19 problems

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…

Li's conjecture. For all $u,v$, (a) $\operatorname{BS}(u,v)=\operatorname{St}(u,v)$; and (b) $\operatorname{BS}(u,v)\leq\max(\\{i\mid\theta_i(u)=1\text{ or }\theta_i(v)=1\\})$.

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

Let ISCD-MPC denote the iterative state- and control-dependent model predictive control technique described in the paper. A chain of integrators is a system formed by integrators o…

A pair of smooth, closed 4-manifolds $X_1$ and $X_2$ is exotic if the two manifolds are homeomorphic but not diffeomorphic. In the oriented case, a single stabilization means takin…

Let $u$ solve the beam equation with positive constants $a$, $b$ and $\alpha$, and with the pointwise damping term $\alpha \frac{\partial u}{\partial t}(\xi,t)\delta_{\xi}$ at…

Stabilizability conjecture. The system is stabilizable under these growth and polynomial-rule conditions.

Let $\mathcal{D}_0$ denote the domain of asymptotic null-controllability, and consider the rectangle and grid used in the numerical example with bounded control constraint set…

Polynomial stabilization conjecture. The remaining cases could be analyzed in the same way, with a slower polynomial decay rate.

Generalized geometric condition conjecture. Uniform stabilization holds if and only if the generalized geometric condition holds.

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 $R$ be a commutative ring with identity, let ${\rm GL}_n(R)$ be the general linear group, and let ${\rm E}_n(R)$ be the subgroup generated by elementary matrices. Consider the…

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…

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…

Let $A_2$ be the Goldberg constant defined as the infimum of the radii $\rho(f)$ over holomorphic functions in the unit disc with finite, pairwise distinct numbers of zeros, poles,…

For the stable Khovanov–Rozansky theory of torus knots, let $n$ denote the rank parameter and let $r,s$ specify the torus knot. DGR stabilization conjecture. The first stable value…