14 problems

Let $n,d,r\in\mathbb{N}$, and consider distributions on $\mathbb{F}_2^n$ generated by distributions of bounded $\operatorname{rank}_d\leqslant r$. The sampling distance is measured…

Let $d\geq K$, let $\Delta$ be the separation parameter, and suppose the mixture means are sampled according to the prior distribution considered in the paper. Let $G^{*,sp}$ be th…

Degree-bounded expected value polynomial conjecture. The polynomial $\mathcal{E}(G,S,q)$ has degree at most $\ell$ if and only if, for every $1\leq k\leq s$,

Let $\F$ be a finite field and let $Q$ be a polynomial of degree $d$ over $\F$. Define $\operatorname{rk}(Q)$ to be $0$ if $Q$ is constant, $\infty$ if $d=1$, and, if $d\geq2$, the…

Surjectivity structure conjecture. There exists a positive integer $R(p,d)$ such that, whenever $P:\mathbb{F}_p^n\rightarrow\mathbb{F}_p$ is not surjective on $S^n$, there is a pol…

Extension conjecture. Theorem still holds for $d\geq p$.

Arjevani et al.'s lower-bound conjecture. For every such $q$, $r$, and $\mu$, some $\nu\in[\mu,\ell]$ satisfies the stated spectral-radius lower bound. This conjecture concerns tig…

Let an honest triangulation be a triangulation in the sense used in the paper, and let its canonical Monsky polynomials be the associated pair of polynomials. A polynomial is posit…

Let $F_p(x)$ be a degree-$p$ polynomial in the polynomial ring $R$, representing an odd integer, and let the Collatz operation be … where $q\geq1$ is the largest positive integer f…

Low-degree variety concentration conjecture. The third condition holds for every collection of tubes pointing in $\delta$-separated directions.

Direction-separated polynomial Wolff conjecture. Every set of tubes pointing in $\delta$-separated directions satisfies the polynomial Wolff axioms; more precisely, it satisfies th…

Kakeya conjecture for the polynomial Wolff axioms. For every dimension $n$, there is a complexity $E$ such that, whenever $\mathbb{T}$ is a set of $\delta$-tubes in $\mathbb{R}^n$…

Non-tightness conjecture. The bound

Generalized finite-field distance conjecture. One has