6 problems

Rational degree conjecture. There exists a $\mathbb{Q}$-valued topological degree ${\rm deg}(f)$ such that

Let $\psi:S^3\to S^3$ be a field in the Skyrme model, with its topological degree interpreted as the corresponding baryonic number. Skyrme's conjecture. Solitons should be interpre…

Brezis's conjecture. There exist constants $C>0$ and $\delta_0>0$ such that the displayed inequality holds for every continuous $f\in C^0({\mathbb S}^1;{\mathbb S}^1)$ and every…

Degree conjecture.

Let $X_1,X_2,X_3\in\operatorname{Mat}(N\times N,\mathbb{C})$ be self-adjoint operators depending on $N$ and satisfying … Assume that there is a constant $c$ such that … for…

Let $f\in C^1(\bold R^n,\bold R^n)$ and $g\in C^0(\bold R\times\bold R^n\times[0,1],\bold R^n)$ be such that $g(t+T,v,\varepsilon)\equiv g(t,v,\varepsilon)$, and suppose that … def…