10 problems

Polterovich's conjecture. There exists a constant $C_M$, depending only on the symplectic manifold $(M,\omega)$, such that for every finite open cover $\mathcal{U}=\{U_i\}$ of $M$,

Let $(M,\omega)$ be a closed symplectic manifold. For a finite open cover $\mathcal{U}=\{U_i\}$ of $M$, let $e(U_i)$ denote the displacement energy of $U_i$ and set … The Poisson b…

Let $F$ and $G$ be algebraically independent polynomials in $\mathbb{C}[x_1,\ldots,x_n]$. Assume that each generates its own centralizer in $\mathbb{C}[x_1,\ldots,x_n]$, that…

Euclidean kernel conjecture. Consider all smooth potentials $V\in C^{\infty}(\mathbb{R}^d)$ with arbitrary $d$. Then

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$, and let $\mathcal{U}$ be an open cover constituted of displaceable sets. For $pb(\mathcal{U})$ as above and the symplec…

Let $(M,c9)$ be a symplectic manifold. For a positive collection $\mathcal{F}$, let $pb(\mathcal{F})$ denote its Poisson bracket invariant, and for an open cover $\mathcal{U}$ let…

Polterovich's conjecture. There exists a constant $C=C(M,\omega)>0$, depending only on the symplectic manifold, such that

Let $(M,\omega)$ be a compact symplectic manifold, and let $\operatorname{pb}(c)$ be the Poisson-bracket function associated with ball capacity $c$. The topology conjecture. Assumi…

Let $(M,\omega)$ be a compact symplectic manifold. For each ball capacity $c$, let $\operatorname{pb}(c)$ denote the Poisson-bracket invariant associated with the corresponding con…

Let $k$ be a field of characteristic $0$, and let $f$ and $g$ be algebraically independent polynomials in $k[x_1,\ldots,x_n]$. Assume that the homogeneous components of maximal deg…