5 problems

Almost rectangular cap characterization conjecture. The almost rectangular caps of ${\mathbb M}$, and hence of ${\mathfrak M}$, are characterized by condition in Lemma.

Parabola decoupling conjecture. For every $1\leq p<6$, there is a constant $C_p$ independent of $\delta$ such that

Let $(\cdot,\cdot)$ be a positive-definite quadratic form of integral coefficients in $r$ variables, let $B$ be a bounded region in $\mathbb{R}^r$, and let $I$ be a bounded interva…

Let ${{\mathbb C}}o^2=\\{(\xi_1,\xi_2,\sqrt{\xi_1^2+\xi_2^2}):1\leq \xi_1^2+\xi_2^2\leq 2\\}$ be the truncated cone, and let $\Gamma(\delta)$ partition its $\delta$-neighborhood in…

Let $\frac12\leq \alpha\leq 1$, and let $\Gamma_\alpha(\delta)$ partition the $\delta$-neighborhood of the truncated parabola … into almost rectangular boxes $\gamma$ of diameter…