9 problems

Let $A$ be an $M$-curve with a deep nest, meaning a nest with at least three levels of nesting, of arbitrary degree. Orevkov's conjecture. $A$ must have some empty outer ovals. Thi…

Diameter upper-bound conjecture. The number of diameters of $\mathcal R$ does not exceed

Asympture conjecture. The first terms of the sequence suggest that $a'_0=\alpha$ is sufficient; under this choice, the exponential-growth constant $\rho$ in the bound for the numbe…

Let $X\subset\mathbb{P}^{n}$ be a smooth, geometrically integral, nondegenerate real curve with $X(\mathbb{R})\neq\emptyset$. A real curve is unramified if every real hyperplane…

Let $C$ be an $M$-curve of even degree $m=2k$, with $p$ even ovals and $n$ odd ovals, and let $\chi=p-n$ be the Euler characteristic of the Ragsdale membrane. Ragsdale's $M$-curve…

Let a smooth real plane curve be of even degree, so that an oval is null-homotopic in $\mathbb RP^2$. Call a scheme rigid when any two curves representing it are rigid-isotopic. On…

Let $C_m$ be any smooth real plane curve of degree $m$, and let its empty ovals be the components whose interiors contain no other real components. Collective contraction conjectur…

Fix an integer $m\ge 1$, and consider real schemes of degree $m$, ordered in the lattice of all real schemes. A scheme is of type I when its real curve is dividing. Rohlin's maxima…

Let $C_9$ be an $M$-curve of degree $9$ with real scheme … Here, the integers $\alpha_i$ count the empty ovals in the three nests. Korchagin's conjecture. At least two of the…