_0(x,y,\lambda)$ such that for every $N\geq N_0$…

Let $(X_{0},X_{1})$ be an exotic pair of closed, simply-connected smooth $4$-manifolds, meaning that they are homeomorphic but not diffeomorphic. Let $X\#Y$ denote the connected su…

Let $B^{4}$ be the smooth $4$-ball. For an exotically knotted pair of closed surfaces $(\Sigma_{1},\Sigma_{2})$ in $B^{4}$, let $d(\Sigma_{1},\Sigma_{2})$ denote the least nonnegat…

Uniform stabilization conjecture. Equation is uniformly stable if and only if $b$ satisfies the Geometric Control Condition and the Turning Point Condition.

Optimality conjecture. For any $N$ and $T$ the polynomials $q(z)$ with $\sigma=\tau=1$ give the solutions to the optimization problem $J_N^{(T)}$.

Univalence conjecture. For any $N$ and $T$ there is a choice of $\gamma$ such that the function $\Phi(z)$ is univalent or typically real in $\mathbb D$. The largest value of…

Let $\phi$ be a gluing and let $A_\phi$ be its assembly map for the homology groups of the graph automorphism groups $\Gamma_{n,s}$. A class is positive-dimensional when it has hom…