16 problems

Let $I$ be an open interval in $\mathbb{R}$ and let $f\in C^{(3)}(I)$ satisfy, for every $x\in I$, constants $c_1>c_2>0$ with $c_1>f'(x)>c_2$ and $f^{\prime\prime}(x)\neq 0$. Defin…

Beresnevich et al.'s conjecture. For every $f\in C^{(3)}(I)$ and every approximating function $\psi$ satisfying

Let $\Gamma(s)$ be a naturally parametrized closed planar curve in the class $BV^1$, with length $L(\Gamma)$ and full rotation $V(\Gamma)$. Its mean absolute curvature is…

Let $I_0$ be a finite open interval, let $f\in C^{(3)}(I_0)$ satisfy … and let $N_f(Q,\psi,I)$ count rational points $\mathbf{p}/q\in\mathbb{Q}^2$ with $q\leq Q$, $p_1/q\in I$, and…

Asymptotic energy quantization conjecture. The energy of a general solution is asymptotically quantized by the number of loops captured in the convergence result, namely equality s…

Lemniscate stability conjecture. As in the circle case, an analogous stability result should hold for the lemniscate under the free elastic flow.

Let $Q$ be an acyclic quiver, let $\alpha$ be a positive real root of its root system, and let $\mathcal{C}(\alpha)$ be the corresponding family of plane curves. Let $M_\alpha$ be…

Let $b>0$ and let $\rho=f_b$ be the power-law curvature function defined by $\rho(r)=r^b$. The associated function $\omega_{f_b}$ is defined on $]0,1[$ as in the preceding analysis…

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow, and let $\kappa$ denote its signed curvature. Write $n_\kappa(t)$ for the number of critical points…

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow. Denote by $n_U(t)$ the number of critical points of its radial function relative to the fixed point…

Akopyan–Vysotsky conjecture.

Let $C$ be the unique compactification with rational components and no other singularities of the curve $\{x^n=y^{dn}\}$. Let … The algebra $A_n$ is the algebra acting on this homo…

Let $C$ be a curve with rational normalization and a unique planar singularity. Let $\mathcal{J}(C)$ be its compactified Jacobian, let $\delta=\dim \mathcal{J}(C)$, and let…

Let $C$ be a complex curve with rational normalization and one planar singularity, and let $\mathcal{J}(C)$ be its compactified Jacobian. Van Straten's vanishing conjecture. The od…

Let a plane graph be reduced by electrical transformations—degree-1 reductions, series-parallel reductions, and $\Delta Y$-transformations—and call a transformation positive when i…

Let $\gamma:[a,b]\rightarrow\mathbb{E}^2$ be an arbitrary continuous rectifiable parametric curve such that, for every $t\in(a,b)$, there is a non-zero derivative vector…